Timescales for stochastic barrier crossing: inferring the potential from nonequilibrium data

TL;DR

This paper investigates early-time nonequilibrium dynamics of barrier crossing in a double-well potential, identifying key timescales and deviations from equilibrium, with implications for experimental and simulation data analysis.

Contribution

It introduces a detailed analysis of nonequilibrium timescales using SE and SPI methods, revealing how potential landscapes evolve over time.

Findings

Short timescale corresponds to well equilibration.

Inflexion point in effective potential appears at t ≈ τ_B.

Density current crossover to equilibrium rate occurs independently of barrier height.

Abstract

Kramers' rate theory forms a cornerstone for thermally activated barrier crossing. However, its reliance on equilibrium quantities excludes analysis of nonequilibrium dynamics at early times. Most works have thus focused on obtaining rates and transition time and path distributions in equilibrium. Instead, here we consider early-time nonequilibrium dynamics in a model system of a particle with overdamped dynamics hopping over the barrier in a double-well potential, using the Smoluchowski equation (SE) and stochastic path integral (SPI) mapping of the Langevin equation. We identify several key timescales relevant to nonequilibrium dynamics and quantify them using the SE and SPI approaches. The shortest timescale corresponds to equilibration in a well at time $t \ll \tau_{\rm B}$, where $\tau_{\rm B}$ is the Brownian diffusion time. The second important timescale is when an inflexion point appears in the effective potential constructed from the density at $t \lessapprox \tau_{\rm B}$. Shortly after, the existence of a second potential well can be inferred from sufficient sampling of the dynamics. Interestingly, this timescale decreases with increasing barrier height. We find significant deviations from the equilibrium limit unless $t \gg \tau_{\rm B}$. We further calculate \red{the} density current at the barrier for bistable and asymmetric potentials and find that it crosses over to that from equilibrium rate theory at a time that does not appear to depend on the barrier height. Our results have important implications for controlling activated processes at finite times and demonstrate the importance of reaching long enough times to faithfully construct potential landscapes from experimental or simulation data.