TY - JOUR A1 - Aydiner, Ekrem A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Wealth distribution, Pareto law, and stretched exponential decay of money BT - Computer simulations analysis of agent-based models JF - Physica : europhysics journal ; A, Statistical mechanics and its applications N2 - 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. KW - Econophysics KW - Wealth and income distribution KW - Pareto law KW - Scaling exponents Y1 - 2017 U6 - https://doi.org/10.1016/j.physa.2017.08.017 SN - 0378-4371 SN - 1873-2119 VL - 490 SP - 278 EP - 288 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Ritschel, Stefan A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Universality of delay-time averages for financial time series BT - analytical results, computer simulations, and analysis of historical stock-market prices JF - Journal of physics. Complexity N2 - 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. KW - econophysics KW - geometric Brownian motion KW - time-series analysis Y1 - 2021 U6 - https://doi.org/10.1088/2632-072X/ac2220 SN - 2632-072X VL - 2 IS - 4 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Bodrova, Anna S. A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Safdari, Hadiseh A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion JF - Scientific reports N2 - It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases. Y1 - 2016 U6 - https://doi.org/10.1038/srep30520 SN - 2045-2322 VL - 6 PB - Nature Publ. Group CY - London ER - TY - GEN A1 - Bodrova, Anna S. A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Safdari, Hadiseh A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Underdamped scaled Brownian motion BT - (non-)existence of the overdamped limit in anomalous diffusion N2 - It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 267 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-97158 ER - TY - JOUR A1 - Bodrova, Anna S. A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Safdari, Hadiseh A1 - Sokolov, Igor M. A1 - Metzler, Ralf T1 - Underdamped scaled Brownian motion BT - (non-)existence of the overdamped limit in anomalous diffusion JF - Scientific reports N2 - It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases. Y1 - 2016 U6 - https://doi.org/10.1038/srep30520 SN - 2045-2322 VL - 6 PB - Nature Publishing Group CY - London ER - TY - JOUR A1 - Bodrova, Anna S. A1 - Chechkin, Aleksei V. A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Ultraslow scaled Brownian motion JF - New journal of physics : the open-access journal for physics N2 - We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form D(t) similar or equal to 1/t. For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations. KW - anomalous diffusion KW - stochastic processes KW - ageing Y1 - 2015 U6 - https://doi.org/10.1088/1367-2630/17/6/063038 SN - 1367-2630 VL - 17 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Metzler, Ralf A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Bodrova, Anna S. T1 - Ultraslow scaled Brownian motion JF - New journal of physics : the open-access journal for physics N2 - We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form . For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations. KW - anomalous diffusion KW - stochastic processes KW - ageing Y1 - 2015 U6 - https://doi.org/10.1088/1367-2630/17/6/063038 SN - 1367-2630 VL - 17 IS - 063038 PB - Dt. Physikalische Ges., IOP CY - Bad Honnef, London ER - TY - GEN A1 - Metzler, Ralf A1 - Cherstvy, Andrey G. A1 - Chechkin, Aleksei V. A1 - Bodrova, Anna S. T1 - Ultraslow scaled Brownian motion T2 - New journal of physics : the open-access journal for physics N2 - We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form . For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 188 KW - anomalous diffusion KW - stochastic processes KW - ageing Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-78618 ER - TY - JOUR A1 - Thapa, Samudrajit A1 - Lukat, Nils A1 - Selhuber-Unkel, Christine A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf T1 - Transient superdiffusion of polydisperse vacuoles in highly motile amoeboid cells JF - The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr N2 - We perform a detailed statistical analysis of diffusive trajectories of membrane-enclosed vesicles (vacuoles) in the supercrowded cytoplasm of living Acanthamoeba castellanii cells. From the vacuole traces recorded in the center-of-area frame of moving amoebae, we examine the statistics of the time-averaged mean-squared displacements of vacuoles, their generalized diffusion coefficients and anomalous scaling exponents, the ergodicity breaking parameter, the non-Gaussian features of displacement distributions of vacuoles, the displacement autocorrelation function, as well as the distributions of speeds and positions of vacuoles inside the amoeba cells. Our findings deliver novel insights into the internal dynamics of cellular structures in these infectious pathogens. Published under license by AIP Publishing. Y1 - 2019 U6 - https://doi.org/10.1063/1.5086269 SN - 0021-9606 SN - 1089-7690 VL - 150 IS - 14 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Vinod, Deepak A1 - Cherstvy, Andrey G. A1 - Metzler, Ralf A1 - Sokolov, Igor M. T1 - Time-averaging and nonergodicity of reset geometric Brownian motion with drift JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevE.106.034137 SN - 2470-0045 SN - 2470-0053 VL - 106 IS - 3 PB - American Physical Society CY - College Park ER -