@article{PetreskaPejovSandevetal.2022,
  author    = {Petreska, Irina and Pejov, Ljupco and Sandev, Trifce and Kocarev, Ljupčo and Metzler, Ralf},
  title     = {Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting},
  series = {Fractal and fractional},
  volume    = {6},
  journal   = {Fractal and fractional},
  number    = {2},
  publisher = {MDPI AG, Fractal Fract Editorial Office},
  address   = {Basel},
  issn      = {2504-3110},
  doi       = {10.3390/fractalfract6020088},
  pages     = {23},
  year      = {2022},
  abstract  = {The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.},
  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{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{XuMetzlerWang2022,
  author    = {Xu, Pengbo and Metzler, Ralf and Wang, Wanli},
  title     = {Infinite density and relaxation for Levy walks in an external potential},
  series = {Physical review},
  volume    = {105},
  journal   = {Physical review},
  number    = {4},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.105.044118},
  pages     = {15},
  year      = {2022},
  abstract  = {Levy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(r ) = v0/./r. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly.},
  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{VinodCherstvyWangetal.2022,
  author    = {Vinod, Deepak and Cherstvy, Andrey G. and Wang, Wei and Metzler, Ralf and Sokolov, Igor M.},
  title     = {Nonergodicity of reset geometric Brownian motion},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {105},
  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.105.L012106},
  pages     = {4},
  year      = {2022},
  abstract  = {We derive. the ensemble-and time-averaged mean-squared displacements (MSD, TAMSD) for Poisson-reset geometric Brownian motion (GBM), in agreement with simulations. We find MSD and TAMSD saturation for frequent resetting, quantify the spread of TAMSDs via the ergodicity-breaking parameter and compute distributions of prices. General MSD-TAMSD nonequivalence proves reset GBM nonergodic.},
  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{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{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}
}
@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}
}