@article{CherstvyNagelBetaetal.2018, author = {Cherstvy, Andrey G. and Nagel, Oliver and Beta, Carsten and Metzler, Ralf}, title = {Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells}, 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 = {35}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c8cp04254c}, pages = {23034 -- 23054}, year = {2018}, abstract = {What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells? Based on the statistical analysis of experimental single-cell tracking data of the two-dimensional motion of the Dictyostelium discoideum amoeboid cells, we quantify their diffusive behaviour based on a number of standard and complementary statistical indicators. We compute the ensemble- and time-averaged mean-squared displacements (MSDs) of the diffusing amoebae cells and observe a pronounced spread of short-time diffusion coefficients and anomalous MSD-scaling exponents for individual cells. The distribution functions of the cell displacements, the long-tailed distribution of instantaneous speeds, and the velocity autocorrelations are also computed. In particular, we observe a systematic superdiffusive short-time behaviour for the ensemble- and time-averaged MSDs of the amoeboid cells. Also, a clear anti-correlation of scaling exponents and generalised diffusivity values for different cells is detected. Most significantly, we demonstrate that the distribution function of the cell displacements has a strongly non-Gaussian shape andusing a rescaled spatio-temporal variablethe cell-displacement data collapse onto a universal master curve. The current analysis of single-cell motions can be implemented for quantifying diffusive behaviours in other living-matter systems, in particular, when effects of active transport, non-Gaussian displacements, and heterogeneity of the population are involved in the dynamics.}, 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{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{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{CherstvyThapaMardoukhietal.2018, author = {Cherstvy, Andrey G. and Thapa, Samudrajit and Mardoukhi, Yousof and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Time averages and their statistical variation for the Ornstein-Uhlenbeck process}, 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.98.022134}, pages = {15}, year = {2018}, abstract = {How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and outof-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.}, language = {en} } @article{AydinerCherstvyMetzler2018, author = {Aydiner, Ekrem and Cherstvy, Andrey G. and Metzler, Ralf}, title = {Wealth distribution, Pareto law, and stretched exponential decay of money}, series = {Physica : europhysics journal ; A, Statistical mechanics and its applications}, volume = {490}, journal = {Physica : europhysics journal ; A, Statistical mechanics and its applications}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0378-4371}, doi = {10.1016/j.physa.2017.08.017}, pages = {278 -- 288}, year = {2018}, abstract = {We study by Monte Carlo simulations a kinetic exchange trading model for both fixed and distributed saving propensities of the agents and rationalize the person and wealth distributions. We show that the newly introduced wealth distribution - that may be more amenable in certain situations - features a different power-law exponent, particularly for distributed saving propensities of the agents. For open agent-based systems, we analyze the person and wealth distributions and find that the presence of trap agents alters their amplitude, leaving however the scaling exponents nearly unaffected. For an open system, we show that the total wealth - for different trap agent densities and saving propensities of the agents - decreases in time according to the classical Kohlrausch-Williams-Watts stretched exponential law. Interestingly, this decay does not depend on the trap agent density, but rather on saving propensities. The system relaxation for fixed and distributed saving schemes are found to be different.}, language = {en} } @article{KrapfMarinariMetzleretal.2018, author = {Krapf, Diego and Marinari, Enzo and Metzler, Ralf and Oshanin, Gleb and Xu, Xinran and Squarcini, Alessio}, title = {Power spectral density of a single Brownian trajectory}, series = {New journal of physics : the open-access journal for physics}, volume = {20}, 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/aaa67c}, pages = {30}, year = {2018}, abstract = {The power spectral density (PSD) of any time-dependent stochastic processX (t) is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X-t over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT -> infinity. Alegitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.}, language = {en} } @article{EstradaDelvenneHatanoetal.2018, author = {Estrada, Ernesto and Delvenne, Jean-Charles and Hatano, Naomichi and Mateos, Jose L. and Metzler, Ralf and Riascos, Alejandro P. and Schaub, Michael T.}, title = {Random multi-hopper model}, series = {Journal of Complex Networks}, volume = {6}, journal = {Journal of Complex Networks}, number = {3}, publisher = {Oxford Univ. Press}, address = {Oxford}, issn = {2051-1310}, doi = {10.1093/comnet/cnx043}, pages = {382 -- 403}, year = {2018}, abstract = {We develop a mathematical model considering a random walker with long-range hops on arbitrary graphs. The random multi-hopper can jump to any node of the graph from an initial position, with a probability that decays as a function of the shortest-path distance between the two nodes in the graph. We consider here two decaying functions in the form of Laplace and Mellin transforms of the shortest-path distances. We prove that when the parameters of these transforms approach zero asymptotically, the hitting time in the multi-hopper approaches the minimum possible value for a normal random walker. We show by computational experiments that the multi-hopper explores a graph with clusters or skewed degree distributions more efficiently than a normal random walker. We provide computational evidences of the advantages of the random multi-hopper model with respect to the normal random walk by studying deterministic, random and real-world networks.}, 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{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{SposiniChechkinSenoetal.2018, author = {Sposini, Vittoria and Chechkin, Aleksei V. and Seno, Flavio and Pagnini, Gianni and Metzler, Ralf}, title = {Random diffusivity from stochastic equations}, series = {New Journal of Physics}, journal = {New Journal of Physics}, publisher = {Deutsche Physikalische Gesellschaft / Institute of Physics}, address = {Bad Honnef und London}, issn = {1367-2630}, doi = {10.1088/1367-2630/aab696}, pages = {1 -- 33}, year = {2018}, abstract = {A considerable number of systems have recently been reported in which Brownian yet non-Gaussian dynamics was observed. These are processes characterised by a linear growth in time of the mean squared displacement, yet the probability density function of the particle displacement is distinctly non-Gaussian, and often of exponential(Laplace) shape. This apparently ubiquitous behaviour observed in very different physical systems has been interpreted as resulting from diffusion in inhomogeneous environments and mathematically represented through a variable, stochastic diffusion coefficient. Indeed different models describing a fluctuating diffusivity have been studied. Here we present a new view of the stochastic basis describing time dependent random diffusivities within a broad spectrum of distributions. Concretely, our study is based on the very generic class of the generalised Gamma distribution. Two models for the particle spreading in such random diffusivity settings are studied. The first belongs to the class of generalised grey Brownian motion while the second follows from the idea of diffusing diffusivities. The two processes exhibit significant characteristics which reproduce experimental results from different biological and physical systems. We promote these two physical models for the description of stochastic particle motion in complex environments.}, language = {en} } @article{ŚlęzakMetzlerMagdziarz2018, author = {Ślęzak, Jakub and Metzler, Ralf and Magdziarz, Marcin}, title = {Superstatistical generalised Langevin equation}, series = {New Journal of Physics}, volume = {20}, journal = {New Journal of Physics}, number = {023026}, publisher = {Deutsche Physikalische Gesellschaft / Institute of Physics}, address = {Bad Honnef und London}, issn = {1367-2630}, doi = {10.1088/1367-2630/aaa3d4}, pages = {1 -- 25}, year = {2018}, abstract = {Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.}, 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{GrebenkovMetzlerOshanin2018, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control}, series = {Communications Chemistry}, volume = {1}, journal = {Communications Chemistry}, publisher = {Macmillan Publishers Limited}, address = {London}, issn = {2399-3669}, doi = {10.1038/s42004-018-0096-x}, pages = {12}, year = {2018}, abstract = {Textbook concepts of diffusion-versus kinetic-control are well-defined for reaction-kinetics involving macroscopic concentrations of diffusive reactants that are adequately described by rate-constants—the inverse of the mean-first-passage-time to the reaction-event. In contradiction, an open important question is whether the mean-first-passage-time alone is a sufficient measure for biochemical reactions that involve nanomolar reactant concentrations. Here, using a simple yet generic, exactly solvable model we study the effect of diffusion and chemical reaction-limitations on the full reaction-time distribution. We show that it has a complex structure with four distinct regimes delineated by three characteristic time scales spanning a window of several decades. Consequently, the reaction-times are defocused: no unique time-scale characterises the reaction-process, diffusion- and kinetic-control can no longer be disentangled, and it is imperative to know the full reaction-time distribution. We introduce the concepts of geometry- and reaction-control, and also quantify each regime by calculating the corresponding reaction depth.}, language = {en} } @article{GrebenkovMetzlerOshanin2018, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {Towards a full quantitative description of single-molecule reaction kinetics in biological cells}, 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 = {24}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c8cp02043d}, pages = {16393 -- 16401}, year = {2018}, abstract = {The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of instances when biomolecules in a biological cell reach their specific binding sites and trigger cellular regulation. Typically, the first-passage properties are given in terms of mean first-passage times. However, modern experiments now monitor single-molecular binding-processes in living cells and thus provide access to the full statistics of the underlying first-passage events, in particular, inherent cell-to-cell fluctuations. We here present a robust explicit approach for obtaining the distribution of FPTs to a small partially reactive target in cylindrical-annulus domains, which represent typical bacterial and neuronal cell shapes. We investigate various asymptotic behaviours of this FPT distribution and show that it is typically very broad in many biological situations, thus, the mean FPT can differ from the most probable FPT by orders of magnitude. The most probable FPT is shown to strongly depend only on the starting position within the geometry and to be almost independent of the target size and reactivity. These findings demonstrate the dramatic relevance of knowing the full distribution of FPTs and thus open new perspectives for a more reliable description of many intracellular processes initiated by the arrival of one or few biomolecules to a small, spatially localised region inside the cell.}, 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{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} }