@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{CherstvyVinodAghionetal.2017, author = {Cherstvy, Andrey G. and Vinod, Deepak and Aghion, Erez and Chechkin, Aleksei and Metzler, Ralf}, title = {Time averaging, ageing and delay analysis of financial time series}, series = {New journal of physics}, volume = {19}, journal = {New journal of physics}, publisher = {IOP}, address = {London}, issn = {1367-2630}, doi = {10.1088/1367-2630/aa7199}, pages = {1 -- 11}, year = {2017}, abstract = {We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black-Scholes-Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.}, language = {en} } @article{CherstvyVinodAghionetal.2017, author = {Cherstvy, Andrey G. and Vinod, Deepak and Aghion, Erez and Chechkin, Aleksei and Metzler, Ralf}, title = {Time averaging, ageing and delay analysis of financial time series}, 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/aa7199}, pages = {135 -- 147}, year = {2017}, abstract = {We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black-Scholes-Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.}, language = {en} } @article{DoerriesChechkinMetzler2022, author = {Doerries, Timo J. and Chechkin, Aleksei 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{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{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{GrebenkovMetzlerOshanin2021, author = {Grebenkov, Denis S. and Metzler, Ralf and Oshanin, Gleb}, title = {A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes}, series = {New Journal of Physics (NJP)}, volume = {23}, journal = {New Journal of Physics (NJP)}, publisher = {IOP - Institute of Physics Publishing}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/ac1e42}, pages = {18}, year = {2021}, abstract = {We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g. a signal transduction proceeds via a series of consecutive 'messengers': the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.}, 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{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{GuggenbergerChechkinMetzler2022, author = {Guggenberger, Tobias and Chechkin, Aleksei and Metzler, Ralf}, title = {Absence of stationary states and non-Boltzmann distributions of fractional Brownian motion in shallow external potentials}, series = {New journal of physics : the open-access journal for physics}, volume = {24}, journal = {New journal of physics : the open-access journal for physics}, number = {7}, publisher = {Dt. Physikalische Ges.}, address = {[Bad Honnef]}, issn = {1367-2630}, doi = {10.1088/1367-2630/ac7b3c}, pages = {18}, year = {2022}, abstract = {We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x|( c ) (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise xi ( alpha )(t) with 0 < alpha <= 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent alpha. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 - 1/alpha) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Levy flights in shallow external potentials.}, language = {en} }