Fluctuation analysis of time-averaged mean-square displacement for the Langevin equation with time-dependent and fluctuating diffusivity

The mean-square displacement (MSD) is widely utilized to study the dynamical properties of stochastic processes. The time-averaged MSD (TAMSD) provides some information on the dynamics which cannot be extracted from the ensemble-averaged MSD. In particular, the relative standard deviation (RSD) of the TAMSD can be utilized to study the long-time relaxation behavior. In this work, we consider a class of Langevin equations which are multiplicatively coupled to time-dependent and fluctuating diffusivities. Various interesting dynamics models such as entangled polymers and supercooled liquids can be interpreted as the Langevin equations with time-dependent and fluctuating diffusivities. We derive a general formula for the RSD of the TAMSD for the Langevin equation with the time-dependent and fluctuating diffusivity. We show that the RSD can be expressed in terms of the correlation function of the diffusivity. The RSD exhibits the crossover at the long time region. The crossover time is related to a weighted average relaxation time for the diffusivity. Thus the crossover time gives some information on the relaxation time of fluctuating diffusivity which cannot be extracted from the ensemble-averaged MSD. We discuss the universality and possible applications of the formula via some simple examples.

Figure 1

The RSD of the TAMSD of a center of mass in the discrete reptation model. The number of tube segments $Z$ is $Z=80$ and the time difference $\mathrm{\Delta }$ is $\mathrm{\Delta }=10{\tau }_l$. ${\tau }_l$ is the characteristic time of the longitudinal motion of a segment along the tube. Symbols represent the kinetic Monte Carlo simulation data. The dotted and dashed curves represent the theoretical prediction [Eq. (46)] and its asymptotic forms [Eq. (44)].

Figure 2

The RSD of the TAMSD of the Markovian two-state model. The diffusion coefficients and transition rates are $D_s=1$, $D_f=10$, $k_s=1$, and $k_f=0.1,1$, and 10. The time difference is $\mathrm{\Delta }=0.001$. Symbols represent the simulation results and solid curves represent the theoretical prediction.

Figure 3

The RSD of the TAMSD of the non-Markovian two-state model. The diffusion coefficients are $D_s=1$ and $D_f=10$. The waiting time distribution for the fast state is given by the exponential distribution with $k_f=1$, and the waiting time distribution for the slow state is given by the power-law type distribution with $k_{s,1}=1$ and $k_{s,0}=0.1,0.01$, and 0.001. The time difference is $\mathrm{\Delta }=0.001$. Symbols represent the simulation results and dashed curves represent the theoretically predicted asymptotic forms.

Figure 4

The RSD and RF of the TAMSD of a center of mass in the discrete reptation model. (The RF data are taken from Ref. [15].) The number of tube segments $Z=80$ and the time difference $\mathrm{\Delta }=10{\tau }_l$, where ${\tau }_l$ is the characteristic time of the longitudinal motion of a segment along the tube. Dashed curves represent the asymptotic forms for the RF [Eq. (B10)].