Velocity Multistability vs. Ergodicity Breaking in a Biased Periodic Potential

Abstract

Multistability, i.e., the coexistence of several attractors for a given set of system parameters, is one of the most important phenomena occurring in dynamical systems. We consider it in the velocity dynamics of a Brownian particle, driven by thermal fluctuations and moving in a biased periodic potential. In doing so, we focus on the impact of ergodicity-A concept which lies at the core of statistical mechanics. The latter implies that a single trajectory of the system is representative for the whole ensemble and, as a consequence, the initial conditions of the dynamics are fully forgotten. The ergodicity of the deterministic counterpart is strongly broken, and we discuss how the velocity multistability depends on the starting position and velocity of the particle. While for non-zero temperatures the ergodicity is, in principle, restored, in the low temperature regime the velocity dynamics is still affected by initial conditions due to weak ergodicity breaking. For moderate and high temperatures, the multistability is robust with respect to the choice of the starting position and velocity of the particle.

Keywords: Brownian motion; ergodicity; multistability; tilted periodic potential.

Figure 1

The probability distribution $p\left(v\right)$ for the instantaneous long time velocity $v=v\left(t\right)$ of the Brownian particle is illustrated for $t={10}^4$ and selected values of temperatures $\theta$ of the system. The used parameters read $\gamma =0.66$ and $f=0.91$.

Figure 2

The basins of attraction for the time averaged velocity $\mathbf{v}$ of the particle. The black colour codes the locked state $\mathbf{v}=0$ whereas the grey part indicates the regime with running solutions $\mathbf{v}\ne 0$. Parameters read $\gamma =0.66$, $f=0.91$ and $\theta =0$.

Figure 3

The probability distribution $p\left(v\right)$ for the instantaneous long time velocity v of the Brownian particle is depicted in the deterministic regime $\theta =0$ for $t={10}^4$ and different choice of the initial conditions for the system. Panel (a): $p_{x_0}\left(x\right)=\delta \left(x\right)$, $p_{v_0}\left(v\right)=\delta \left(v\right)$; (b): $p_{x_0}\left(x\right)=\mathsf{U}(0,2\pi )$, $p_{v_0}\left(v\right)=\delta \left(v\right)$; (c): $p_{x_0}\left(x\right)=\delta \left(x\right)$, $p_{v_0}\left(v\right)=\mathsf{U}(-2,2)$; (d): $p_{x_0}\left(x\right)=\delta (x-\pi )$, $p_{v_0}\left(v\right)=\mathsf{U}(-2,2)$; (e): $p_{x_0}\left(x\right)=\mathsf{U}(0,2\pi )$, $p_{v_0}\left(v\right)=\mathsf{U}(-2,2)$; and (f): $p_{x_0}\left(x\right)=\mathsf{N}(0,1)$, $p_{v_0}\left(v\right)=\mathsf{N}(0,1)$, where $\mathsf{U}(a,b)$ indicates the uniform distribution over the interval $[a,b]$. Likewise, $\mathsf{N}(\mu ,{\sigma }^2)$ is the Gaussian distribution with the mean $\mu$ and the variance ${\sigma }^2$. In the inset, the corresponding probability distribution $P\left(\mathbf{v}\right)$ for the time averaged velocity $\mathbf{v}$ is shown. Parameters read $\gamma =0.66$, $f=0.91$, and $\theta =0$.

Figure 4

The probability distribution $p\left(v\right)$ for the instantaneous long time velocity v of the Brownian particle is depicted for $t={10}^4$ and different initial conditions of the system. The red solid line indicates $p_{x_0}\left(x\right)=\delta \left(x\right)$, $p_{v_0}\left(v\right)=\delta \left(v\right)$. The blue dotted line corresponds to $p_{x_0}\left(x\right)=\mathsf{U}(0,2\pi )$, $p_{v_0}\left(v\right)=\mathsf{U}(-2,2)$. The green dashed line denotes $p_{x_0}\left(x\right)=\mathsf{N}(0,1)$, $p_{v_0}\left(v\right)=\mathsf{N}(0,1)$. In panel (a) temperature is $\theta =0.0001$ while in (b) $\theta =0.05$. Other parameters read $\gamma =0.66$, $f=0.91$.