Random diffusivity processes in an external force field

Brownian yet non-Gaussian processes have recently been observed in numerous biological systems, and corresponding theories have been constructed based on random diffusivity models. Considering the particularity of random diffusivity, this paper studies the effect of an external force acting on two kinds of random diffusivity models whose difference is embodied in whether the fluctuation-dissipation theorem is valid. Based on the two random diffusivity models, we derive the Fokker-Planck equations with an arbitrary external force, and we analyze various observables in the case with a constant force, including the Einstein relation, the moments, the kurtosis, and the asymptotic behaviors of the probability density function of particle displacement at different timescales. Both the theoretical results and numerical simulations of these observables show a significant difference between the two kinds of random diffusivity models, which implies the important role of the fluctuation-dissipation theorem in random diffusivity systems.

Figure 1

Moments $\langle x^n\left(t\right)\rangle$ with $n=1,2,3,4$. Model I (in red) and Model II (in blue) represent the Langevin systems in Eqs. (28) and (39), respectively. The circle and star markers denote the simulation results, while the solid and dashed lines denote the theoretical results in Eqs. (31) and (47), respectively. Based on Eqs. (31) and (47), the two lines with the same $n$ are parallel for two models, i.e., ${\langle x^n\left(t\right)\rangle }_{\text{I}}=2^n{\langle x^n\left(t\right)\rangle }_{\text{II}}$. Correspondingly, each set of two lines (or markers) from the bottom to the top represent the first, second, third, and fourth moments, respectively. Parameters: $T={10}^3$, $F=2$, and ${10}^3$ samples are used for ensemble average.

Figure 2

Kurtosis [defined in Eq. (33)] in Model I (in red) and Model II (in blue), which represent the Langevin systems in Eqs. (28) and (39), respectively. For the two models, the circle and star markers denote the simulation results, while the solid and dashed lines denote the theoretical results in Eqs. (C1) and (C2), respectively. The kurtosis in Eqs. (34) and (51) both have the same asymptotics as the force-free case. In contrast to the monotone decreasing behavior of the kurtosis line of Model I, that of Model II has a maximum value around $t=0.5$. Parameters: $T={10}^2$, $F=2$, and ${10}^6$ samples are used for ensemble average.

Figure 3

Short-time PDFs in Model I (in red) and Model II (in blue), which represent the Langevin systems in Eqs. (28) and (39), respectively. For the two models, the circle and star markers denote the simulation results, while the solid and dashed lines denote the theoretical results in Eqs. (36) and (55), respectively. The PDF of Model I is a symmetric exponential distribution with the center at $x=Ft$, while the PDF of Model II is an asymmetric skewed exponential distribution. Parameters: $T=0.1$, $F=1$, and ${10}^7$ samples are used for ensemble average.

Figure 4

Long-time PDFs in Model I (in red) and Model II (in blue), which represent the Langevin systems in Eqs. (28) and (39), respectively. For the two models, the circle and star markers denote the simulation results, while the solid and dashed lines denote the theoretical results in Eqs. (35) and (61), respectively. Both of the PDFs of the two models are Gaussian shapes. The PDF of Model I has the mean $Ft$ and the variance $t$, while that of Model II has a smaller mean $Ft/2$ but a larger variance $(F^2/2+1)t$. Parameters: $T=20$, $F=1$, and ${10}^7$ samples are used for ensemble average.