@article{EliazarMetzler2012,
  author    = {Eliazar, Iddo and Metzler, Ralf},
  title     = {The RARE model a generalized approach to random relaxation processes in disordered systems},
  series = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr},
  volume    = {137},
  journal   = {The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr},
  number    = {23},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {0021-9606},
  doi       = {10.1063/1.4770266},
  pages     = {9},
  year      = {2012},
  abstract  = {This paper introduces and analyses a general statistical model, termed the RAndom RElaxations (RARE) model, of random relaxation processes in disordered systems. The model considers excitations that are randomly scattered around a reaction center in a general embedding space. The model's input quantities are the spatial scattering statistics of the excitations around the reaction center, and the chemical reaction rates between the excitations and the reaction center as a function of their mutual distance. The framework of the RARE model is versatile and a detailed stochastic analysis of the random relaxation processes is established. Analytic results regarding the duration and the range of the random relaxation processes, as well as the model's thermodynamic limit, are obtained in closed form. In particular, the case of power-law inputs, which turn out to yield stretched exponential relaxation patterns and asymptotically Paretian relaxation ranges, is addressed in detail.},
  language  = {en}
}
@article{MetzlerJeon2012,
  author    = {Metzler, Ralf and Jeon, Jae-Hyung},
  title     = {The role of ergodicity in anomalous stochastic processes - analysis of single-particle trajectories},
  series = {Physica scripta : an international journal for experimental and theoretical physics},
  volume    = {86},
  journal   = {Physica scripta : an international journal for experimental and theoretical physics},
  number    = {5},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0031-8949},
  doi       = {10.1088/0031-8949/86/05/058510},
  pages     = {5},
  year      = {2012},
  abstract  = {Single-particle experiments produce time series x(t) of individual particle trajectories, frequently revealing anomalous diffusion behaviour. Typically, individual x(t) are evaluated in terms of time-averaged quantities instead of ensemble averages. Here we discuss the behaviour of the time-averaged mean squared displacement of different stochastic processes giving rise to anomalous diffusion. In particular, we pay attention to the ergodic properties of these processes, i.e. the (non)equivalence of time and ensemble averages.},
  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{WangCherstvyKantzetal.2021,
  author    = {Wang, Wei and Cherstvy, Andrey G. and Kantz, Holger and Metzler, Ralf and Sokolov, Igor M.},
  title     = {Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes},
  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 Institute of Physics},
  address   = {Woodbury, NY},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.104.024105},
  pages     = {27},
  year      = {2021},
  abstract  = {How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x) = D0|x|gamma and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicitybreaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB similar to(1/r )-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.},
  language  = {en}
}
@article{CherstvyVinodAghionetal.2017,
  author    = {Cherstvy, Andrey G. and Vinod, Deepak and Aghion, Erez and Chechkin, Aleksei V. 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 V. 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{VinodCherstvyMetzleretal.2022,
  author    = {Vinod, Deepak and Cherstvy, Andrey G. and Metzler, Ralf and Sokolov, Igor M.},
  title     = {Time-averaging and nonergodicity of reset geometric Brownian motion with drift},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {106},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {3},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.106.034137},
  pages     = {36},
  year      = {2022},
  abstract  = {How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Delta) not equal TAMSD(Delta) and Variance(Delta) not equal TAMSD(Delta) at short lag times Delta and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Delta/T << 1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.},
  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{SposiniKrapfMarinarietal.2022,
  author    = {Sposini, Vittoria and Krapf, Diego and Marinari, Enzo and Sunyer, Raimon and Ritort, Felix and Taheri, Fereydoon and Selhuber-Unkel, Christine and Benelli, Rebecca and Weiss, Matthias and Metzler, Ralf and Oshanin, Gleb},
  title     = {Towards a robust criterion of anomalous diffusion},
  series = {Communications Physics},
  volume    = {5},
  journal   = {Communications Physics},
  publisher = {Springer Nature},
  address   = {London},
  issn      = {2399-3650},
  doi       = {10.1038/s42005-022-01079-8},
  pages     = {10},
  year      = {2022},
  abstract  = {Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian—or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching consequences, e.g., in predictions for reaction- or relaxation-laws. Demonstrating that a system exhibits normal- or anomalous-diffusion is highly desirable for a vast host of applications. Here, we present a criterion for anomalous-diffusion based on the method of power-spectral analysis of single trajectories. The robustness of this criterion is studied for trajectories of fractional-Brownian-motion, a ubiquitous stochastic process for the description of anomalous-diffusion, in the presence of two types of measurement errors. In particular, we find that our criterion is very robust for subdiffusion. Various tests on surrogate data in absence or presence of additional positional noise demonstrate the efficacy of this method in practical contexts. Finally, we provide a proof-of-concept based on diverse experiments exhibiting both normal and anomalous-diffusion.},
  language  = {en}
}
@article{KursaweSchulzMetzler2013,
  author    = {Kursawe, Jochen and Schulz, Johannes H. P. and Metzler, Ralf},
  title     = {Transient aging in fractional brownian and langevin-equation motion},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {88},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {6},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.88.062124},
  pages     = {13},
  year      = {2013},
  abstract  = {Stochastic processes driven by stationary fractional Gaussian noise, that is, fractional Brownian motion and fractional Langevin-equation motion, are usually considered to be ergodic in the sense that, after an algebraic relaxation, time and ensemble averages of physical observables coincide. Recently it was demonstrated that fractional Brownian motion and fractional Langevin-equation motion under external confinement are transiently nonergodic-time and ensemble averages behave differently-from the moment when the particle starts to sense the confinement. Here we show that these processes also exhibit transient aging, that is, physical observables such as the time-averaged mean-squared displacement depend on the time lag between the initiation of the system at time t = 0 and the start of the measurement at the aging time t(a). In particular, it turns out that for fractional Langevin-equation motion the aging dependence on ta is different between the cases of free and confined motion. We obtain explicit analytical expressions for the aged moments of the particle position as well as the time-averaged mean-squared displacement and present a numerical analysis of this transient aging phenomenon.},
  language  = {en}
}