@article{KruesemannGodecMetzler2015, author = {Kr{\"u}semann, Henning and Godec, Aljaz and Metzler, Ralf}, title = {Ageing first passage time density in continuous time random walks and quenched energy landscapes}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {48}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {28}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/48/28/285001}, pages = {20}, year = {2015}, abstract = {We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function Psi (t) similar or equal to t(-1-alpha) (0 <= alpha <= 2). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time t(a), the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs with 0 < alpha < 1 describe systems such as charge carriers in amorphous semiconducters, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range 1 < alpha < 2, for sufficiently long ta. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the ageing CTRW results. With small bias or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.}, language = {en} } @article{KruesemannSchwarzlMetzler2016, author = {Kruesemann, Henning and Schwarzl, Richard and Metzler, Ralf}, title = {Ageing Scher-Montroll Transport}, series = {Transport in Porous Media}, volume = {115}, journal = {Transport in Porous Media}, publisher = {Springer}, address = {New York}, issn = {0169-3913}, doi = {10.1007/s11242-016-0686-y}, pages = {327 -- 344}, year = {2016}, abstract = {We study the properties of ageing Scher-Montroll transport in terms of a biased subdiffusive continuous time random walk in which the waiting times between consecutive jumps of the charge carriers are distributed according to the power law probability with . As we show, the dynamical properties of the Scher-Montroll transport depend on the ageing time span between the initial preparation of the system and the start of the observation. The Scher-Montroll transport theory was originally shown to describe the photocurrent in amorphous solids in the presence of an external electric field, but it has since been used in many other fields of physical sciences, in particular also in the geophysical context for the description of the transport of tracer particles in subsurface aquifers. In the absence of ageing () the photocurrent of the classical Scher-Montroll model or the breakthrough curves in the groundwater context exhibit a crossover between two power law regimes in time with the scaling exponents and . In the presence of ageing a new power law regime and an initial plateau regime of the current emerge. We derive the different power law regimes and crossover times of the ageing Scher-Montroll transport and show excellent agreement with simulations of the process. Experimental data of ageing Scher-Montroll transport in polymeric semiconductors are shown to agree well with the predictions of our theory.}, language = {en} } @article{MetzlerSandersLomholtetal.2014, author = {Metzler, Ralf and Sanders, L. and Lomholt, M. A. and Lizana, L. and Fogelmark, K. and Ambjoernsson, Tobias}, title = {Ageing single file motion}, series = {The European physical journal}, volume = {223}, journal = {The European physical journal}, number = {14}, publisher = {Springer}, address = {Heidelberg}, issn = {1951-6355}, doi = {10.1140/epjst/e2014-02333-5}, pages = {3287 -- 3293}, year = {2014}, language = {en} } @article{SchulzBarkaiMetzler2013, author = {Schulz, Johannes H. P. and Barkai, Eli and Metzler, Ralf}, title = {Aging effects and population splitting in single-particle trajectoryaverages}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.020602}, pages = {5}, year = {2013}, abstract = {We study time averages of single particle trajectories in scale-free anomalous diffusion processes, in which the measurement starts at some time t(a) > 0 after initiation of the process at t = 0. Using aging renewal theory, we show that for such nonstationary processes a large class of observables are affected by a unique aging function, which is independent of boundary conditions or the external forces. Moreover, we discuss the implications of aging induced population splitting: with growing age ta of the process, an increasing fraction of particles remains motionless in a measurement of fixed duration. Consequences for single biomolecule tracking in live cells are discussed.}, language = {en} } @article{SchulzBarkaiMetzler2014, author = {Schulz, Johannes H. P. and Barkai, Eli and Metzler, Ralf}, title = {Aging renewal theory and application to random walks}, series = {Physical review : X, Expanding access}, volume = {4}, journal = {Physical review : X, Expanding access}, number = {1}, publisher = {American Physical Society}, address = {College Park}, issn = {2160-3308}, doi = {10.1103/PhysRevX.4.011028}, pages = {24}, year = {2014}, abstract = {We discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long-time properties, this process exhibits aging: events counted initially in a time interval [0, t] statistically strongly differ from those observed at later times [t(a,) t(a) + t]. The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous-time random walk. By this, we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking, along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics.}, language = {en} } @article{SafdariChechkinJafarietal.2015, author = {Safdari, Hadiseh and Chechkin, Aleksei V. and Jafari, Gholamreza R. and Metzler, Ralf}, title = {Aging scaled Brownian motion}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {91}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.91.042107}, pages = {9}, year = {2015}, abstract = {Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.}, language = {en} } @article{SafdariCherstvyChechkinetal.2017, author = {Safdari, Hadiseh and Cherstvy, Andrey G. and Chechkin, Aleksei V. and Bodrova, Anna and Metzler, Ralf}, title = {Aging underdamped scaled Brownian motion}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {95}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.95.012120}, pages = {15}, year = {2017}, abstract = {We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble-and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.}, language = {en} } @article{JavanainenHammarenMonticellietal.2013, author = {Javanainen, Matti and Hammaren, Henrik and Monticelli, Luca and Jeon, Jae-Hyung and Miettinen, Markus S. and Martinez-Seara, Hector and Metzler, Ralf and Vattulainen, Ilpo}, title = {Anomalous and normal diffusion of proteins and lipids in crowded lipid membranes}, series = {Faraday discussions}, volume = {161}, journal = {Faraday discussions}, number = {1}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1359-6640}, doi = {10.1039/c2fd20085f}, pages = {397 -- 417}, year = {2013}, abstract = {Lateral diffusion plays a crucial role in numerous processes that take place in cell membranes, yet it is quite poorly understood in native membranes characterized by, e.g., domain formation and large concentration of proteins. In this article, we use atomistic and coarse-grained simulations to consider how packing of membranes and crowding with proteins affect the lateral dynamics of lipids and membrane proteins. We find that both packing and protein crowding have a profound effect on lateral diffusion, slowing it down. Anomalous diffusion is observed to be an inherent property in both protein-free and protein-rich membranes, and the time scales of anomalous diffusion and the exponent associated with anomalous diffusion are found to strongly depend on packing and crowding. Crowding with proteins also has a striking effect on the decay rate of dynamical correlations associated with lateral single-particle motion, as the transition from anomalous to normal diffusion is found to take place at macroscopic time scales: while in protein-poor conditions normal diffusion is typically observed in hundreds of nanoseconds, in protein-rich conditions the onset of normal diffusion is tens of microseconds, and in the most crowded systems as large as milliseconds. The computational challenge which results from these time scales is not easy to deal with, not even in coarse-grained simulations. We also briefly discuss the physical limits of protein motion. Our results suggest that protein concentration is anything but constant in the plane of cell membranes. Instead, it is strongly dependent on proteins' preference for aggregation.}, language = {en} } @article{CherstvyChechkinMetzler2013, author = {Cherstvy, Andrey G. and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes}, series = {New journal of physics : the open-access journal for physics}, volume = {15}, journal = {New journal of physics : the open-access journal for physics}, number = {15}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/15/8/083039}, pages = {13}, year = {2013}, abstract = {We demonstrate the non-ergodicity of a simple Markovian stochastic process with space-dependent diffusion coefficient D(x). For power-law forms D(x) similar or equal to vertical bar x vertical bar(alpha), this process yields anomalous diffusion of the form < x(2)(t)> similar or equal to t(2/(2-alpha)). Interestingly, in both the sub- and superdiffusive regimes we observe weak ergodicity breaking: the scaling of the time-averaged mean-squared displacement <(delta(2)(Delta))over bar> remains linear in the lag time Delta and thus differs from the corresponding ensemble average < x(2)(t)>. We analyse the non-ergodic behaviour of this process in terms of the time-averaged mean- squared displacement (delta(2)) over bar and its random features, i.e. the statistical distribution of (delta(2)) over bar and the ergodicity breaking parameters. The heterogeneous diffusion model represents an alternative approach to non- ergodic, anomalous diffusion that might be particularly relevant for diffusion in heterogeneous media.}, language = {en} } @article{WangCherstvyLiuetal.2020, author = {Wang, Wei and Cherstvy, Andrey G. and Liu, Xianbin and Metzler, Ralf}, title = {Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {102}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {1}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.102.012146}, pages = {012146-1 -- 012146-16}, year = {2020}, abstract = {Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x) = D-0|x|(alpha). Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble-and time-averaged mean-squared displacements couple the scaling exponents alpha of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y similar to |x|(1/(2/(2-alpha)))/t(H) coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), PHDP-FBM(y) = e(-y2)/root pi. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.}, language = {en} } @article{JeonLeijnseOddershedeetal.2013, author = {Jeon, Jae-Hyung and Leijnse, Natascha and Oddershede, Lene B. and Metzler, Ralf}, title = {Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions}, series = {New journal of physics : the open-access journal for physics}, volume = {15}, journal = {New journal of physics : the open-access journal for physics}, number = {4}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/15/4/045011}, pages = {16}, year = {2013}, abstract = {We report the results of single tracer particle tracking by optical tweezers and video microscopy in micellar solutions. From careful analysis in terms of different stochastic models, we show that the polystyrene tracer beads of size 0.52-2.5 mu m after short-time normal diffusion turn over to perform anomalous diffusion of the form < r(2)(t)> similar or equal to t(alpha) with alpha approximate to 0.3. This free anomalous diffusion is ergodic and consistent with a description in terms of the generalized Langevin equation with a power-law memory kernel. With optical tweezers tracking, we unveil a power-law relaxation over several decades in time to the thermal plateau value under the confinement of the harmonic tweezer potential, as predicted previously (Phys. Rev. E 85 021147 (2012)). After the subdiffusive motion in the millisecond range, the motion becomes faster and turns either back to normal Brownian diffusion or to even faster superdiffusion, depending on the size of the tracer beads.}, language = {en} } @article{CherstvyMetzler2016, author = {Cherstvy, Andrey G. and Metzler, Ralf}, title = {Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes}, series = {Physical chemistry, chemical physics : PCCP ; a journal of European chemical societies}, volume = {18}, journal = {Physical chemistry, chemical physics : PCCP ; a journal of European chemical societies}, publisher = {RSC Publ.}, address = {Cambridge}, issn = {1463-9084}, doi = {10.1039/C6CP03101C}, pages = {23840 -- 23852}, year = {2016}, abstract = {We investigate the ensemble and time averaged mean squared displacements for particle diffusion in a simple model for disordered media by assuming that the local diffusivity is both fluctuating in time and has a deterministic average growth or decay in time. In this study we compare computer simulations of the stochastic Langevin equation for this random diffusion process with analytical results. We explore the regimes of normal Brownian motion as well as anomalous diffusion in the sub- and superdiffusive regimes. We also consider effects of the inertial term on the particle motion. The investigation of the resulting diffusion is performed for unconfined and confined motion.}, language = {en} } @article{CherstvyMetzler2016, author = {Cherstvy, Andrey G. and Metzler, Ralf}, title = {Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {18}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c6cp03101c}, pages = {23840 -- 23852}, year = {2016}, abstract = {We investigate the ensemble and time averaged mean squared displacements for particle diffusion in a simple model for disordered media by assuming that the local diffusivity is both fluctuating in time and has a deterministic average growth or decay in time. In this study we compare computer simulations of the stochastic Langevin equation for this random diffusion process with analytical results. We explore the regimes of normal Brownian motion as well as anomalous diffusion in the sub- and superdiffusive regimes. We also consider effects of the inertial term on the particle motion. The investigation of the resulting diffusion is performed for unconfined and confined motion.}, language = {en} } @article{MetzlerJeonCherstvyetal.2014, author = {Metzler, Ralf and Jeon, Jae-Hyung and Cherstvy, Andrey G. and Barkai, Eli}, title = {Anomalous diffusion models and their properties}, series = {physical chemistry, chemical physics : PCCP}, volume = {2014}, journal = {physical chemistry, chemical physics : PCCP}, number = {16}, issn = {1463-9076}, doi = {10.1039/c4cp03465a}, pages = {24128 -- 24164}, year = {2014}, abstract = {Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.}, language = {en} } @article{MetzlerJeonCherstvyetal.2014, author = {Metzler, Ralf and Jeon, Jae-Hyung and Cherstvy, Andrey G. and Barkai, Eli}, title = {Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {16}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, number = {44}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c4cp03465a}, pages = {24128 -- 24164}, year = {2014}, abstract = {Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.}, language = {en} } @article{JeonMonneJavanainenetal.2012, author = {Jeon, Jae-Hyung and Monne, Hector Martinez-Seara and Javanainen, Matti and Metzler, Ralf}, title = {Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins}, series = {Physical review letters}, volume = {109}, journal = {Physical review letters}, number = {18}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.109.188103}, pages = {5}, year = {2012}, abstract = {Combining extensive molecular dynamics simulations of lipid bilayer systems of varying chemical compositions with single-trajectory analyses, we systematically elucidate the stochastic nature of the lipid motion. We observe subdiffusion over more than 4 orders of magnitude in time, clearly stretching into the submicrosecond domain. The lipid motion depends on the lipid chemistry, the lipid phase, and especially the presence of cholesterol. We demonstrate that fractional Langevin equation motion universally describes the lipid motion in all phases, including the gel phase, and in the presence of cholesterol. The results underline the relevance of anomalous diffusion in lipid bilayers and the strong effects of the membrane composition.}, language = {en} } @article{WangMetzlerCherstvy2022, author = {Wang, Wei and Metzler, Ralf and Cherstvy, Andrey G.}, title = {Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models}, series = {Physical chemistry, chemical physics : PCCP ; a journal of European chemical societies}, volume = {24}, journal = {Physical chemistry, chemical physics : PCCP ; a journal of European chemical societies}, number = {31}, publisher = {RSC Publ.}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/d2cp01741e}, pages = {18482 -- 18504}, year = {2022}, abstract = {How does a systematic time-dependence of the diffusion coefficient D(t) affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we answer this question via studying the characteristics of a set of standard statistical quantifiers relevant to single-particle-tracking (SPT) experiments. We examine, for instance, how the behavior of the ensemble- and time-averaged mean-squared displacements-denoted as the standard MSD < x(2)(Delta)> and TAMSD <<(delta(2)(Delta))over bar>> quantifiers-of FBM featuring < x(2) (Delta >> = <<(delta(2)(Delta >)over bar>> proportional to Delta(2H) (where H is the Hurst exponent and Delta is the [lag] time) changes in the presence of a power-law deterministically varying diffusivity D-proportional to(t) proportional to t(alpha-1) -germane to the process of scaled Brownian motion (SBM)-determining the strength of fractional Gaussian noise. The resulting compound "scaled-fractional" Brownian motion or FBM-SBM is found to be nonergodic, with < x(2)(Delta >> proportional to Delta(alpha+)(2H)(-1) and <(delta 2(Delta >) over bar > proportional to Delta(2H). We also detect a stalling behavior of the MSDs for very subdiffusive SBM and FBM, when alpha + 2H - 1 < 0. The distribution of particle displacements for FBM-SBM remains Gaussian, as that for the parent processes of FBM and SBM, in the entire region of scaling exponents (0 < alpha < 2 and 0 < H < 1). The FBM-SBM process is aging in a manner similar to SBM. The velocity autocorrelation function (ACF) of particle increments of FBM-SBM exhibits a dip when the parent FBM process is subdiffusive. Both for sub- and superdiffusive FBM contributions to the FBM-SBM process, the SBM exponent affects the long-time decay exponent of the ACF. Applications of the FBM-SBM-amalgamated process to the analysis of SPT data are discussed. A comparative tabulated overview of recent experimental (mainly SPT) and computational datasets amenable for interpretation in terms of FBM-, SBM-, and FBM-SBM-like models of diffusion culminates the presentation. The statistical aspects of the dynamics of a wide range of biological systems is compared in the table, from nanosized beads in living cells, to chromosomal loci, to water diffusion in the brain, and, finally, to patterns of animal movements.}, language = {en} } @article{CherstvySafdariMetzler2021, author = {Cherstvy, Andrey G. and Safdari, Hadiseh and Metzler, Ralf}, title = {Anomalous diffusion, nonergodicity, and ageing for exponentially and logarithmically time-dependent diffusivity}, series = {Journal of physics. D, Applied physics}, volume = {54}, journal = {Journal of physics. D, Applied physics}, number = {19}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {0022-3727}, doi = {10.1088/1361-6463/abdff0}, pages = {18}, year = {2021}, abstract = {We investigate a diffusion process with a time-dependent diffusion coefficient, both exponentially increasing and decreasing in time, D(t)=D-0(e +/- 2 alpha t). For this (hypothetical) nonstationary diffusion process we compute-both analytically and from extensive stochastic simulations-the behavior of the ensemble- and time-averaged mean-squared displacements (MSDs) of the particles, both in the over- and underdamped limits. Simple asymptotic relations derived for the short- and long-time behaviors are shown to be in excellent agreement with the results of simulations. The diffusive characteristics in the presence of ageing are also considered, with dramatic differences of the over- versus underdamped regime. Our results for D(t)=D-0(e +/- 2 alpha t) extend and generalize the class of diffusive systems obeying scaled Brownian motion featuring a power-law-like variation of the diffusivity with time, D(t) similar to t(alpha-1). We also examine the logarithmically increasing diffusivity, D(t)=D(0)log[t/tau(0)], as another fundamental functional dependence (in addition to the power-law and exponential) and as an example of diffusivity slowly varying in time. One of the main conclusions is that the behavior of the massive particles is predominantly ergodic, while weak ergodicity breaking is repeatedly found for the time-dependent diffusion of the massless particles at short times. The latter manifests itself in the nonequivalence of the (both nonaged and aged) MSD and the mean time-averaged MSD. The current findings are potentially applicable to a class of physical systems out of thermal equilibrium where a rapid increase or decrease of the particles' diffusivity is inherently realized. One biological system potentially featuring all three types of time-dependent diffusion (power-law-like, exponential, and logarithmic) is water diffusion in the brain tissues, as we thoroughly discuss in the end.}, language = {en} } @article{EliazarMetzler2013, author = {Eliazar, Iddo and Metzler, Ralf}, title = {Anomalous statistics of random relaxations in random environments}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {87}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.87.022141}, pages = {12}, year = {2013}, abstract = {We comprehensively analyze the emergence of anomalous statistics in the context of the random relaxation ( RARE) model [Eliazar and Metzler, J. Chem. Phys. 137, 234106 ( 2012)], a recently introduced versatile model of random relaxations in random environments. The RARE model considers excitations scattered randomly across a metric space around a reaction center. The excitations react randomly with the center, the reaction rates depending on the excitations' distances from this center. Relaxation occurs upon the first reaction between an excitation and the center. Addressing both the relaxation time and the relaxation range, we explore when these random variables display anomalous statistics, namely, heavy tails at zero and at infinity that manifest, respectively, exceptionally high occurrence probabilities of very small and very large outliers. A cohesive set of closed-form analytic results is established, determining precisely when such anomalous statistics emerge.}, language = {en} } @article{GhoshCherstvyGrebenkovetal.2016, author = {Ghosh, Surya K. and Cherstvy, Andrey G. and Grebenkov, Denis S. and Metzler, Ralf}, title = {Anomalous, non-Gaussian tracer diffusion in crowded two-dimensional environments}, series = {NEW JOURNAL OF PHYSICS}, volume = {18}, journal = {NEW JOURNAL OF PHYSICS}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/18/1/013027}, pages = {16}, year = {2016}, abstract = {A topic of intense current investigation pursues the question of how the highly crowded environment of biological cells affects the dynamic properties of passively diffusing particles. Motivated by recent experiments we report results of extensive simulations of the motion of a finite sized tracer particle in a heterogeneously crowded environment made up of quenched distributions of monodisperse crowders of varying sizes in finite circular two-dimensional domains. For given spatial distributions of monodisperse crowders we demonstrate how anomalous diffusion with strongly non-Gaussian features arises in this model system. We investigate both biologically relevant situations of particles released either at the surface of an inner domain or at the outer boundary, exhibiting distinctly different features of the observed anomalous diffusion for heterogeneous distributions of crowders. Specifically we reveal an asymmetric spreading of tracers even at moderate crowding. In addition to the mean squared displacement (MSD) and local diffusion exponent we investigate the magnitude and the amplitude scatter of the time averaged MSD of individual tracer trajectories, the non-Gaussianity parameter, and the van Hove correlation function. We also quantify how the average tracer diffusivity varies with the position in the domain with a heterogeneous radial distribution of crowders and examine the behaviour of the survival probability and the dynamics of the tracer survival probability. Inter alia, the systems we investigate are related to the passive transport of lipid molecules and proteins in two-dimensional crowded membranes or the motion in colloidal solutions or emulsions in effectively two-dimensional geometries, as well as inside supercrowded, surface adhered cells.}, language = {en} } @article{DoerriesChechkinMetzler2022, author = {Doerries, Timo J. and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Apparent anomalous diffusion and non-Gaussian distributions in a simple mobile-immobile transport model with Poissonian switching}, series = {Interface : journal of the Royal Society}, volume = {19}, journal = {Interface : journal of the Royal Society}, number = {192}, publisher = {Royal Society}, address = {London}, issn = {1742-5689}, doi = {10.1098/rsif.2022.0233}, pages = {14}, year = {2022}, abstract = {We analyse mobile-immobile transport of particles that switch between the mobile and immobile phases with finite rates. Despite this seemingly simple assumption of Poissonian switching, we unveil a rich transport dynamics including significant transient anomalous diffusion and non-Gaussian displacement distributions. Our discussion is based on experimental parameters for tau proteins in neuronal cells, but the results obtained here are expected to be of relevance for a broad class of processes in complex systems. Specifically, we obtain that, when the mean binding time is significantly longer than the mean mobile time, transient anomalous diffusion is observed at short and intermediate time scales, with a strong dependence on the fraction of initially mobile and immobile particles. We unveil a Laplace distribution of particle displacements at relevant intermediate time scales. For any initial fraction of mobile particles, the respective mean squared displacement (MSD) displays a plateau. Moreover, we demonstrate a short-time cubic time dependence of the MSD for immobile tracers when initially all particles are immobile.}, language = {en} } @article{VahabiSchulzShokrietal.2013, author = {Vahabi, Mahsa and Schulz, Johannes H. P. and Shokri, Babak and Metzler, Ralf}, title = {Area coverage of radial Levy flights with periodic boundary conditions}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {87}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.87.042136}, pages = {10}, year = {2013}, abstract = {We consider the area coverage of radial Levy flights in a finite square area with periodic boundary conditions. From simulations we show how the fractal path dimension d(f) and thus the degree of area coverage depends on the number of steps of the trajectory, the size of the area, and the resolution of the applied box counting algorithm. For sufficiently long trajectories and not too high resolution, the fractal dimension returned by the box counting method equals two, and in that sense the Levy flight fully covers the area. Otherwise, the determined fractal dimension equals the stable index of the distribution of jump lengths of the Levy flight. We provide mathematical expressions for the turnover between these two scaling regimes. As complementary methods to analyze confined Levy flights we investigate fractional order moments of the position for which we also provide scaling arguments. Finally, we study the time evolution of the probability density function and the first passage time density of Levy flights in a square area. Our findings are of interest for a general understanding of Levy flights as well as for the analysis of recorded trajectories of animals searching for food or for human motion patterns.}, language = {en} } @article{PadashSandevKantzetal.2022, author = {Padash, Amin and Sandev, Trifce and Kantz, Holger and Metzler, Ralf and Chechkin, Aleksei}, title = {Asymmetric Levy flights are more efficient in random search}, series = {Fractal and fractional}, volume = {6}, journal = {Fractal and fractional}, number = {5}, publisher = {MDPI}, address = {Basel}, issn = {2504-3110}, doi = {10.3390/fractalfract6050260}, pages = {23}, year = {2022}, abstract = {We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Levy flights by leveraging the Fokker-Planck equation with a delta-sink and an asymmetric space-fractional derivative operator with stable index alpha and asymmetry (skewness) parameter beta. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Levy flights with beta <= 0 (with a rightward bias) for short initial distances, while for beta>0 (with a leftward bias) Levy flights with alpha -> 1 are more efficient. When increasing the initial distance of the searcher to the target, Levy flight search (except for alpha=1 with beta=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Levy search compared to symmetric Levy flights at both short and long distances, and the effect is more pronounced for stable indices alpha close to unity.}, language = {en} } @article{ThapaLomholtKrogetal.2018, author = {Thapa, Samudrajit and Lomholt, Michael Andersen and Krog, Jens and Cherstvy, Andrey G. and Metzler, Ralf}, title = {Bayesian analysis of single-particle tracking data using the nested-sampling algorithm: maximum-likelihood model selection applied to stochastic-diffusivity data}, series = {Physical chemistry, chemical physics : PCCP ; a journal of European Chemical Societies}, volume = {20}, journal = {Physical chemistry, chemical physics : PCCP ; a journal of European Chemical Societies}, number = {46}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c8cp04043e}, pages = {29018 -- 29037}, year = {2018}, abstract = {We employ Bayesian statistics using the nested-sampling algorithm to compare and rank multiple models of ergodic diffusion (including anomalous diffusion) as well as to assess their optimal parameters for in silico-generated and real time-series. We focus on the recently-introduced model of Brownian motion with "diffusing diffusivity'-giving rise to widely-observed non-Gaussian displacement statistics-and its comparison to Brownian and fractional Brownian motion, also for the time-series with some measurement noise. We conduct this model-assessment analysis using Bayesian statistics and the nested-sampling algorithm on the level of individual particle trajectories. We evaluate relative model probabilities and compute best-parameter sets for each diffusion model, comparing the estimated parameters to the true ones. We test the performance of the nested-sampling algorithm and its predictive power both for computer-generated (idealised) trajectories as well as for real single-particle-tracking trajectories. Our approach delivers new important insight into the objective selection of the most suitable stochastic model for a given time-series. We also present first model-ranking results in application to experimental data of tracer diffusion in polymer-based hydrogels.}, language = {en} } @article{SecklerMetzler2022, author = {Seckler, Henrik and Metzler, Ralf}, title = {Bayesian deep learning for error estimation in the analysis of anomalous diffusion}, series = {Nature Communnications}, volume = {13}, journal = {Nature Communnications}, publisher = {Nature Publishing Group UK}, address = {London}, issn = {2041-1723}, doi = {10.1038/s41467-022-34305-6}, pages = {13}, year = {2022}, abstract = {Modern single-particle-tracking techniques produce extensive time-series of diffusive motion in a wide variety of systems, from single-molecule motion in living-cells to movement ecology. The quest is to decipher the physical mechanisms encoded in the data and thus to better understand the probed systems. We here augment recently proposed machine-learning techniques for decoding anomalous-diffusion data to include an uncertainty estimate in addition to the predicted output. To avoid the Black-Box-Problem a Bayesian-Deep-Learning technique named Stochastic-Weight-Averaging-Gaussian is used to train models for both the classification of the diffusionmodel and the regression of the anomalous diffusion exponent of single-particle-trajectories. Evaluating their performance, we find that these models can achieve a wellcalibrated error estimate while maintaining high prediction accuracies. In the analysis of the output uncertainty predictions we relate these to properties of the underlying diffusion models, thus providing insights into the learning process of the machine and the relevance of the output.}, language = {en} } @article{SecklerMetzler2022, author = {Seckler, Henrik and Metzler, Ralf}, title = {Bayesian deep learning for error estimation in the analysis of anomalous diffusion}, series = {Nature Communications}, volume = {13}, journal = {Nature Communications}, number = {1}, publisher = {Nature portfolio}, address = {Berlin}, issn = {2041-1723}, doi = {10.1038/s41467-022-34305-6}, pages = {13}, year = {2022}, abstract = {Modern single-particle-tracking techniques produce extensive time-series of diffusive motion in a wide variety of systems, from single-molecule motion in living-cells to movement ecology. The quest is to decipher the physical mechanisms encoded in the data and thus to better understand the probed systems. We here augment recently proposed machine-learning techniques for decoding anomalous-diffusion data to include an uncertainty estimate in addition to the predicted output. To avoid the Black-Box-Problem a Bayesian-Deep-Learning technique named Stochastic-Weight-Averaging-Gaussian is used to train models for both the classification of the diffusion model and the regression of the anomalous diffusion exponent of single-particle-trajectories. Evaluating their performance, we find that these models can achieve a well-calibrated error estimate while maintaining high prediction accuracies. In the analysis of the output uncertainty predictions we relate these to properties of the underlying diffusion models, thus providing insights into the learning process of the machine and the relevance of the output.
Diffusive motions in complex environments such as living biological cells or soft matter systems can be analyzed with single-particle-tracking approaches, where accuracy of output may vary. The authors involve a machine-learning technique for decoding anomalous-diffusion data and provide an uncertainty estimate together with predicted output.}, language = {en} } @article{ThapaParkKimetal.2022, author = {Thapa, Samudrajit and Park, Seongyu and Kim, Yeongjin and Jeon, Jae-Hyung and Metzler, Ralf and Lomholt, Michael A.}, title = {Bayesian inference of scaled versus fractional Brownian motion}, series = {Journal of physics : A, mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, mathematical and theoretical}, number = {19}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac60e7}, pages = {21}, year = {2022}, abstract = {We present a Bayesian inference scheme for scaled Brownian motion, and investigate its performance on synthetic data for parameter estimation and model selection in a combined inference with fractional Brownian motion. We include the possibility of measurement noise in both models. We find that for trajectories of a few hundred time points the procedure is able to resolve well the true model and parameters. Using the prior of the synthetic data generation process also for the inference, the approach is optimal based on decision theory. We include a comparison with inference using a prior different from the data generating one.}, language = {en} } @article{SandevSokolovMetzleretal.2017, author = {Sandev, Trifce and Sokolov, Igor M. and Metzler, Ralf and Chechkin, Aleksei V.}, title = {Beyond monofractional kinetics}, series = {Chaos, solitons \& fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science}, volume = {102}, journal = {Chaos, solitons \& fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science}, publisher = {Elsevier}, address = {Oxford}, issn = {0960-0779}, doi = {10.1016/j.chaos.2017.05.001}, pages = {210 -- 217}, year = {2017}, abstract = {We discuss generalized integro-differential diffusion equations whose integral kernels are not of a simple power law form, and thus these equations themselves do not belong to the family of fractional diffusion equations exhibiting a monoscaling behavior. They instead generate a broad class of anomalous nonscaling patterns, which correspond either to crossovers between different power laws, or to a non-power-law behavior as exemplified by the logarithmic growth of the width of the distribution. We consider normal and modified forms of these generalized diffusion equations and provide a brief discussion of three generic types of integral kernels for each form, namely, distributed order, truncated power law and truncated distributed order kernels. For each of the cases considered we prove the non-negativity of the solution of the corresponding generalized diffusion equation and calculate the mean squared displacement. (C) 2017 Elsevier Ltd. All rights reserved.}, language = {en} } @article{HouCherstvyMetzleretal.2018, author = {Hou, Ru and Cherstvy, Andrey G. and Metzler, Ralf and Akimoto, Takuma}, title = {Biased continuous-time random walks for ordinary and equilibrium cases}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {20}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, number = {32}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c8cp01863d}, pages = {20827 -- 20848}, year = {2018}, abstract = {We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent alpha are considered for the WTD psi(t) similar to 1/t(1+alpha). We treat continuous-time random walks (CTRWs) with 0 < alpha < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < alpha < 2 and alpha > 2. We demonstrate that in the presence of a drift CTRWs with alpha < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < alpha < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with alpha > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < alpha < 2 and alpha > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with alpha > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent alpha.}, language = {en} } @article{TalukderSenChakrabortietal.2014, author = {Talukder, Srijeeta and Sen, Shrabani and Chakraborti, Prantik and Metzler, Ralf and Banik, Suman K. and Chaudhury, Pinaki}, title = {Breathing dynamics based parameter sensitivity analysis of hetero-polymeric DNA}, series = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, volume = {140}, journal = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, number = {12}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0021-9606}, doi = {10.1063/1.4869112}, pages = {10}, year = {2014}, abstract = {We study the parameter sensitivity of hetero-polymeric DNA within the purview of DNA breathing dynamics. The degree of correlation between the mean bubble size and the model parameters is estimated for this purpose for three different DNA sequences. The analysis leads us to a better understanding of the sequence dependent nature of the breathing dynamics of hetero-polymeric DNA. Out of the 14 model parameters for DNA stability in the statistical Poland-Scheraga approach, the hydrogen bond interaction epsilon(hb)(AT) for an AT base pair and the ring factor. turn out to be the most sensitive parameters. In addition, the stacking interaction epsilon(st)(TA-TA) for an TA-TA nearest neighbor pair of base-pairs is found to be the most sensitive one among all stacking interactions. Moreover, we also establish that the nature of stacking interaction has a deciding effect on the DNA breathing dynamics, not the number of times a particular stacking interaction appears in a sequence. We show that the sensitivity analysis can be used as an effective measure to guide a stochastic optimization technique to find the kinetic rate constants related to the dynamics as opposed to the case where the rate constants are measured using the conventional unbiased way of optimization. (c) 2014 AIP Publishing LLC.}, language = {en} } @article{Metzler2019, author = {Metzler, Ralf}, title = {Brownian motion and beyond: first-passage, power spectrum, non-Gaussianity, and anomalous diffusion}, series = {Journal of statistical mechanics: theory and experiment}, volume = {2019}, journal = {Journal of statistical mechanics: theory and experiment}, number = {11}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/ab4988}, pages = {18}, year = {2019}, abstract = {Brownian motion is a ubiquitous physical phenomenon across the sciences. After its discovery by Brown and intensive study since the first half of the 20th century, many different aspects of Brownian motion and stochastic processes in general have been addressed in Statistical Physics. In particular, there now exists a very large range of applications of stochastic processes in various disciplines. Here we provide a summary of some of the recent developments in the field of stochastic processes, highlighting both the experimental findings and theoretical frameworks.}, language = {en} } @article{ChechkinSenoMetzleretal.2017, author = {Chechkin, Aleksei V. and Seno, Flavio and Metzler, Ralf and Sokolov, Igor M.}, title = {Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities}, series = {Physical review : X, Expanding access}, volume = {7}, journal = {Physical review : X, Expanding access}, publisher = {American Physical Society}, address = {College Park}, issn = {2160-3308}, doi = {10.1103/PhysRevX.7.021002}, pages = {20}, year = {2017}, abstract = {A growing number of biological, soft, and active matter systems are observed to exhibit normal diffusive dynamics with a linear growth of the mean-squared displacement, yet with a non-Gaussian distribution of increments. Based on the Chubinsky-Slater idea of a diffusing diffusivity, we here establish and analyze a minimal model framework of diffusion processes with fluctuating diffusivity. In particular, we demonstrate the equivalence of the diffusing diffusivity process with a superstatistical approach with a distribution of diffusivities, at times shorter than the diffusivity correlation time. At longer times, a crossover to a Gaussian distribution with an effective diffusivity emerges. Specifically, we establish a subordination picture of Brownian but non-Gaussian diffusion processes, which can be used for a wide class of diffusivity fluctuation statistics. Our results are shown to be in excellent agreement with simulations and numerical evaluations.}, language = {en} } @article{EmanuelCherstvyMetzleretal.2020, author = {Emanuel, Marc D. and Cherstvy, Andrey G. and Metzler, Ralf and Gompper, Gerhard}, title = {Buckling transitions and soft-phase invasion of two-component icosahedral shells}, series = {Physical review / publ. by The American Physical Society. E, Statistical, nonlinear, and soft matter physics}, volume = {102}, journal = {Physical review / publ. by The American Physical Society. E, Statistical, nonlinear, and soft matter physics}, number = {6}, publisher = {Woodbury}, address = {New York}, issn = {2470-0045}, doi = {10.1103/PhysRevE.102.062104}, pages = {26}, year = {2020}, abstract = {What is the optimal distribution of two types of crystalline phases on the surface of icosahedral shells, such as of many viral capsids? We here investigate the distribution of a thin layer of soft material on a crystalline convex icosahedral shell. We demonstrate how the shapes of spherical viruses can be understood from the perspective of elasticity theory of thin two-component shells. We develop a theory of shape transformations of an icosahedral shell upon addition of a softer, but still crystalline, material onto its surface. We show how the soft component "invades" the regions with the highest elastic energy and stress imposed by the 12 topological defects on the surface. We explore the phase diagram as a function of the surface fraction of the soft material, the shell size, and the incommensurability of the elastic moduli of the rigid and soft phases. We find that, as expected, progressive filling of the rigid shell by the soft phase starts from the most deformed regions of the icosahedron. With a progressively increasing soft-phase coverage, the spherical segments of domes are filled first (12 vertices of the shell), then the cylindrical segments connecting the domes (30 edges) are invaded, and, ultimately, the 20 flat faces of the icosahedral shell tend to be occupied by the soft material. We present a detailed theoretical investigation of the first two stages of this invasion process and develop a model of morphological changes of the cone structure that permits noncircular cross sections. In conclusion, we discuss the biological relevance of some structures predicted from our calculations, in particular for the shape of viral capsids.}, language = {en} } @article{ChechkinZaidLomholtetal.2012, author = {Chechkin, Aleksei V. and Zaid, Irwin M. and Lomholt, Michael A. and Sokolov, Igor M. and Metzler, Ralf}, title = {Bulk-mediated diffusion on a planar surface full solution}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {86}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.86.041101}, pages = {11}, year = {2012}, abstract = {We consider the effective surface motion of a particle that intermittently unbinds from a planar surface and performs bulk excursions. Based on a random-walk approach, we derive the diffusion equations for surface and bulk diffusion including the surface-bulk coupling. From these exact dynamic equations, we analytically obtain the propagator of the effective surface motion. This approach allows us to deduce a superdiffusive, Cauchy-type behavior on the surface, together with exact cutoffs limiting the Cauchy form. Moreover, we study the long-time dynamics for the surface motion.}, language = {en} } @article{ChechkinZaidLomholtetal.2013, author = {Chechkin, Aleksei V. and Zaid, I. M. and Lomholt, M. A. and Sokolov, Igor M. and Metzler, Ralf}, title = {Bulk-mediated surface diffusion on a cylinder in the fast exchange limit}, series = {Mathematical modelling of natural phenomena}, volume = {8}, journal = {Mathematical modelling of natural phenomena}, number = {2}, publisher = {EDP Sciences}, address = {Les Ulis}, issn = {0973-5348}, doi = {10.1051/mmnp/20138208}, pages = {114 -- 126}, year = {2013}, abstract = {In various biological systems and small scale technological applications particles transiently bind to a cylindrical surface. Upon unbinding the particles diffuse in the vicinal bulk before rebinding to the surface. Such bulk-mediated excursions give rise to an effective surface translation, for which we here derive and discuss the dynamic equations, including additional surface diffusion. We discuss the time evolution of the number of surface-bound particles, the effective surface mean squared displacement, and the surface propagator. In particular, we observe sub- and superdiffusive regimes. A plateau of the surface mean-squared displacement reflects a stalling of the surface diffusion at longer times. Finally, the corresponding first passage problem for the cylindrical geometry is analysed.}, language = {en} } @article{KindlerPulkkinenCherstvyetal.2019, author = {Kindler, Oliver and Pulkkinen, Otto and Cherstvy, Andrey G. and Metzler, Ralf}, title = {Burst Statistics in an Early Biofilm Quorum Sensing Mode}, series = {Scientific Reports}, volume = {9}, journal = {Scientific Reports}, publisher = {Macmillan Publishers Limited part of Springer Nature}, address = {London}, issn = {2045-2322}, doi = {10.1038/s41598-019-48525-2}, pages = {19}, year = {2019}, abstract = {Quorum-sensing bacteria in a growing colony of cells send out signalling molecules (so-called "autoinducers") and themselves sense the autoinducer concentration in their vicinity. Once—due to increased local cell density inside a "cluster" of the growing colony—the concentration of autoinducers exceeds a threshold value, cells in this clusters get "induced" into a communal, multi-cell biofilm-forming mode in a cluster-wide burst event. We analyse quantitatively the influence of spatial disorder, the local heterogeneity of the spatial distribution of cells in the colony, and additional physical parameters such as the autoinducer signal range on the induction dynamics of the cell colony. Spatial inhomogeneity with higher local cell concentrations in clusters leads to earlier but more localised induction events, while homogeneous distributions lead to comparatively delayed but more concerted induction of the cell colony, and, thus, a behaviour close to the mean-field dynamics. We quantify the induction dynamics with quantifiers such as the time series of induction events and burst sizes, the grouping into induction families, and the mean autoinducer concentration levels. Consequences for different scenarios of biofilm growth are discussed, providing possible cues for biofilm control in both health care and biotechnology.}, language = {en} } @article{VargheseChechkinMetzleretal.2021, author = {Varghese, Alan J. and Chechkin, Aleksei and Metzler, Ralf and Sujith, Raman I.}, title = {Capturing multifractality of pressure fluctuations in thermoacoustic systems using fractional-order derivatives}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {31}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {3}, publisher = {American Institute of Physics, AIP}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/5.0032585}, pages = {9}, year = {2021}, abstract = {The stable operation of a turbulent combustor is not completely silent; instead, there is a background of small amplitude aperiodic acoustic fluctuations known as combustion noise. Pressure fluctuations during this state of combustion noise are multifractal due to the presence of multiple temporal scales that contribute to its dynamics. However, existing models are unable to capture the multifractality in the pressure fluctuations. We conjecture an underlying fractional dynamics for the thermoacoustic system and obtain a fractional-order model for pressure fluctuations. The data from this model has remarkable visual similarity to the experimental data and also has a wide multifractal spectrum during the state of combustion noise. Quantitative similarity with the experimental data in terms of the Hurst exponent and the multifractal spectrum is observed during the state of combustion noise. This model is also able to produce pressure fluctuations that are qualitatively similar to the experimental data acquired during intermittency and thermoacoustic instability. Furthermore, we argue that the fractional dynamics vanish as we approach the state of thermoacoustic instability.}, language = {en} } @article{MutothyaXuLietal.2021, author = {Mutothya, Nicholas Mwilu and Xu, Yong and Li, Yongge and Metzler, Ralf}, title = {Characterising stochastic motion in heterogeneous media driven by coloured non-Gaussian noise}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {54}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {29}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/abfba6}, pages = {31}, year = {2021}, abstract = {We study the stochastic motion of a test particle in a heterogeneous medium in terms of a position dependent diffusion coefficient mimicking measured deterministic diffusivity gradients in biological cells or the inherent heterogeneity of geophysical systems. Compared to previous studies we here investigate the effect of the interplay of anomalous diffusion effected by position dependent diffusion coefficients and coloured non-Gaussian noise. The latter is chosen to be distributed according to Tsallis' q-distribution, representing a popular example for a non-extensive statistic. We obtain the ensemble and time averaged mean squared displacements for this generalised process and establish its non-ergodic properties as well as analyse the non-Gaussian nature of the associated displacement distribution. We consider both non-stratified and stratified environments.}, language = {en} } @article{VilkAghionNathanetal.2022, author = {Vilk, Ohad and Aghion, Erez and Nathan, Ran and Toledo, Sivan and Metzler, Ralf and Assaf, Michael}, title = {Classification of anomalous diffusion in animal movement data using power spectral analysis}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {33}, publisher = {IOP Publishing}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac7e8f}, pages = {16}, year = {2022}, abstract = {The field of movement ecology has seen a rapid increase in high-resolution data in recent years, leading to the development of numerous statistical and numerical methods to analyse relocation trajectories. Data are often collected at the level of the individual and for long periods that may encompass a range of behaviours. Here, we use the power spectral density (PSD) to characterise the random movement patterns of a black-winged kite (Elanus caeruleus) and a white stork (Ciconia ciconia). The tracks are first segmented and clustered into different behaviours (movement modes), and for each mode we measure the PSD and the ageing properties of the process. For the foraging kite we find 1/f noise, previously reported in ecological systems mainly in the context of population dynamics, but not for movement data. We further suggest plausible models for each of the behavioural modes by comparing both the measured PSD exponents and the distribution of the single-trajectory PSD to known theoretical results and simulations.}, language = {en} } @article{AwadMetzler2022, author = {Awad, Emad and Metzler, Ralf}, title = {Closed-form multi-dimensional solutions and asymptotic behaviours for subdiffusive processes with crossovers: II. Accelerating case}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {20}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac5a90}, pages = {29}, year = {2022}, abstract = {Anomalous diffusion with a power-law time dependence vertical bar R vertical bar(2)(t) similar or equal to t(alpha i) of the mean squared displacement occurs quite ubiquitously in numerous complex systems. Often, this anomalous diffusion is characterised by crossovers between regimes with different anomalous diffusion exponents alpha(i). Here we consider the case when such a crossover occurs from a first regime with alpha(1) to a second regime with alpha(2) such that alpha(2) > alpha(1), i.e., accelerating anomalous diffusion. A widely used framework to describe such crossovers in a one-dimensional setting is the bi-fractional diffusion equation of the so-called modified type, involving two time-fractional derivatives defined in the Riemann-Liouville sense. We here generalise this bi-fractional diffusion equation to higher dimensions and derive its multidimensional propagator (Green's function) for the general case when also a space fractional derivative is present, taking into consideration long-ranged jumps (Levy flights). We derive the asymptotic behaviours for this propagator in both the short- and long-time as well the short- and long-distance regimes. Finally, we also calculate the mean squared displacement, skewness and kurtosis in all dimensions, demonstrating that in the general case the non-Gaussian shape of the probability density function changes.}, language = {en} } @article{ŚlęzakMetzlerMagdziarz2019, author = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin}, title = {Codifference can detect ergodicity breaking and non-Gaussianity}, series = {New Journal of Physics}, volume = {21}, journal = {New Journal of Physics}, publisher = {Deutsche Physikalische Gesellschaft}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab13f3}, pages = {25}, year = {2019}, abstract = {We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a nonlinear analogue of the mean squared displacement.}, language = {en} } @article{GodecBauerMetzler2014, author = {Godec, Aljaz and Bauer, Maximilian and Metzler, Ralf}, title = {Collective dynamics effect transient subdiffusion of inert tracers in flexible gel networks}, series = {New journal of physics : the open-access journal for physics}, volume = {16}, journal = {New journal of physics : the open-access journal for physics}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/16/9/092002}, pages = {13}, year = {2014}, abstract = {Based on extensive Brownian dynamics simulations we study the thermal motion of a tracer bead in a cross-linked, flexible gel in the limit when the tracer particle size is comparable to or even larger than the equilibrium mesh size of the gel. The analysis of long individual trajectories of the tracer demonstrates the existence of pronounced transient anomalous diffusion. From the time averaged mean squared displacement and the time averaged van Hove correlation functions we elucidate the many-body origin of the non-Brownian tracer bead dynamics. Our results shed new light onto the ongoing debate over the physical origin of steric tracer interactions with structured environments.}, language = {en} } @article{SandevIominKantzetal.2016, author = {Sandev, Trifce and Iomin, Alexander and Kantz, Holger and Metzler, Ralf and Chechkin, Aleksei V.}, title = {Comb Model with Slow and Ultraslow Diffusion}, series = {Mathematical modelling of natural phenomena}, volume = {11}, journal = {Mathematical modelling of natural phenomena}, publisher = {EDP Sciences}, address = {Les Ulis}, issn = {0973-5348}, doi = {10.1051/mmnp/201611302}, pages = {18 -- 33}, year = {2016}, abstract = {We consider a generalized diffusion equation in two dimensions for modeling diffusion on a comb-like structures. We analyze the probability distribution functions and we derive the mean squared displacement in x and y directions. Different forms of the memory kernels (Dirac delta, power-law, and distributed order) are considered. It is shown that anomalous diffusion may occur along both x and y directions. Ultraslow diffusion and some more general diffusive processes are observed as well. We give the corresponding continuous time random walk model for the considered two dimensional diffusion-like equation on a comb, and we derive the probability distribution functions which subordinate the process governed by this equation to the Wiener process.}, language = {en} } @article{PalyulinMantsevichKlagesetal.2017, author = {Palyulin, Vladimir V. and Mantsevich, Vladimir N. and Klages, Rainer and Metzler, Ralf and Chechkin, Aleksei V.}, title = {Comparison of pure and combined search strategies for single and multiple targets}, series = {The European physical journal : B, Condensed matter and complex systems}, volume = {90}, journal = {The European physical journal : B, Condensed matter and complex systems}, publisher = {Springer}, address = {New York}, issn = {1434-6028}, doi = {10.1140/epjb/e2017-80372-4}, pages = {20 -- 37}, year = {2017}, abstract = {We address the generic problem of random search for a point-like target on a line. Using the measures of search reliability and efficiency to quantify the random search quality, we compare Brownian search with Levy search based on long-tailed jump length distributions. We then compare these results with a search process combined of two different long-tailed jump length distributions. Moreover, we study the case of multiple targets located by a Levy searcher.}, language = {en} } @article{BlavatskaMetzler2015, author = {Blavatska, Viktoria and Metzler, Ralf}, title = {Conformational properties of complex polymers: rosette versus star-like structures}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {48}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {13}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/48/13/135001}, pages = {14}, year = {2015}, abstract = {Multiple loop formation in polymer macromolecules is an important feature of the chromatin organization and DNA compactification in the nuclei. We analyse the size and shape characteristics of complex polymer structures, containing in general f(1) loops (petals) and f(2) linear chains (branches). Within the frames of continuous model of Gaussian macromolecule, we apply the path integration method and obtain the estimates for gyration radius R-g and asphericity (A) over cap of typical conformation as functions of parameters f(1), f(2). In particular, our results qualitatively reveal the extent of anisotropy of star-like topologies as compared to the rosette structures of the same total molecular weight.}, language = {en} } @article{DybiecCapalaChechkinetal.2018, author = {Dybiec, Bartlomiej and Capala, Karol and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Conservative random walks in confining potentials}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {1}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aaefc2}, pages = {25}, year = {2018}, abstract = {Levy walks are continuous time random walks with spatio-temporal coupling of jump lengths and waiting times, often used to model superdiffusive spreading processes such as animals searching for food, tracer motion in weakly chaotic systems, or even the dynamics in quantum systems such as cold atoms. In the simplest version Levy walks move with a finite speed. Here, we present an extension of the Levy walk scenario for the case when external force fields influence the motion. The resulting motion is a combination of the response to the deterministic force acting on the particle, changing its velocity according to the principle of total energy conservation, and random velocity reversals governed by the distribution of waiting times. For the fact that the motion stays conservative, that is, on a constant energy surface, our scenario is fundamentally different from thermal motion in the same external potentials. In particular, we present results for the velocity and position distributions for single well potentials of different steepness. The observed dynamics with its continuous velocity changes enriches the theory of Levy walk processes and will be of use in a variety of systems, for which the particles are externally confined.}, language = {en} } @article{GranadoAbadMetzleretal.2020, author = {Granado, Felipe Le Vot and Abad, Enrique and Metzler, Ralf and Yuste, Santos B.}, title = {Continuous time random walk in a velocity field}, series = {New Journal of Physics}, volume = {22}, journal = {New Journal of Physics}, publisher = {Dt. Physikalische Ges.}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab9ae2}, pages = {27}, year = {2020}, abstract = {We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a L{\´e}vy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of L{\´e}vy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.}, language = {en} } @article{SchulzChechkinMetzler2013, author = {Schulz, Johannes H. P. and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Correlated continuous time random walks - combining scale-invariance with long-range memory for spatial and temporal dynamics}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {46}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {47}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/46/47/475001}, pages = {22}, year = {2013}, abstract = {Standard continuous time random walk (CTRW) models are renewal processes in the sense that at each jump a new, independent pair of jump length and waiting time are chosen. Globally, anomalous diffusion emerges through scale-free forms of the jump length and/or waiting time distributions by virtue of the generalized central limit theorem. Here we present a modified version of recently proposed correlated CTRW processes, where we incorporate a power-law correlated noise on the level of both jump length and waiting time dynamics. We obtain a very general stochastic model, that encompasses key features of several paradigmatic models of anomalous diffusion: discontinuous, scale-free displacements as in Levy flights, scale-free waiting times as in subdiffusive CTRWs, and the long-range temporal correlations of fractional Brownian motion (FBM). We derive the exact solutions for the single-time probability density functions and extract the scaling behaviours. Interestingly, we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws. Our model will be useful for describing recent experimental single particle tracking data that feature a combination of CTRW and FBM properties.}, language = {en} } @article{MagdziarzMetzlerSzczotkaetal.2012, author = {Magdziarz, Marcin and Metzler, Ralf and Szczotka, Wladyslaw and Zebrowski, Piotr}, title = {Correlated continuous-time random walks in external force fields}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {85}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {5}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.85.051103}, pages = {5}, year = {2012}, abstract = {We study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated waiting times. In this model the current waiting time T-i is equal to the previous waiting time Ti-1 plus a small increment. Based on the associated coupled Langevin equations the force field is systematically introduced. We show that in a confining potential the relaxation dynamics follows power-law or stretched exponential pattern, depending on the model parameters. The process obeys a generalized Einstein-Stokes-Smoluchowski relation and observes the second Einstein relation. The stationary solution is of Boltzmann-Gibbs form. The case of an harmonic potential is discussed in some detail. We also show that the process exhibits aging and ergodicity breaking.}, language = {en} } @article{MagdziarzMetzlerSzczotkaetal.2012, author = {Magdziarz, Marcin and Metzler, Ralf and Szczotka, Wladyslaw and Zebrowski, Piotr}, title = {Correlated continuous-time random walks-scaling limits and Langevin picture}, series = {Journal of statistical mechanics: theory and experiment}, journal = {Journal of statistical mechanics: theory and experiment}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/2012/04/P04010}, pages = {18}, year = {2012}, abstract = {In this paper we analyze correlated continuous-time random walks introduced recently by Tejedor and Metzler (2010 J. Phys. A: Math. Theor. 43 082002). We obtain the Langevin equations associated with this process and the corresponding scaling limits of their solutions. We prove that the limit processes are self-similar and display anomalous dynamics. Moreover, we extend the model to include external forces. Our results are confirmed by Monte Carlo simulations.}, language = {en} } @article{SandevMetzlerTomovski2014, author = {Sandev, Trifce and Metzler, Ralf and Tomovski, Zivorad}, title = {Correlation functions for the fractional generalized Langevin equation in the presence of internal and external noise}, series = {Journal of mathematical physics}, volume = {55}, journal = {Journal of mathematical physics}, number = {2}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0022-2488}, doi = {10.1063/1.4863478}, pages = {23}, year = {2014}, abstract = {We study generalized fractional Langevin equations in the presence of a harmonic potential. General expressions for the mean velocity and particle displacement, the mean squared displacement, position and velocity correlation functions, as well as normalized displacement correlation function are derived. We report exact results for the cases of internal and external friction, that is, when the driving noise is either internal and thus the fluctuation-dissipation relation is fulfilled or when the noise is external. The asymptotic behavior of the generalized stochastic oscillator is investigated, and the case of high viscous damping (overdamped limit) is considered. Additional behaviors of the normalized displacement correlation functions different from those for the regular damped harmonic oscillator are observed. In addition, the cases of a constant external force and the force free case are obtained. The validity of the generalized Einstein relation for this process is discussed. The considered fractional generalized Langevin equation may be used to model anomalous diffusive processes including single file-type diffusion.}, language = {en} } @article{TeomyMetzler2019, author = {Teomy, Eial and Metzler, Ralf}, title = {Correlations and transport in exclusion processes with general finite memory}, series = {Journal of statistical mechanics: theory and experiment}, volume = {2019}, journal = {Journal of statistical mechanics: theory and experiment}, number = {10}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/ab47fb}, pages = {31}, year = {2019}, language = {en} } @article{CaetanoCarvalhoMetzleretal.2020, author = {Caetano, Daniel L. Z. and Carvalho, Sidney Jurado de and Metzler, Ralf and Cherstvy, Andrey G.}, title = {Critical adsorption of multiple polyelectrolytes onto a nanosphere}, series = {Interface : journal of the Royal Society}, volume = {17}, journal = {Interface : journal of the Royal Society}, number = {167}, publisher = {Royal Society}, address = {London}, issn = {1742-5689}, doi = {10.1098/rsif.2020.0199}, pages = {10}, year = {2020}, abstract = {Employing extensive Monte Carlo computer simulations, we investigate in detail the properties of multichain adsorption of charged flexible polyelectrolytes (PEs) onto oppositely charged spherical nanoparticles (SNPs). We quantify the conditions of critical adsorption-the phase-separation curve between the adsorbed and desorbed states of the PEs-as a function of the SNP surface-charge density and the concentration of added salt. We study the degree of fluctuations of the PE-SNP electrostatic binding energy, which we use to quantify the emergence of the phase subtransitions, including a series of partially adsorbed PE configurations. We demonstrate how the phase-separation adsorption-desorption boundary shifts and splits into multiple subtransitions at low-salt conditions, thereby generalizing and extending the results for critical adsorption of a single PE onto the SNP. The current findings are relevant for finite concentrations of PEs around the attracting SNP, such as the conditions for PE adsorption onto globular proteins carrying opposite electric charges.}, language = {en} } @article{CaetanodeCarvalhoMetzleretal.2017, author = {Caetano, Daniel L. Z. and de Carvalho, Sidney J. and Metzler, Ralf and Cherstvy, Andrey G.}, title = {Critical adsorption of periodic and random polyampholytes onto charged surfaces}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {19}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c7cp04040g}, pages = {23397 -- 23413}, year = {2017}, abstract = {How different are the properties of critical adsorption of polyampholytes and polyelectrolytes onto charged surfaces? How important are the details of polyampholyte charge distribution on the onset of critical adsorption transition? What are the scaling relations governing the dependence of critical surface charge density on salt concentration in the surrounding solution? Here, we employ Metropolis Monte Carlo simulations and uncover the scaling relations for critical adsorption for quenched periodic and random charge distributions along the polyampholyte chains. We also evaluate and discuss the dependence of the adsorbed layer width on solution salinity and details of the charge distribution. We contrast our findings to the known results for polyelectrolyte adsorption onto oppositely charged surfaces, in particular, their dependence on electrolyte concentration.}, language = {en} } @article{deCarvalhoMetzlerCherstvy2014, author = {de Carvalho, Sidney J. and Metzler, Ralf and Cherstvy, Andrey G.}, title = {Critical adsorption of polyelectrolytes onto charged Janus nanospheres}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {16}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, number = {29}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c4cp02207f}, pages = {15539 -- 15550}, year = {2014}, abstract = {Based on extensive Monte Carlo simulations and analytical considerations we study the electrostatically driven adsorption of flexible polyelectrolyte chains onto charged Janus nanospheres. These net-neutral colloids are composed of two equally but oppositely charged hemispheres. The critical binding conditions for polyelectrolyte chains are analysed as function of the radius of the Janus particle and its surface charge density, as well as the salt concentration in the ambient solution. Specifically for the adsorption of finite-length polyelectrolyte chains onto Janus nanoparticles, we demonstrate that the critical adsorption conditions drastically differ when the size of the Janus particle or the screening length of the electrolyte are varied. We compare the scaling laws obtained for the adsorption-desorption threshold to the known results for uniformly charged spherical particles, observing significant disparities. We also contrast the changes to the polyelectrolyte chain conformations close to the surface of the Janus nanoparticles as compared to those for simple spherical particles. Finally, we discuss experimentally relevant physicochemical systems for which our simulations results may become important. In particular, we observe similar trends with polyelectrolyte complexation with oppositely but heterogeneously charged proteins.}, language = {en} } @article{deCarvalhoMetzlerCherstvy2016, author = {de Carvalho, Sidney J. and Metzler, Ralf and Cherstvy, Andrey G.}, title = {Critical adsorption of polyelectrolytes onto planar and convex highly charged surfaces}, series = {New journal of physics : the open-access journal for physics}, volume = {18}, journal = {New journal of physics : the open-access journal for physics}, publisher = {IOP Publ.}, address = {London}, issn = {1367-2630}, doi = {10.1088/1367-2630/18/8/083037}, year = {2016}, abstract = {We study the adsorption-desorption transition of polyelectrolyte chains onto planar, cylindrical and spherical surfaces with arbitrarily high surface charge densities by massive Monte Carlo computer simulations. We examine in detail how the well known scaling relations for the threshold transition—demarcating the adsorbed and desorbed domains of a polyelectrolyte near weakly charged surfaces—are altered for highly charged interfaces. In virtue of high surface potentials and large surface charge densities, the Debye-H{\"u}ckel approximation is often not feasible and the nonlinear Poisson-Boltzmann approach should be implemented. At low salt conditions, for instance, the electrostatic potential from the nonlinear Poisson-Boltzmann equation is smaller than the Debye-H{\"u}ckel result, such that the required critical surface charge density for polyelectrolyte adsorption σc increases. The nonlinear relation between the surface charge density and electrostatic potential leads to a sharply increasing critical surface charge density with growing ionic strength, imposing an additional limit to the critical salt concentration above which no polyelectrolyte adsorption occurs at all. We contrast our simulations findings with the known scaling results for weak critical polyelectrolyte adsorption onto oppositely charged surfaces for the three standard geometries. Finally, we discuss some applications of our results for some physical-chemical and biophysical systems.}, language = {en} } @article{deCarvalhoMetzlerCherstvy2016, author = {de Carvalho, Sidney J. and Metzler, Ralf and Cherstvy, Andrey G.}, title = {Critical adsorption of polyelectrolytes onto planar and convex highly charged surfaces: the nonlinear Poisson-Boltzmann approach}, series = {NEW JOURNAL OF PHYSICS}, volume = {18}, journal = {NEW JOURNAL OF PHYSICS}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/18/8/083037}, pages = {17}, year = {2016}, abstract = {We study the adsorption-desorption transition of polyelectrolyte chains onto planar, cylindrical and spherical surfaces with arbitrarily high surface charge densities by massive Monte Carlo computer simulations. We examine in detail how the well known scaling relations for the threshold transition demarcating the adsorbed and desorbed domains of a polyelectrolyte near weakly charged surfaces-are altered for highly charged interfaces. In virtue of high surface potentials and large surface charge densities, the Debye-Huckel approximation is often not feasible and the nonlinear Poisson-Boltzmann approach should be implemented. At low salt conditions, for instance, the electrostatic potential from the nonlinear Poisson-Boltzmann equation is smaller than the Debye-Huckel result, such that the required critical surface charge density for polyelectrolyte adsorption sigma(c) increases. The nonlinear relation between the surface charge density and electrostatic potential leads to a sharply increasing critical surface charge density with growing ionic strength, imposing an additional limit to the critical salt concentration above which no polyelectrolyte adsorption occurs at all. We contrast our simulations findings with the known scaling results for weak critical polyelectrolyte adsorption onto oppositely charged surfaces for the three standard geometries. Finally, we discuss some applications of our results for some physical-chemical and biophysical systems.}, language = {en} } @article{AwadMetzler2020, author = {Awad, Emad and Metzler, Ralf}, title = {Crossover dynamics from superdiffusion to subdiffusion}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {23}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {1}, publisher = {De Gruyter}, address = {Berlin ; Boston}, issn = {1311-0454}, doi = {10.1515/fca-2020-0003}, pages = {55 -- 102}, year = {2020}, abstract = {The Cattaneo or telegrapher's equation describes the crossover from initial ballistic to normal diffusion. Here we study and survey time-fractional generalisations of this equation that are shown to produce the crossover of the mean squared displacement from superdiffusion to subdiffusion. Conditional solutions are derived in terms of Fox H-functions and the dth-order moments as well as the diffusive flux of the different models are derived. Moreover, the concept of the distribution-like is proposed as an alternative to the probability density function.}, language = {en} } @article{MolinaGarciaSandevSafdarietal.2018, author = {Molina-Garcia, Daniel and Sandev, Trifce and Safdari, Hadiseh and Pagnini, Gianni and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Crossover from anomalous to normal diffusion}, series = {New Journal of Physics}, volume = {20}, journal = {New Journal of Physics}, publisher = {IOP Publishing Ltd}, address = {London und Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/aae4b2}, pages = {28}, year = {2018}, abstract = {Abstract The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive-diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive-diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion-diffusion and subdiffusion-subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling.}, language = {en} } @article{GhoshCherstvyMetzler2014, author = {Ghosh, Surya K. and Cherstvy, Andrey G. and Metzler, Ralf}, title = {Deformation propagation in responsive polymer network films}, series = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, volume = {141}, journal = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, number = {7}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0021-9606}, doi = {10.1063/1.4893056}, pages = {9}, year = {2014}, abstract = {We study the elastic deformations in a cross-linked polymer network film triggered by the binding of submicron particles with a sticky surface, mimicking the interactions of viral pathogens with thin films of stimulus-responsive polymeric materials such as hydrogels. From extensive Langevin Dynamics simulations we quantify how far the network deformations propagate depending on the elasticity parameters of the network and the adhesion strength of the particles. We examine the dynamics of the collective area shrinkage of the network and obtain some simple relations for the associated characteristic decay lengths. A detailed analysis elucidates how the elastic energy of the network is distributed between stretching and compression modes in response to the particle binding. We also examine the force-distance curves of the repulsion or attraction interactions for a pair of sticky particles in the polymer network film as a function of the particle-particle separation. The results of this computational study provide new insight into collective phenomena in soft polymer network films and may, in particular, be applied to applications for visual detection of pathogens such as viruses via a macroscopic response of thin films of cross-linked hydrogels. (C) 2014 AIP Publishing LLC.}, language = {en} } @article{SandevChechkinKantzetal.2015, author = {Sandev, Trifce and Chechkin, Aleksei V. and Kantz, Holger and Metzler, Ralf}, title = {Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {18}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {4}, publisher = {De Gruyter}, address = {Berlin}, issn = {1311-0454}, doi = {10.1515/fca-2015-0059}, pages = {1006 -- 1038}, year = {2015}, abstract = {We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck-Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.}, language = {en} } @article{LeijnseJeonLoftetal.2012, author = {Leijnse, N. and Jeon, J. -H. and Loft, S. and Metzler, Ralf and Oddershede, L. B.}, title = {Diffusion inside living human cells}, series = {European physical journal special topics}, volume = {204}, journal = {European physical journal special topics}, number = {1}, publisher = {Springer}, address = {Heidelberg}, issn = {1951-6355}, doi = {10.1140/epjst/e2012-01553-y}, pages = {75 -- 84}, year = {2012}, abstract = {Naturally occurring lipid granules diffuse in the cytoplasm and can be used as tracers to map out the viscoelastic landscape inside living cells. Using optical trapping and single particle tracking we found that lipid granules exhibit anomalous diffusion inside human umbilical vein endothelial cells. For these cells the exact diffusional pattern of a particular granule depends on the physiological state of the cell and on the localization of the granule within the cytoplasm. Granules located close to the actin rich periphery of the cell move less than those located towards to the center of the cell or within the nucleus. Also, granules in cells which are stressed by intense laser illumination or which have attached to a surface for a long period of time move in a more restricted fashion than those within healthy cells. For granules diffusing in healthy cells, in regions away from the cell periphery, occurrences of weak ergodicity breaking are observed, similar to the recent observations inside living fission yeast cells [1].}, language = {en} } @article{KosztolowiczMetzler2020, author = {Kosztolowicz, Tadeusz and Metzler, Ralf}, title = {Diffusion of antibiotics through a biofilm in the presence of diffusion and absorption barriers}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {102}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {3}, publisher = {American Physical Society}, address = {Melville, NY}, issn = {2470-0045}, doi = {10.1103/PhysRevE.102.032408}, pages = {11}, year = {2020}, abstract = {We propose a model of antibiotic diffusion through a bacterial biofilm when diffusion and/or absorption barriers develop in the biofilm. The idea of this model is: We deduce details of the diffusion process in a medium in which direct experimental study is difficult, based on probing diffusion in external regions. Since a biofilm has a gel-like consistency, we suppose that subdiffusion of particles in the biofilm may occur. To describe this process we use a fractional subdiffusion-absorption equation with an adjustable anomalous diffusion exponent. The boundary conditions at the boundaries of the biofilm are derived by means of a particle random walk model on a discrete lattice leading to an expression involving a fractional time derivative. We show that the temporal evolution of the total amount of substance that has diffused through the biofilm explicitly depends on whether there is antibiotic absorption in the biofilm. This fact is used to experimentally check for antibiotic absorption in the biofilm and if subdiffusion and absorption parameters of the biofilm change over time. We propose a four-stage model of antibiotic diffusion in biofilm based on the following physical characteristics: whether there is absorption of the antibiotic in the biofilm and whether all biofilm parameters remain unchanged over time. The biological interpretation of the stages, in particular their relation with the bacterial defense mechanisms, is discussed. Theoretical results are compared with empirical results of ciprofloxacin diffusion through Pseudomonas aeruginosa biofilm, and ciprofloxacin and gentamicin diffusion through Proteus mirabilis biofilm.}, language = {en} } @article{BauerGodecMetzler2014, author = {Bauer, Maximilian and Godec, Aljaz and Metzler, Ralf}, title = {Diffusion of finite-size particles in two-dimensional channels with random wall configurations}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {16}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, number = {13}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c3cp55160a}, pages = {6118 -- 6128}, year = {2014}, abstract = {Diffusion of chemicals or tracer molecules through complex systems containing irregularly shaped channels is important in many applications. Most theoretical studies based on the famed Fick-Jacobs equation focus on the idealised case of infinitely small particles and reflecting boundaries. In this study we use numerical simulations to consider the transport of finite-size particles through asymmetrical two-dimensional channels. Additionally, we examine transient binding of the molecules to the channel walls by applying sticky boundary conditions. We consider an ensemble of particles diffusing in independent channels, which are characterised by common structural parameters. We compare our results for the long-time effective diffusion coefficient with a recent theoretical formula obtained by Dagdug and Pineda}, language = {en} } @article{BauerGodecMetzler2014, author = {Bauer, Maximilian and Godec, Aljaž and Metzler, Ralf}, title = {Diffusion of finite-size particles in two-dimensional channels with random wall configurations}, series = {Physical chemistry, chemical physics : PCCP ; a journal of European chemical societies}, volume = {16}, journal = {Physical chemistry, chemical physics : PCCP ; a journal of European chemical societies}, number = {13}, publisher = {RSC Publications}, address = {Cambridge}, issn = {1463-9084}, doi = {10.1039/C3CP55160A}, pages = {6118 -- 6128}, year = {2014}, abstract = {Diffusion of chemicals or tracer molecules through complex systems containing irregularly shaped channels is important in many applications. Most theoretical studies based on the famed Fick-Jacobs equation focus on the idealised case of infinitely small particles and reflecting boundaries. In this study we use numerical simulations to consider the transport of finite-size particles through asymmetrical two-dimensional channels. Additionally, we examine transient binding of the molecules to the channel walls by applying sticky boundary conditions. We consider an ensemble of particles diffusing in independent channels, which are characterised by common structural parameters. We compare our results for the long-time effective diffusion coefficient with a recent theoretical formula obtained by Dagdug and Pineda [J. Chem. Phys., 2012, 137, 024107].}, language = {en} } @article{JavanainenMartinezSearaMetzleretal.2017, author = {Javanainen, Matti and Martinez-Seara, Hector and Metzler, Ralf and Vattulainen, Ilpo}, title = {Diffusion of Integral Membrane Proteins in Protein-Rich Membranes}, series = {The journal of physical chemistry letters}, volume = {8}, journal = {The journal of physical chemistry letters}, publisher = {American Chemical Society}, address = {Washington}, issn = {1948-7185}, doi = {10.1021/acs.jpclett.7b01758}, pages = {4308 -- 4313}, year = {2017}, abstract = {The lateral diffusion of embedded proteins along lipid membranes in protein-poor conditions has been successfully described in terms of the Saffman-Delbruck (SD) model, which predicts that the protein diffusion coefficient D is weakly dependent on its radius R as D proportional to ln(1/R). However, instead of being protein-poor, native cell membranes are extremely crowded with proteins. On the basis of extensive molecular simulations, we here demonstrate that protein crowding of the membrane at physiological levels leads to deviations from the SD relation and to the emergence of a stronger Stokes-like dependence D proportional to 1/R. We propose that this 1/R law mainly arises due to geometrical factors: smaller proteins are able to avoid confinement effects much better than their larger counterparts. The results highlight that the lateral dynamics in the crowded setting found in native membranes is radically different from protein-poor conditions and plays a significant role in formation of functional multiprotein complexes.}, language = {en} } @article{PulkkinenMetzler2013, author = {Pulkkinen, Otto and Metzler, Ralf}, title = {Distance matters the impact of gene proximity in bacterial gene regulation}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {19}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.198101}, pages = {5}, year = {2013}, abstract = {Following recent discoveries of colocalization of downstream-regulating genes in living cells, the impact of the spatial distance between such genes on the kinetics of gene product formation is increasingly recognized. We here show from analytical and numerical analysis that the distance between a transcription factor (TF) gene and its target gene drastically affects the speed and reliability of transcriptional regulation in bacterial cells. For an explicit model system, we develop a general theory for the interactions between a TF and a transcription unit. The observed variations in regulation efficiency are linked to the magnitude of the variation of the TF concentration peaks as a function of the binding site distance from the signal source. Our results support the role of rapid binding site search for gene colocalization and emphasize the role of local concentration differences.}, language = {en} } @article{SandevChechkinKorabeletal.2015, author = {Sandev, Trifce and Chechkin, Aleksei V. and Korabel, Nickolay and Kantz, Holger and Sokolov, Igor M. and Metzler, Ralf}, title = {Distributed-order diffusion equations and multifractality: Models and solutions}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {92}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.92.042117}, pages = {19}, year = {2015}, abstract = {We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.}, language = {en} } @article{GrebenkovMetzlerOshanin2021, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Distribution of first-reaction times with target regions on boundaries of shell-like domains}, series = {New Journal of Physics (NJP)}, volume = {2021}, journal = {New Journal of Physics (NJP)}, edition = {23}, publisher = {IOP Publishing}, address = {London}, issn = {1367-2630}, doi = {10.1088/1367-2630/ac4282}, pages = {1 -- 23}, year = {2021}, abstract = {We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted 'onion-shell' geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes.}, language = {en} } @article{SarabadaniMetzlerAlaNissila2022, author = {Sarabadani, Jalal and Metzler, Ralf and Ala-Nissila, Tapio}, title = {Driven polymer translocation into a channel: Isoflux tension propagation theory and Langevin dynamics simulations}, series = {Physical Review Research}, volume = {4}, journal = {Physical Review Research}, edition = {3}, publisher = {American Physical Society}, address = {College Park, Maryland, USA}, issn = {2643-1564}, doi = {10.1103/PhysRevResearch.4.033003}, pages = {033003-1 -- 033003-14}, year = {2022}, abstract = {Isoflux tension propagation (IFTP) theory and Langevin dynamics (LD) simulations are employed to study the dynamics of channel-driven polymer translocation in which a polymer translocates into a narrow channel and the monomers in the channel experience a driving force fc. In the high driving force limit, regardless of the channel width, IFTP theory predicts τ ∝ f βc for the translocation time, where β = -1 is the force scaling exponent. Moreover, LD data show that for a very narrow channel fitting only a single file of monomers, the entropic force due to the subchain inside the channel does not play a significant role in the translocation dynamics and the force exponent β = -1 regardless of the force magnitude. As the channel width increases the number of possible spatial configurations of the subchain inside the channel becomes significant and the resulting entropic force causes the force exponent to drop below unity.}, language = {en} } @article{MejiaMonasterioMetzlerVollmer2020, author = {Mejia-Monasterio, Carlos and Metzler, Ralf and Vollmer, J{\"u}rgen}, title = {Editorial: anomalous transport}, series = {Frontiers in Physics}, volume = {8}, journal = {Frontiers in Physics}, publisher = {Frontiers Media}, address = {Lausanne}, issn = {2296-424X}, doi = {10.3389/fphy.2020.622417}, pages = {4}, year = {2020}, language = {en} } @article{GrebenkovMetzlerOshanin2017, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains}, series = {New journal of physics : the open-access journal for physics}, volume = {19}, journal = {New journal of physics : the open-access journal for physics}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/aa8ed9}, pages = {11}, year = {2017}, abstract = {We study the mean first passage time (MFPT) to a reaction event on a specific site in a cylindrical geometry-characteristic, for instance, for bacterial cells, with a concentric inner cylinder representing the nuclear region of the bacterial cell. A similar problem emerges in the description of a diffusive search by a transcription factor protein for a specific binding region on a single strand of DNA. We develop a unified theoretical approach to study the underlying boundary value problem which is based on a self-consistent approximation of the mixed boundary condition. Our approach permits us to derive explicit, novel, closed-form expressions for the MFPT valid for a generic setting with an arbitrary relation between the system parameters. We analyse this general result in the asymptotic limits appropriate for the above-mentioned biophysical problems. Our investigation reveals the crucial role of the target aspect ratio and of the intrinsic reactivity of the binding region, which were disregarded in previous studies. Theoretical predictions are confirmed by numerical simulations.}, language = {en} } @article{GrebenkovMetzlerOshanin2017, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Effects of the target aspect ratio and intrinsic reactivity onto diffusive search in bounded domains}, series = {New journal of physics}, volume = {19}, journal = {New journal of physics}, publisher = {IOP}, address = {London}, issn = {1367-2630}, doi = {10.1088/1367-2630/aa8ed9}, pages = {1 -- 11}, year = {2017}, abstract = {Westudy the mean first passage time (MFPT) to a reaction event on a specific site in a cylindrical geometry—characteristic, for instance, for bacterial cells, with a concentric inner cylinder representing the nuclear region of the bacterial cell. Asimilar problem emerges in the description of a diffusive search by a transcription factor protein for a specific binding region on a single strand of DNA.We develop a unified theoretical approach to study the underlying boundary value problem which is based on a self-consistent approximation of the mixed boundary condition. Our approach permits us to derive explicit, novel, closed-form expressions for the MFPT valid for a generic setting with an arbitrary relation between the system parameters.Weanalyse this general result in the asymptotic limits appropriate for the above-mentioned biophysical problems. Our investigation reveals the crucial role of the target aspect ratio and of the intrinsic reactivity of the binding region, which were disregarded in previous studies. Theoretical predictions are confirmed by numerical simulations.}, language = {en} } @article{CherstvyMetzler2015, author = {Cherstvy, Andrey G. and Metzler, Ralf}, title = {Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes}, series = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, volume = {142}, journal = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, number = {14}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0021-9606}, doi = {10.1063/1.4917077}, pages = {11}, year = {2015}, abstract = {We study noisy heterogeneous diffusion processes with a position dependent diffusivity of the form D(x) similar to D-0 vertical bar x vertical bar (alpha 0) in the presence of annealed and quenched disorder of the environment, corresponding to an effective variation of the exponent a in time and space. In the case of annealed disorder, for which effectively alpha(0) = alpha(0)(t), we show how the long time scaling of the ensemble mean squared displacement (MSD) and the amplitude variation of individual realizations of the time averaged MSD are affected by the disorder strength. For the case of quenched disorder, the long time behavior becomes effectively Brownian after a number of jumps between the domains of a stratified medium. In the latter situation, the averages are taken over both an ensemble of particles and different realizations of the disorder. As physical observables, we analyze in detail the ensemble and time averaged MSDs, the ergodicity breaking parameter, and higher order moments of the time averages. (C) 2015 AIP Publishing LLC.}, language = {en} } @article{CherstvyMetzler2015, author = {Cherstvy, Andrey G. and Metzler, Ralf}, title = {Ergodicity breaking, ageing, and confinement in generalized diffusion processes with position and time dependent diffusivity}, series = {Journal of statistical mechanics: theory and experiment}, journal = {Journal of statistical mechanics: theory and experiment}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1742-5468}, doi = {10.1088/1742-5468/2015/05/P05010}, pages = {20}, year = {2015}, abstract = {We study generalized anomalous diffusion processes whose diffusion coefficient D(x, t) similar to D-0x(alpha)t(beta) depends on both the position x of the test particle and the process time t. This process thus combines the features of scaled Brownian motion and heterogeneous diffusion parent processes. We compute the ensemble and time averaged mean squared displacements of this generalized diffusion process. The scaling exponent of the ensemble averaged mean squared displacement is shown to be the product of the critical exponents of the parent processes, and describes both subdiffusive and superdiffusive systems. We quantify the amplitude fluctuations of the time averaged mean squared displacement as function of the length of the time series and the lag time. In particular, we observe a weak ergodicity breaking of this generalized diffusion process: even in the long time limit the ensemble and time averaged mean squared displacements are strictly disparate. When we start to observe this process some time after its initiation we observe distinct features of ageing. We derive a universal ageing factor for the time averaged mean squared displacement containing all information on the ageing time and the measurement time. External confinement is shown to alter the magnitudes and statistics of the ensemble and time averaged mean squared displacements.}, language = {en} } @article{AkimotoCherstvyMetzler2018, author = {Akimoto, Takuma and Cherstvy, Andrey G. and Metzler, Ralf}, title = {Ergodicity, rejuvenation, enhancement, and slow relaxation of diffusion in biased continuous-time random walks}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {98}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.98.022105}, pages = {6}, year = {2018}, abstract = {Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we report on an interesting phenomenon for the enhancement of diffusion by the start of the measurement in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case, the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. Moreover, the diffusivity can be increased by the aging time when the initial ensemble is not in equilibrium. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance; i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay unexpectedly slowly. Our findings provide a rejuvenation phenomenon in the superdiffusive regime, that is, the diffusivity for a nonequilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed.}, language = {en} } @article{GrebenkovSposiniMetzleretal.2020, author = {Grebenkov, Denis S. and Sposini, Vittoria and Metzler, Ralf and Oshanin, Gleb and Seno, Flavio}, title = {Exact distributions of the maximum and range of random diffusivity processes}, series = {New Journal of Physics}, volume = {23}, journal = {New Journal of Physics}, publisher = {Dt. Physikalische Ges.}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/abd313}, pages = {23}, year = {2020}, abstract = {We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic'diffusivity', defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(-Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.}, language = {en} } @article{ScottWeissSelhuberUnkeletal.2022, author = {Scott, Shane and Weiss, Matthias and Selhuber-Unkel, Christine and Barooji, Younes F. and Sabri, Adal and Erler, Janine T. and Metzler, Ralf and Oddershede, Lene B.}, title = {Extracting, quantifying, and comparing dynamical and biomechanical properties of living matter through single particle tracking}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {25}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, number = {3}, publisher = {RSC Publ.}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/d2cp01384c}, pages = {1513 -- 1537}, year = {2022}, abstract = {A panoply of new tools for tracking single particles and molecules has led to an explosion of experimental data, leading to novel insights into physical properties of living matter governing cellular development and function, health and disease. In this Perspective, we present tools to investigate the dynamics and mechanics of living systems from the molecular to cellular scale via single-particle techniques. In particular, we focus on methods to measure, interpret, and analyse complex data sets that are associated with forces, materials properties, transport, and emergent organisation phenomena within biological and soft-matter systems. Current approaches, challenges, and existing solutions in the associated fields are outlined in order to support the growing community of researchers at the interface of physics and the life sciences. Each section focuses not only on the general physical principles and the potential for understanding living matter, but also on details of practical data extraction and analysis, discussing limitations, interpretation, and comparison across different experimental realisations and theoretical frameworks. Particularly relevant results are introduced as examples. While this Perspective describes living matter from a physical perspective, highlighting experimental and theoretical physics techniques relevant for such systems, it is also meant to serve as a solid starting point for researchers in the life sciences interested in the implementation of biophysical methods.}, language = {en} } @article{LiuCherstvyMetzler2017, author = {Liu, Lin and Cherstvy, Andrey G. and Metzler, Ralf}, title = {Facilitated Diffusion of Transcription Factor Proteins with Anomalous Bulk Diffusion}, series = {The journal of physical chemistry : B, Condensed matter, materials, surfaces, interfaces \& biophysical chemistry}, volume = {121}, journal = {The journal of physical chemistry : B, Condensed matter, materials, surfaces, interfaces \& biophysical chemistry}, publisher = {American Chemical Society}, address = {Washington}, issn = {1520-6106}, doi = {10.1021/acs.jpcb.6b12413}, pages = {1284 -- 1289}, year = {2017}, abstract = {What are the physical laws of the diffusive search of proteins for their specific binding sites on DNA in the presence of the macromolecular crowding in cells? We performed extensive computer simulations to elucidate the protein target search on DNA. The novel feature is the viscoelastic non-Brownian protein bulk diffusion recently observed experimentally. We examine the influence of the protein-DNA binding affinity and the anomalous diffusion exponent on the target search time. In all cases an optimal search time is found. The relative contribution of intermittent three-dimensional bulk diffusion and one-dimensional sliding of proteins along the DNA is quantified. Our results are discussed in the light of recent single molecule tracking experiments, aiming at a better understanding of the influence of anomalous kinetics of proteins on the facilitated diffusion mechanism.}, language = {en} } @article{ShinCherstvyKimetal.2015, author = {Shin, Jaeoh and Cherstvy, Andrey G. and Kim, Won Kyu and Metzler, Ralf}, title = {Facilitation of polymer looping and giant polymer diffusivity in crowded solutions of active particles}, series = {New journal of physics : the open-access journal for physics}, volume = {17}, journal = {New journal of physics : the open-access journal for physics}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/17/11/113008}, pages = {12}, year = {2015}, abstract = {We study the dynamics of polymer chains in a bath of self-propelled particles (SPP) by extensive Langevin dynamics simulations in a two-dimensional model system. Specifically, we analyse the polymer looping properties versus the SPP activity and investigate how the presence of the active particles alters the chain conformational statistics. We find that SPPs tend to extend flexible polymer chains, while they rather compactify stiffer semiflexible polymers, in agreement with previous results. Here we show that higher activities of SPPs yield a higher effective temperature of the bath and thus facilitate the looping kinetics of a passive polymer chain. We explicitly compute the looping probability and looping time in a wide range of the model parameters. We also analyse the motion of a monomeric tracer particle and the polymer's centre of mass in the presence of the active particles in terms of the time averaged mean squared displacement, revealing a giant diffusivity enhancement for the polymer chain via SPP pooling. Our results are applicable to rationalising the dimensions and looping kinetics of biopolymers at constantly fluctuating and often actively driven conditions inside biological cells or in suspensions of active colloidal particles or bacteria cells.}, language = {en} } @article{GodecMetzler2013, author = {Godec, Aljaz and Metzler, Ralf}, title = {Finite-Time effects and ultraweak ergodicity breaking in superdiffusive dynamics}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.020603}, pages = {5}, year = {2013}, abstract = {We study the ergodic properties of superdiffusive, spatiotemporally coupled Levy walk processes. For trajectories of finite duration, we reveal a distinct scatter of the scaling exponents of the time averaged mean squared displacement (delta x(2)) over bar around the ensemble value 3 - alpha (1 < alpha < 2) ranging from ballistic motion to subdiffusion, in strong contrast to the behavior of subdiffusive processes. In addition we find a significant dependence of the average of (delta x(2)) over bar over an ensemble of trajectories as a function of the finite measurement time. This so-called finite-time amplitude depression and the scatter of the scaling exponent is vital in the quantitative evaluation of superdiffusive processes. Comparing the long time average of the second moment with the ensemble mean squared displacement, these only differ by a constant factor, an ultraweak ergodicity breaking.}, language = {en} } @article{PalyulinBlackburnLomholtetal.2019, author = {Palyulin, Vladimir V. and Blackburn, George and Lomholt, Michael A. and Watkins, Nicholas W. and Metzler, Ralf and Klages, Rainer and Chechkin, Aleksei V.}, title = {First passage and first hitting times of Levy flights and Levy walks}, series = {New journal of physics : the open-access journal for physics}, volume = {21}, journal = {New journal of physics : the open-access journal for physics}, number = {10}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab41bb}, pages = {23}, year = {2019}, abstract = {For both L{\´e}vy flight and L{\´e}vy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For L{\´e}vy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the L{\´e}vy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.}, language = {en} } @article{PalyulinBlackburnLomholtetal.2019, author = {Palyulin, Vladimir V and Blackburn, George and Lomholt, Michael A and Watkins, Nicholas W and Metzler, Ralf and Klages, Rainer and Chechkin, Aleksei V.}, title = {First passage and first hitting times of L{\´e}vy flights and L{\´e}vy walks}, series = {New Journal of Physics}, volume = {21}, journal = {New Journal of Physics}, publisher = {Dt. Physikalische Ges.}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab41bb}, pages = {24}, year = {2019}, abstract = {For both L{\´e}vy flight and L{\´e}vy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For L{\´e}vy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ('leapovers'), these two definitions lead to significantly different results. We study the first-passage and first-hitting time distributions as functions of the L{\´e}vy stable index, highlighting the different behaviour for the cases when the first absolute moment of the jump length distribution is finite or infinite. In particular we examine the limits of short and long times. Our results will find their application in the mathematical modelling of random search processes as well as computer algorithms.}, language = {en} } @article{MutothyaXuLietal.2021, author = {Mutothya, Nicholas Mwilu and Xu, Yong and Li, Yongge and Metzler, Ralf and Mutua, Nicholas Muthama}, title = {First passage dynamics of stochastic motion in heterogeneous media driven by correlated white Gaussian and coloured non-Gaussian noises}, series = {Journal of physics. Complexity}, volume = {2}, journal = {Journal of physics. Complexity}, publisher = {IOP Publishing}, address = {Bristol}, issn = {2632-072X}, doi = {10.1088/2632-072X/ac35b5}, pages = {24}, year = {2021}, abstract = {We study the first passage dynamics for a diffusing particle experiencing a spatially varying diffusion coefficient while driven by correlated additive Gaussian white noise and multiplicative coloured non-Gaussian noise. We consider three functional forms for position dependence of the diffusion coefficient: power-law, exponential, and logarithmic. The coloured non-Gaussian noise is distributed according to Tsallis' q-distribution. Tracks of the non-Markovian systems are numerically simulated by using the fourth-order Runge-Kutta algorithm and the first passage times (FPTs) are recorded. The FPT density is determined along with the mean FPT (MFPT). Effects of the noise intensity and self-correlation of the multiplicative noise, the intensity of the additive noise, the cross-correlation strength, and the non-extensivity parameter on the MFPT are discussed.}, language = {en} } @article{SposiniChechkinMetzler2018, author = {Sposini, Vittoria and Chechkin, Aleksei V. and Metzler, Ralf}, title = {First passage statistics for diffusing diffusivity}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {4}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aaf6ff}, pages = {11}, year = {2018}, abstract = {A rapidly increasing number of systems is identified in which the stochastic motion of tracer particles follows the Brownian law < r(2)(t)> similar or equal to Dt yet the distribution of particle displacements is strongly non-Gaussian. A central approach to describe this effect is the diffusing diffusivity (DD) model in which the diffusion coefficient itself is a stochastic quantity, mimicking heterogeneities of the environment encountered by the tracer particle on its path. We here quantify in terms of analytical and numerical approaches the first passage behaviour of the DD model. We observe significant modifications compared to Brownian-Gaussian diffusion, in particular that the DD model may have a faster first passage dynamics. Moreover we find a universal crossover point of the survival probability independent of the initial condition.}, language = {en} } @article{GodecMetzler2016, author = {Godec, Aljaz and Metzler, Ralf}, title = {First passage time distribution in heterogeneity controlled kinetics: going beyond the mean first passage time}, series = {Scientific reports}, volume = {6}, journal = {Scientific reports}, publisher = {Nature Publ. Group}, address = {London}, issn = {2045-2322}, doi = {10.1038/srep20349}, pages = {11}, year = {2016}, abstract = {The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold (target) for the first time. The last decade saw an enlightening series of new results focusing mostly on the so-called mean and global first passage time (MFPT and GFPT, respectively) of such processes. Here we push the understanding of first passage processes one step further. For a simple heterogeneous system we derive rigorously the complete distribution of first passage times (FPTs). Our results demonstrate that the typical FPT significantly differs from the MFPT, which corresponds to the long time behaviour of the FPT distribution. Conversely, the short time behaviour is shown to correspond to trajectories connecting directly from the initial value to the target. Remarkably, we reveal a previously overlooked third characteristic time scale of the first passage dynamics mirroring brief excursion away from the target.}, language = {en} } @article{AbdolvahabMetzlerEjtehadi2011, author = {Abdolvahab, Rouhollah Haji and Metzler, Ralf and Ejtehadi, Mohammad Reza}, title = {First passage time distribution of chaperone driven polymer translocation through a nanopore homopolymer and heteropolymer cases}, series = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, volume = {135}, journal = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr}, number = {24}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0021-9606}, doi = {10.1063/1.3669427}, pages = {8}, year = {2011}, abstract = {Combining the advection-diffusion equation approach with Monte Carlo simulations we study chaperone driven polymer translocation of a stiff polymer through a nanopore. We demonstrate that the probability density function of first passage times across the pore depends solely on the Peclet number, a dimensionless parameter comparing drift strength and diffusivity. Moreover it is shown that the characteristic exponent in the power-law dependence of the translocation time on the chain length, a function of the chaperone-polymer binding energy, the chaperone concentration, and the chain length, is also effectively determined by the Peclet number. We investigate the effect of the chaperone size on the translocation process. In particular, for large chaperone size, the translocation progress and the mean waiting time as function of the reaction coordinate exhibit pronounced sawtooth-shapes. The effects of a heterogeneous polymer sequence on the translocation dynamics is studied in terms of the translocation velocity, the probability distribution for the translocation progress, and the monomer waiting times. (C) 2011 American Institute of Physics.}, language = {en} } @article{GodecMetzler2017, author = {Godec, Aljaž and Metzler, Ralf}, title = {First passage time statistics for two-channel diffusion}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {50}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {8}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aa5204}, pages = {17}, year = {2017}, abstract = {We present rigorous results for the mean first passage time and first passage time statistics for two-channel Markov additive diffusion in a 3-dimensional spherical domain. Inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes, which is shown to effect a non-trivial first passage behaviour. We also address the scenario of intermittent immobilisation. In both cases we prove that despite the perfectly non-recurrent motion of two-channel Markov additive diffusion in 3 dimensions the first passage statistics at long times do not display Poisson-like behaviour if none of the phases has a vanishing diffusion coefficient. This stands in stark contrast to the standard (one-channel) Markov diffusion counterpart. We also discuss the relevance of our results in the context of cellular signalling.}, language = {en} } @article{MattosMejiaMonasterioMetzleretal.2012, author = {Mattos, Thiago G. and Mejia-Monasterio, Carlos and Metzler, Ralf and Oshanin, Gleb}, title = {First passages in bounded domains When is the mean first passage time meaningful?}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {86}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {3}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.86.031143}, pages = {8}, year = {2012}, abstract = {We study the first passage statistics to adsorbing boundaries of a Brownian motion in bounded two-dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries. From extensive numerical analysis we obtain the probability P(omega) distribution of the random variable omega = tau(1)/(tau(1) + tau(2)), which is a measure for how similar the first passage times tau(1) and tau(2) are of two independent realizations of a Brownian walk starting at the same location. We construct a chart for each domain, determining whether P(omega) represents a unimodal, bell-shaped form, or a bimodal, M-shaped behavior. While in the former case the mean first passage time (MFPT) is a valid characteristic of the first passage behavior, in the latter case it is an insufficient measure for the process. Strikingly we find a distinct turnover between the two modes of P(omega), characteristic for the domain shape and the respective location of absorbing and reflective boundaries. Our results demonstrate that large fluctuations of the first passage times may occur frequently in two-dimensional domains, rendering quite vague the general use of the MFPT as a robust measure of the actual behavior even in bounded domains, in which all moments of the first passage distribution exist.}, language = {en} } @article{PadashChechkinDybiecetal.2019, author = {Padash, Amin and Chechkin, Aleksei V. and Dybiec, Bartlomiej and Pavlyukevich, Ilya and Shokri, Babak and Metzler, Ralf}, title = {First-passage properties of asymmetric Levy flights}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {45}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ab493e}, pages = {48}, year = {2019}, abstract = {L{\´e}vy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the 'jump lengths'—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of L{\´e}vy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering L{\´e}vy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of L{\´e}vy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.}, language = {en} } @article{KruesemannGodecMetzler2014, author = {Kr{\"u}semann, Henning and Godec, Aljaz and Metzler, Ralf}, title = {First-passage statistics for aging diffusion in systems with annealed and quenched disorder}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {89}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.89.040101}, pages = {5}, year = {2014}, abstract = {Aging, the dependence of the dynamics of a physical process on the time t(a) since its original preparation, is observed in systems ranging from the motion of charge carriers in amorphous semiconductors over the blinking dynamics of quantum dots to the tracer dispersion in living biological cells. Here we study the effects of aging on one of the most fundamental properties of a stochastic process, the first-passage dynamics. We find that for an aging continuous time random walk process, the scaling exponent of the density of first-passage times changes twice as the aging progresses and reveals an intermediate scaling regime. The first-passage dynamics depends on t(a) differently for intermediate and strong aging. Similar crossovers are obtained for the first-passage dynamics for a confined and driven particle. Comparison to the motion of an aged particle in the quenched trap model with a bias shows excellent agreement with our analytical findings. Our results demonstrate how first-passage measurements can be used to unravel the age t(a) of a physical system.}, language = {en} } @article{MardoukhiJeonChechkinetal.2018, author = {Mardoukhi, Yousof and Jeon, Jae-Hyung and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Fluctuations of random walks in critical random environments}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {20}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, number = {31}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c8cp03212b}, pages = {20427 -- 20438}, year = {2018}, abstract = {Percolation networks have been widely used in the description of porous media but are now found to be relevant to understand the motion of particles in cellular membranes or the nucleus of biological cells. Random walks on the infinite cluster at criticality of a percolation network are asymptotically ergodic. On any finite size cluster of the network stationarity is reached at finite times, depending on the cluster's size. Despite of this we here demonstrate by combination of analytical calculations and simulations that at criticality the disorder and cluster size average of the ensemble of clusters leads to a non-vanishing variance of the time averaged mean squared displacement, regardless of the measurement time. Fluctuations of this relevant experimental quantity due to the disorder average of such ensembles are thus persistent and non-negligible. The relevance of our results for single particle tracking analysis in complex and biological systems is discussed.}, language = {en} } @article{GuggenbergerPagniniVojtaetal.2019, author = {Guggenberger, Tobias and Pagnini, Gianni and Vojta, Thomas and Metzler, Ralf}, title = {Fractional Brownian motion in a finite interval}, series = {New Journal of Physics}, volume = {21}, journal = {New Journal of Physics}, publisher = {Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics}, address = {Bad Honnef und London}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab075f}, pages = {13}, year = {2019}, abstract = {Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) 〈X² (t)〉 ⋍ tᵅ with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.}, language = {en} } @article{GuggenbergerChechkinMetzler2021, author = {Guggenberger, Tobias and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Fractional Brownian motion in superharmonic potentials and non-Boltzmann stationary distributions}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {54}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {29}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac019b}, pages = {17}, year = {2021}, abstract = {We study the stochastic motion of particles driven by long-range correlated fractional Gaussian noise (FGN) in a superharmonic external potential of the form U(x) proportional to x(2n) (n is an element of N). When the noise is considered to be external, the resulting overdamped motion is described by the non-Markovian Langevin equation for fractional Brownian motion. For this case we show the existence of long time, stationary probability density functions (PDFs) the shape of which strongly deviates from the naively expected Boltzmann PDF in the confining potential U(x). We analyse in detail the temporal approach to stationarity as well as the shape of the non-Boltzmann stationary PDF. A typical characteristic is that subdiffusive, antipersistent (with negative autocorrelation) motion tends to effect an accumulation of probability close to the origin as compared to the corresponding Boltzmann distribution while the opposite trend occurs for superdiffusive (persistent) motion. For this latter case this leads to distinct bimodal shapes of the PDF. This property is compared to a similar phenomenon observed for Markovian Levy flights in superharmonic potentials. We also demonstrate that the motion encoded in the fractional Langevin equation driven by FGN always relaxes to the Boltzmann distribution, as in this case the fluctuation-dissipation theorem is fulfilled.}, language = {en} } @article{WangCherstvyChechkinetal.2020, author = {Wang, Wei and Cherstvy, Andrey G. and Chechkin, Aleksei V. and Thapa, Samudrajit and Seno, Flavio and Liu, Xianbin and Metzler, Ralf}, title = {Fractional Brownian motion with random diffusivity}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {53}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {47}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aba467}, pages = {34}, year = {2020}, abstract = {Numerous examples for a priori unexpected non-Gaussian behaviour for normal and anomalous diffusion have recently been reported in single-particle tracking experiments. Here, we address the case of non-Gaussian anomalous diffusion in terms of a random-diffusivity mechanism in the presence of power-law correlated fractional Gaussian noise. We study the ergodic properties of this model via examining the ensemble- and time-averaged mean-squared displacements as well as the ergodicity breaking parameter EB quantifying the trajectory-to-trajectory fluctuations of the latter. For long measurement times, interesting crossover behaviour is found as function of the correlation time tau characterising the diffusivity dynamics. We unveil that at short lag times the EB parameter reaches a universal plateau. The corresponding residual value of EB is shown to depend only on tau and the trajectory length. The EB parameter at long lag times, however, follows the same power-law scaling as for fractional Brownian motion. We also determine a corresponding plateau at short lag times for the discrete representation of fractional Brownian motion, absent in the continuous-time formulation. These analytical predictions are in excellent agreement with results of computer simulations of the underlying stochastic processes. Our findings can help distinguishing and categorising certain nonergodic and non-Gaussian features of particle displacements, as observed in recent single-particle tracking experiments.}, language = {en} } @article{TomovskiMetzlerGerhold2022, author = {Tomovski, Živorad and Metzler, Ralf and Gerhold, Stefan}, title = {Fractional characteristic functions, and a fractional calculus approach for moments of random variables}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {25}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {4}, publisher = {De Gruyter}, address = {Berlin ; Boston}, issn = {1314-2224}, doi = {10.1007/s13540-022-00047-x}, pages = {1307 -- 1323}, year = {2022}, abstract = {In this paper we introduce a fractional variant of the characteristic function of a random variable. It exists on the whole real line, and is uniformly continuous. We show that fractional moments can be expressed in terms of Riemann-Liouville integrals and derivatives of the fractional characteristic function. The fractional moments are of interest in particular for distributions whose integer moments do not exist. Some illustrative examples for particular distributions are also presented.}, language = {en} } @article{SandevMetzlerChechkin2018, author = {Sandev, Trifce and Metzler, Ralf and Chechkin, Aleksei V.}, title = {From continuous time random walks to the generalized diffusion equation}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {21}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {1}, publisher = {De Gruyter}, address = {Berlin}, issn = {1311-0454}, doi = {10.1515/fca-2018-0002}, pages = {10 -- 28}, year = {2018}, abstract = {We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.}, language = {en} } @article{GrebenkovMetzlerOshanin2020, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {From single-particle stochastic kinetics to macroscopic reaction rates}, series = {New Journal of Physics}, volume = {22}, journal = {New Journal of Physics}, publisher = {Dt. Physikalische Ges.}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/abb1de}, pages = {28}, year = {2020}, abstract = {We consider the first-passage problem for N identical independent particles that are initially released uniformly in a finite domain Ω and then diffuse toward a reactive area Γ, which can be part of the outer boundary of Ω or a reaction centre in the interior of Ω. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the N particles reacts with Γ. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fFPT with the particle number N, namely, a much stronger dependence (1/N and 1/N² for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.}, language = {en} } @article{GrebenkovMetzlerOshanin2019, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Full distribution of first exit times in the narrow escape problem}, series = {New Journal of Physics}, volume = {21}, journal = {New Journal of Physics}, publisher = {Dt. Physikalische Ges.}, address = {Bad Honnef}, issn = {1367-2630}, doi = {10.1088/1367-2630/ab5de4}, pages = {23}, year = {2019}, abstract = {In the scenario of the narrow escape problem (NEP) a particle diffuses in a finite container and eventually leaves it through a small 'escape window' in the otherwise impermeable boundary, once it arrives to this window and crosses an entropic barrier at the entrance to it. This generic problem is mathematically identical to that of a diffusion-mediated reaction with a partially-reactive site on the container's boundary. Considerable knowledge is available on the dependence of the mean first-reaction time (FRT) on the pertinent parameters. We here go a distinct step further and derive the full FRT distribution for the NEP. We demonstrate that typical FRTs may be orders of magnitude shorter than the mean one, thus resulting in a strong defocusing of characteristic temporal scales. We unveil the geometry-control of the typical times, emphasising the role of the initial distance to the target as a decisive parameter. A crucial finding is the further FRT defocusing due to the barrier, necessitating repeated escape or reaction attempts interspersed with bulk excursions. These results add new perspectives and offer a broad comprehension of various features of the by-now classical NEP that are relevant for numerous biological and technological systems.}, language = {en} } @article{StojkoskiSandevBasnarkovetal.2020, author = {Stojkoski, Viktor and Sandev, Trifce and Basnarkov, Lasko and Kocarev, Ljupco and Metzler, Ralf}, title = {Generalised geometric Brownian motion}, series = {Entropy}, volume = {22}, journal = {Entropy}, number = {12}, publisher = {MDPI}, address = {Basel}, issn = {1099-4300}, doi = {10.3390/e22121432}, pages = {34}, year = {2020}, abstract = {Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.}, language = {en} }