Arcsine law and multistable Brownian dynamics in a tilted periodic potential

Multistability is one of the most important phenomena in dynamical systems, e.g., bistability enables the implementation of logic gates and therefore computation. Recently multistability has attracted a greatly renewed interest related to memristors and graphene structures, to name only a few. We investigate tristability in velocity dynamics of a Brownian particle subjected to a tilted periodic potential. It is demonstrated that the origin of this effect is attributed to the arcsine law for the velocity dynamics at the zero temperature limit. We analyze the impact of thermal fluctuations and construct the phase diagram for the stability of the velocity dynamics. It suggests an efficient strategy to control the multistability by changing solely the force acting on the particle or temperature of the system. Our findings for the paradigmatic model of nonequilibrium statistical physics apply to, inter alia, Brownian motors, Josephson junctions, cold atoms dwelling in optical lattices, and colloidal systems.

Figure 1

Phase diagram for occurrence of the velocity bistability phenomenon in the deterministic counterpart of the system (1) presented in the dimensionless parameter plane $(\gamma ,f)$. The critical forces $f_1\left(\gamma \right)$ as well as $f_3$ are indicated by the corresponding lines. Velocity bistability is observed only if the friction and force values lie in the marked gray area.

Figure 2

The probability distribution $p\left(v\right)$ for the instantaneous long time velocity $v$ of the Brownian particle is depicted in (a) for selected values of the dimensionless temperature $\theta$. Parameters are $\gamma =0.66$ and $f=0.91$. In (b) the same characteristic is shown but for fixed temperature $\theta =0.001$ and varied bias $f$ values.

Figure 3

The probability distribution $p\left(v\right)$ for the instantaneous long time velocity $v$ of the deterministic system $\theta =0$. The maxima of $p\left(v\right)$ are at $v_1=0$, $v_m=0.59$, and $v_M=1.98$. In the inset we display the corresponding distribution $P\left(\mathbf{v}\right)$ for the time-averaged velocity $\mathbf{v}$. The parameters are $\gamma =0.66$ and $f=0.91$.

Figure 4

The exemplary deterministic trajectories of the particle coordinate $x\left(t\right)$ and velocity $\dot{x}\left(t\right)=v\left(t\right)$ (in the inset) vs time for two different initial conditions $x_0$ corresponding to the running state. Other parameters are $\gamma =0.66$, $f=0.91$, $\theta =0$, and $v_0=0$.

Figure 5

(a) The cumulative distribution function $F\left(v\right)$ for the instantaneous long time velocity $v$ corresponding to the $U$-shaped part of the density $p\left(v\right)$ is compared with the arcsine distribution $\mathcal{F}\left(v\right)=(2/\pi )arcsin\left[\sqrt{(v-v_m)/(v_M-v_m)}\right]$. Inset: The solution $v\left(t\right)$ for $x_0=0.6$ and $v_0=0$ is compared with the function $V\left(t\right)=Asin(\omega t+\varphi )+c$, where $A=0.695$, $\omega =1.18$, $\varphi =-5.6$, and $c=1.28$. Other parameters are $\gamma =0.66$, $f=0.91$ and $\theta =0$. (b) The probability density $g\left(\varphi \right)$ for the phase difference $\varphi$ between the trajectories $v\left(t\right)$ in the long time regime. Inset: The cumulative distribution function $G\left(\varphi \right)$ corresponding to the density $g\left(\varphi \right)=dG\left(\varphi \right)/d\varphi$ is compared with the cumulative distribution function of the uniformly distributed random variable (the green curve). Parameters are $\gamma =0.66$, $f=0.91$ and $\theta =0$.

Figure 6

Influence of temperature on the probability distribution $p\left(v\right)$ of the instantaneous long time velocity of the Brownian particle depicted for different parameter $(\gamma ,f)$ regimes. (a) Velocity bistability (coexistence of the locked and running states) for $\gamma =0.15$ and $f=0.45$; (b) velocity monostability (the running state) for $\gamma =0.15$ and $f=0.95$; (c) velocity bistability (induced by the running solutions) for $\gamma =0.7$ and $f=0.99$. Velocity multistability (coexistence of one locked and two running peaks) is exemplified for $\gamma =0.66$ and $f=0.91$ in Fig. 2.

Figure 7

Phase diagram for the long time velocity dynamics of the Brownian particle dwelling in the tilted periodic potential shown in the parameter plane $(\gamma ,f)$ for selected values of dimensionless temperature $\theta$. (a) $\theta =0.001$; (b) $\theta =0.01$; (c) $\theta =0.02$; (d) $\theta =0.03$; (e) $\theta =0.05$; (f) $\theta =0.1$. Different regimes of the velocity states are marked with the corresponding color: blue, the monostability (the single locked state); orange, the monostability (the single running state); green, the bistability (coexistence of the locked and running states); red, the bistability (concurrence of two running peaks); yellow, the multistability (coexistence of one locked and two running peaks).

Figure 8

(a) The fraction of basins of attraction for the time-averaged velocity $\mathbf{v}$ of the particle. The blue region indicates the locked state whereas red stands for the running states. (b) Snapshot of the instantaneous velocity $v$ in the long time regime. (c) Phase difference $\varphi$ between the trajectories $v\left(t\right)$ in the long time limit. All presented versus the initial conditions $\{x_0,v_0\}$ of the system. Other parameters read $\gamma =0.66$, $f=0.91$, and $\theta =0$.