Relaxation functions of the Ornstein-Uhlenbeck process with fluctuating diffusivity

We study the relaxation behavior of the Ornstein-Uhlenbeck (OU) process with time-dependent and fluctuating diffusivity. In this process, the dynamics of the 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 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.

Figure 1

Example of a realization of the stochastic process which obeys the OUFD in one dimension $x\left(t\right)$. The solid black curve represent the position $x\left(t\right)$ at time $t$ and the background colors represents the diffusivity $D\left(t\right)$. The position is fluctuating around the origin $x=0$ (dashed black line) due to the restoring force.

Figure 2

Relaxation function $\mathrm{\Phi }\left(t\right)$ for the OUFD by the Markovian two-state model. (a) The relaxation rate is ${\mu }_s/{\mu }_f={10}^{-2}$ and the transition rate varies. The solid curves represent data for $\kappa =k_f/{\mu }_f=k_s/{\mu }_f={10}^{-2}$, ${10}^{-1}$, 1, and ${10}^1$. The dotted gray curves represent asymptotic forms [Eqs. (32) and (34)]. (b) The transition rate is constant $\kappa ={10}^{-2}$ and the relaxation rate varies. The solid curves represent data for ${\mu }_s/{\mu }_f=1$, ${10}^{-1}$, ${10}^{-2}$, and ${10}^{-3}$. The dotted gray curves represent the asymptotic forms.

Figure 3

Relaxation function $\mathrm{\Phi }\left(t\right)$ for the OUFD. The noise coefficient obeys the OU model. The solid curves represent the relaxation functions for $k/\mu ={10}^{-6}$, ${10}^{-4}$, ${10}^{-2}$, 1, and ${10}^2$, calculated by Eq. (44). The dotted gray curves represent the asymptotic forms for $k/\mu \gg 1$ [Eq. (48), left] and $k/\mu \ll 1$ [Eq. (50), right].