Relaxation Functions of Ornstein-Uhlenbeck Process with Fluctuating Diffusivity

Abstract

We study a relaxation behavior of an Ornstein-Uhlenbeck (OU) process with a time-dependent and fluctuating diffusivity. In this process, the dynamics of a position vector is modeled by the Langevin equation with a linear restoring force and a fluctuating diffusivity (FD). This process can be interpreted as a simple model of the relaxational dynamics with internal degrees of freedom or in a heterogeneous environment. By utilizing the functional integral expression and the transfer matrix method, we show that the relaxation function can be expressed in terms of the eigenvalues and eigenfunctions of the transfer matrix, for general FD processes. We apply our general theory to two simple FD processes, where the FD is described by the Markovian two-state model or an OU type process. We show analytic expressions of the relaxation functions in these models, and their asymptotic forms. We also show that the relaxation behavior of the OU process with an FD is qualitatively different from those obtained from conventional models such as the generalized Langevin equation.

arXiv comment22 pages, 3 figures, to appear in Phys. Rev. E

References in corpus (14)

• Stochastic thermodynamics, fluctuation theorems, and molecular machines
• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Dynamics of Highly Supercooled Liquids:Heterogeneity, Rheology, and Diffusion
• Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities
• "Diffusing diffusivity": A model for anomalous and "anomalous yet Brownian" diffusion
• Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes
• Heterogeneous Diffusion in Highly Supercooled Liquids
• Nonergodic Subdiffusion from Brownian Motion in an Inhomogeneous Medium
• Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity
• Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes
• Fluctuation Analysis of Time-Averaged Mean-Square Displacement for Langevin Equation with Time-Dependent and Fluctuating Diffusivity
• Langevin equation with fluctuating diffusivity: a two-state model
• Distributional Behaviors of Time-averaged Observables in Langevin Equation with Fluctuating Diffusivity: Normal Diffusion but Anomalous Fluctuations
• Elucidating fluctuating diffusivity in center-of-mass motion of polymer models with time-averaged mean-square-displacement tensor

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