@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{CherstvyVinodAghionetal.2021,
  author    = {Cherstvy, Andrey G. and Vinod, Deepak and Aghion, Erez and Sokolov, Igor M. and Metzler, Ralf},
  title     = {Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {103},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {6},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.103.062127},
  pages     = {11},
  year      = {2021},
  abstract  = {Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble- and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, sigma(t) similar to t(alpha), named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times Delta. The proportionality factor between these the two averages of the time series is Delta/T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at Delta/T << 1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with s (t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.},
  language  = {en}
}
@article{CherstvyWangMetzleretal.2021,
  author    = {Cherstvy, Andrey G. and Wang, Wei and Metzler, Ralf and Sokolov, Igor M.},
  title     = {Inertia triggers nonergodicity of fractional Brownian motion},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {104},
  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.104.024115},
  pages     = {12},
  year      = {2021},
  abstract  = {How related are the ergodic properties of the over- and underdamped Langevin equations driven by fractional Gaussian noise? We here find that for massive particles performing fractional Brownian motion (FBM) inertial effects not only destroy the stylized fact of the equivalence of the ensemble-averaged mean-squared displacement (MSD) to the time-averaged MSD (TAMSD) of overdamped or massless FBM, but also dramatically alter the values of the ergodicity-breaking parameter (EB). Our theoretical results for the behavior of EB for underdamped or massive FBM for varying particle mass m, Hurst exponent H, and trace length T are in excellent agreement with the findings of stochastic computer simulations. The current results can be of interest for the experimental community employing various single-particle-tracking techniques and aiming at assessing the degree of nonergodicity for the recorded time series (studying, e.g., the behavior of EB versus lag time). To infer FBM as a realizable model of anomalous diffusion for a set single-particle-tracking data when massive particles are being tracked, the EBs from the data should be compared to EBs of massive (rather than massless) FBM.},
  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{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{KlettCherstvyShinetal.2021,
  author    = {Klett, Kolja and Cherstvy, Andrey G. and Shin, Jaeoh and Sokolov, Igor M. and Metzler, Ralf},
  title     = {Non-Gaussian, transiently anomalous, and ergodic self-diffusion of flexible dumbbells in crowded two-dimensional environments},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {104},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {6},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.104.064603},
  pages     = {18},
  year      = {2021},
  abstract  = {We employ Langevin-dynamics simulations to unveil non-Brownian and non-Gaussian center-of-mass self-diffusion of massive flexible dumbbell-shaped particles in crowded two-dimensional solutions. We study the intradumbbell dynamics of the relative motion of the two constituent elastically coupled disks. Our main focus is on effects of the crowding fraction phi and of the particle structure on the diffusion characteristics. We evaluate the time-averaged mean-squared displacement (TAMSD), the displacement probability-density function (PDF), and the displacement autocorrelation function (ACF) of the dimers. For the TAMSD at highly crowded conditions of dumbbells, e.g., we observe a transition from the short-time ballistic behavior, via an intermediate subdiffusive regime, to long-time Brownian-like spreading dynamics. The crowded system of dimers exhibits two distinct diffusion regimes distinguished by the scaling exponent of the TAMSD, the dependence of the diffusivity on phi, and the features of the displacement-ACF. We attribute these regimes to a crowding-induced transition from viscous to viscoelastic diffusion upon growing phi. We also analyze the relative motion in the dimers, finding that larger phi suppress their vibrations and yield strongly non-Gaussian PDFs of rotational displacements. For the diffusion coefficients D(phi) of translational and rotational motion of the dumbbells an exponential decay with phi for weak and a power-law variation D(phi) proportional to (phi - phi(star))(2.4) for strong crowding is found. A comparison of simulation results with theoretical predictions for D(phi) is discussed and some relevant experimental systems are overviewed.},
  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{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{RitschelCherstvyMetzler2021,
  author    = {Ritschel, Stefan and Cherstvy, Andrey G. and Metzler, Ralf},
  title     = {Universality of delay-time averages for financial time series},
  series = {Journal of physics. Complexity},
  volume    = {2},
  journal   = {Journal of physics. Complexity},
  number    = {4},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {2632-072X},
  doi       = {10.1088/2632-072X/ac2220},
  pages     = {30},
  year      = {2021},
  abstract  = {We analyze historical data of stock-market prices for multiple financial indices using the concept of delay-time averaging for the financial time series (FTS). The region of validity of our recent theoretical predictions [Cherstvy A G et al 2017 New J. Phys. 19 063045] for the standard and delayed time-averaged mean-squared 'displacements' (TAMSDs) of the historical FTS is extended to all lag times. As the first novel element, we perform extensive computer simulations of the stochastic differential equation describing geometric Brownian motion (GBM) which demonstrate a quantitative agreement with the analytical long-term price-evolution predictions in terms of the delayed TAMSD (for all stock-market indices in crisis-free times). Secondly, we present a robust procedure of determination of the model parameters of GBM via fitting the features of the price-evolution dynamics in the FTS for stocks and cryptocurrencies. The employed concept of single-trajectory-based time averaging can serve as a predictive tool (proxy) for a mathematically based assessment and rationalization of probabilistic trends in the evolution of stock-market prices.},
  language  = {en}
}