Statistical Physics http://hdl.handle.net/20.500.11824/19 2022-05-28T16:45:59Z Fractional Diffusion and Medium Heterogeneity: The Case of the Continuos Time Random Walk http://hdl.handle.net/20.500.11824/1333 Fractional Diffusion and Medium Heterogeneity: The Case of the Continuos Time Random Walk Sposini, V.; Vitali, S.; Paradisi, P.; Pagnini, G. In this contribution we show that fractional diffusion emerges from a simple Markovian Gaussian random walk when the medium displays a power-law heterogeneity. Within the framework of the continuous time random walk, the heterogeneity of the medium is represented by the selection, at any jump, of a different time-scale for an exponential survival probability. The resulting process is a non-Markovian non-Gaussian random walk. In particular, for a power-law distribution of the time-scales, the resulting random walk corresponds to a time-fractional diffusion process. We relates the power-law of the medium heterogeneity to the fractional order of the diffusion. This relation provides an interpretation and an estimation of the fractional order of derivation in terms of environment heterogeneity. The results are supported by simulations. 2021-07-24T00:00:00Z Stochastic Properties of Colliding Hard Spheres in a Non-equilibrium Thermal Bath http://hdl.handle.net/20.500.11824/1332 Stochastic Properties of Colliding Hard Spheres in a Non-equilibrium Thermal Bath Bazzani, A.; Vitali, S.; Montanari, C.; Monti, M.; Rambaldi, S.; Castellani, G. We consider the problem of describing the dynamics of a test particle moving in a thermal bath using the stochastic differential equations. We briefly recall the stochastic approach to the Brownian based on the statistical properties of collision theory for a gas of elastic particles and the molecular chaos hypothesis. The mathematical formulation of the Brownian motion leads to the formulation of the Ornstein-Uhlenbeck equation that provides a stationary solution consistent with the Maxwell-Boltzmann distribution. According to the stochastic thermodynamics, we assume that the stochastic differential equations allow to describe the transient states of the test particle dynamics in a thermal bath and it extends their application to the study of the non-equilibrium statistical physics. Then we consider the problem of the dynamics of a test massive particle in a non homogeneous thermal bath where a gradient of temperature is present. We discuss as the existence of a local thermodynamics equilibrium is consistent with a Stratonovich interpretation of the stochastic differential equations with a multiplicative noise. The stochastic model applied to the test particle dynamics implies the existence of a long transient state during which the particle shows a net drift toward the cold region of the system. This effect recalls the thermophoresis phenomenon performed by large molecule in a solution in response to a macroscopic temperature gradient and it can be explained as an effect of the non-locality character of the collision interactions between the test particle and the thermal bath particles. To validate the stochastic model assumptions we analyze the simulation results of the 2-dimensional hard sphere gas obtained by using an event-based computer code, that solves exactly the sphere dynamics. The temperature gradient is simulated by the presence of two reflecting boundary conditions at different temperature. The simulations suggest that existence of a local thermodynamic equilibrium is justified and highlight the presence of a drift in the average dynamics of an ensemble of massive particles. The results of the paper could be relevant for the applications of stochastic dynamical systems to the non-equilibrium statistical physics that is a key issue for the Complex Systems Physics. 2021-07-24T00:00:00Z Stochastic resetting by a random amplitude http://hdl.handle.net/20.500.11824/1293 Stochastic resetting by a random amplitude Dahlenburg, M.; Chechkin, A. V.; Schumer, R.; Metzler, R. Stochastic resetting, a diffusive process whose amplitude is reset to the origin at random times, is a vividly studied strategy to optimize encounter dynamics, e.g., in chemical reactions. Here we generalize the resetting step by introducing a random resetting amplitude such that the diffusing particle may be only partially reset towards the trajectory origin or even overshoot the origin in a resetting step. We introduce different scenarios for the random-amplitude stochastic resetting process and discuss the resulting dynamics. Direct applications are geophysical layering (stratigraphy) and population dynamics or financial markets, as well as generic search processes. 2021-05-18T00:00:00Z Exact distributions of the maximum and range of random diffusivity processes http://hdl.handle.net/20.500.11824/1287 Exact distributions of the maximum and range of random diffusivity processes Grebenkov, D. S.; Sposini, V.; Metzler, R.; Oshanin, G.; Seno, F. We study the extremal properties of a stochastic process $x_t$ defined by the Langevin equation ${\dot {x}}_{t}=\sqrt{2{D}_{t}}\enspace {\xi }_{t}$, in which $\xi_t$ is a Gaussian white noise with zero mean and $D_t$ is a stochastic 'diffusivity', defined as a functional of independent Brownian motion $B_t$. We focus on three choices for the random diffusivity $D_t$: cut-off Brownian motion, $D_t \sim \Theta(B_t)$, where $\Theta(x)$ is the Heaviside step function; geometric Brownian motion, $D_t \sim exp(−B_t)$; and a superdiffusive process based on squared Brownian motion, ${D}_{t}\sim {B}_{t}^{2}$. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process $x_t$ on the time interval $t \in (0, T)$. 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 ($D_t = D_0$) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process. 2021-02-09T00:00:00Z