Nonequilibrium steady state of Brownian motion in an intermittent potential

Abstract: We calculate the steady state distribution $P_{\text{SSD}}(\boldsymbol{X})$ of the position of a Brownian particle under an intermittent confining potential that switches on and off with a constant rate $\gamma$. We assume the external potential $U(\boldsymbol{x})$ to be smooth and have a unique global minimum at $\boldsymbol{x} = \boldsymbol{x}0$, and in dimension $d>1$ we additionally assume that $U(\boldsymbol{x})$ is central. We focus on the rapid-switching limit $\gamma \to \infty$. Typical fluctuations follow a Boltzmann distribution $P{\text{SSD}}(\boldsymbol{X}) \sim e^{- U_{\text{eff}}(\boldsymbol{X}) / D}$, with an effective potential $U_{\text{eff}}(\boldsymbol{X}) = U(\boldsymbol{X})/2$, where $D$ is the diffusion coefficient. However, we also calculate the tails of $P_{\text{SSD}}(\boldsymbol{X})$ which behave very differently. In the far tails $|\boldsymbol{X}| \to \infty$, a universal behavior $P_{\text{SSD}}\left(\boldsymbol{X}\right)\sim e^{-\sqrt{\gamma/D} \, \left|\boldsymbol{X}-\boldsymbol{x}{0}\right|}$ emerges, that is independent of the trapping potential. The mean first-passage time to reach position $\boldsymbol{X}$ is given, in the leading order, by $\sim 1/P{\text{SSD}}(\boldsymbol{X})$. This coincides with the Arrhenius law (for the effective potential $U_{\text{eff}}$) for $\boldsymbol{X} \simeq \boldsymbol{x}0$, but deviates from it elsewhere. We give explicit results for the harmonic potential. Finally, we extend our results to periodic one-dimensional systems. Here we find that in the limit of $\gamma \to \infty$ and $D \to 0$, the logarithm of $P{\text{SSD}}(X)$ exhibits a singularity which we interpret as a first-order dynamical phase transition (DPT). This DPT occurs in absence of any external drift. We also calculate the nonzero probability current in the steady state that is a result of the nonequilibrium nature of the system.