Nonequilibrium steady state of Brownian motion in an intermittent potential

We calculate the steady state distribution P SSD ( X ) of the position of a Brownian particle under an intermittent confining potential that switches on and off with a constant rate γ. We assume the external potential U ( x ) to be smooth and to have a unique global minimum at x = x 0 , and in dimension d > 1 we additionally assume that U ( x ) is central. We focus on the rapid-switching limit γ → ∞ . Typical fluctuations follow a Boltzmann distribution P SSD ( X ) ∼ e − U eff ( X ) / D , with an effective potential U eff ( X ) = U ( X ) / 2 , where D is the diffusion coefficient. However, we also calculate the tails of P SSD ( X ) which behave very differently. In the far tails | X | → ∞ , a universal behavior P SSD ( X ) ∼ e − γ / D | X − x 0 | emerges, that is independent of the trapping potential. The mean first-passage time to reach position X is given, in the leading order, by ∼ 1 / P SSD ( X ) . This coincides with the Arrhenius law (for the effective potential U eff ) for X ≃ 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 γ → ∞ and D → 0, the logarithm of P SSD ( X ) exhibits a singularity which we interpret as a first-order dynamical phase transition (DPT). This DPT occurs in the 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.