Conundrum of weak-noise limit for diffusion in a tilted periodic potential

The weak-noise limit of dissipative dynamical systems is often the most fascinating one. In such a case fluctuations can interact with a rich complexity, frequently hidden in deterministic systems, to give rise to phenomena that are absent for both noiseless and strong fluctuations regimes. Unfortunately, this limit is also notoriously hard to approach analytically or numerically. We reinvestigate in this context the paradigmatic model of nonequilibrium statistical physics consisting of inertial Brownian particles diffusing in a tilted periodic potential by exploiting state-of-the-art computer simulations of an extremely long timescale. In contrast to previous results on this longstanding problem, we draw an inference that in the parameter regime for which the particle velocity is bistable the lifetime of ballistic diffusion diverges to infinity when the thermal noise intensity tends to zero, i.e., an everlasting ballistic diffusion emerges. As a consequence, the diffusion coefficient does not reach its stationary constant value.

Figure 1

Phase diagram for the occurrence of the velocity bistability phenomenon in the deterministic system presented in the dimensionless parameter plane $(\gamma ,f)$. Due to this effect, in the gray area, the ballistic diffusion emerges in the deterministic limit of vanishing temperature. The dashed horizontal and vertical lines correspond to $f=0.9$ and $\gamma =0.66$, respectively. This regime is analyzed in the paper.

Figure 2

Diffusion coefficient $D$ versus bias $f$ for different values of temperature $\theta$ of the system. The gray shaded region shows the interval $[f_1,f_3]$ where, in the deterministic system, the bistability of the velocity dynamics occurs. The orange regions shows the critical bias $f$ range $[f_{gd,-},f_{gd,+}]$ for which, according to the two-state theory presented in [50], the diffusion coefficient tends to infinity $D\to \infty$ when temperature vanishes $\theta \to 0$. The dimensionless friction coefficient is $\gamma =0.66$.

Figure 3

Trajectory of the diffusion coefficient $D\left(t\right)$ depicted for different (color-coded) values of the dimensionless temperature $\theta$ of the system, with (a) subcritical force $f=0.85\in [f_1,f_{gd,-}]$, (b) critical bias $f=0.9\in [f_{gd,-},f_{gd,+}]$, and (c) supercritical force $f=0.94\in [f_{gd,+},f_3]$ (cf. Fig. 2). The friction coefficient is $\gamma =0.66$.

Figure 4

Crossover times ${\tau }_1$ (solid lines) and ${\tau }_2$ (dashed lines) of the ballistic motion and subdiffusion, respectively, versus the temperature $\theta$ of the system for the subcritical $f=0.85$, critical $f=0.9$, and supercritical $f=0.94$ biases.

Figure 5

Difference $Q=1-|p_l-p_r|$ between the stationary probability $p_l$ and $p_r$ of finding the particle in the locked ($\mathbb{E}\left[v\right]=0$) and running ($\mathbb{E}\left[v\right]>0$) states, respectively, versus the bias $f$ for different values of the temperature $\theta$ of the system. The gray shaded region shows the interval $[f_1,f_3]$ where, in the deterministic system, the bistability of the velocity dynamics occurs. The orange region shows the critical bias $f$ range $[f_{gd,-},f_{gd,+}]$ for which, according to the two-state theory presented in [50], the diffusion coefficient tends to infinity $D\to \infty$ when temperature vanishes $\theta \to 0$. The dimensionless friction coefficient is $\gamma =0.66$.

Figure 6

Finite-time diffusion coefficient $D(t=t_i={10}^6)$ versus the temperature $\theta$ of the system for the subcritical, critical, and supercritical biases $f$.