Response to a small external force and fluctuations of a passive particle in a one-dimensional diffusive environment

We investigate the long-time behavior of a passive particle evolving in a one-dimensional diffusive random environment, with diffusion constant $D$. We consider two cases: (a) The particle is pulled forward by a small external constant force and (b) there is no systematic bias. Theoretical arguments and numerical simulations provide evidence that the particle is eventually trapped by the environment. This is diagnosed in two ways: The asymptotic speed of the particle scales quadratically with the external force as it goes to zero, and the fluctuations scale diffusively in the unbiased environment, up to possible logarithmic corrections in both cases. Moreover, in the large $D$ limit (homogenized regime), we find an important transient region giving rise to other, finite-size scalings, and we describe the crossover to the true asymptotic behavior.

Figure 1

Evolution of the passive particle in the potential $V(x,t)-Fx$.

Figure 2

Time to stationarity: $W\left(\tau \right)$ for various values of $F$ and $\lambda$. Average over 1000 realizations at least.

Figure 3

Drift: $v\left(F\right)/\lambda F$ (upper panel) and $v\left(F\right)/F^2$ (lower panel) as a function of ${log}_2(F/\lambda )$ for various values of $\lambda$. $v\left(F\right)$ is approximated by $\langle X_{t\left(F\right)}\rangle /t\left(F\right)$ with $t\left(F\right)=2^{10}{F}^{-4}$ for $F=2^{-1},\cdots ,2^{-5}$ (see the discussion on the time to stationarity), while slightly long times are taken for $F=2^0,\cdots ,2^3$ (these points are obviously not the most relevant to determine the asymptotic $F\to 0)$. Average over 1000 realizations at least.

Figure 4

Fluctuations: $\langle X_t^2\rangle /{\lambda }^2{t}^{3/2}$ (upper panel), $\langle X_t^2\rangle /t$ (lower panel), and $\langle X_t^2\rangle /{\left(\lambda t\right)}^{4/3}$ (inset) as a function of ${log}_2\left({\lambda }^{4}t\right)$ for various values of $\lambda$. Dotted line in the upper panel: See the main text. Average over 8000 realizations at least.