Evaluation of transition rates from nonequilibrium instantons

TL;DR

This work tackles nonequilibrium rare transitions where equilibrium rate theories fail. It develops a nonequilibrium instanton rate theory NEQI that uses a weak-noise path-integral and an efficient Laplace approximation to obtain rate constants $k$ from the minimum-action path $S_N$ and its fluctuations, with $k ~ exp(-S_N/epsilon)$ in the weak-noise limit. The method is validated on a driven particle model and on an active field theory, showing close agreement with numerically exact results and revealing how activity reshapes instanton paths and speeds. These results enable quantitative predictions of reaction rates in driven, dissipative systems and offer a route to extend classical nucleation concepts to nonequilibrium settings.

Abstract

Equilibrium rate theories play a crucial role in understanding rare, reactive events. However, they are inapplicable to a range of irreversible processes in systems driven far from thermodynamic equilibrium like active and biological matter. Here, we develop an efficient numerical method to compute the rate constant of rare nonequilibrium events in the weak-noise limit based on an instanton approximation to the stochastic path integral and illustrate its wide range of application. We demonstrate excellent agreement of the instanton rates with numerically exact results for a particle under a non-conservative force. We also study phase transitions in an active field theory. We elucidate how activity alters the stability of the two phases and their rates of interconversion in a manner that can be well described by modifying classical nucleation theory,

Figures (12)

• Figure 1: Instantons at different values of $\nu$ on a color map of the potential defined in the text and a stream plot depicting the conservative force field. The forward (solid) and backward (transparent) instantons, whose direction is indicated by the black arrows, do not coincide out of equilibrium ($\nu\neq 0$).
• Figure 2: Rate constants for the model defined in Eq. (\ref{['equ:force']}) over a range of noise strengths $\epsilon$ and driving strengths $\nu$ computed from nonequilibrium instanton theory (NEQI) and a numerically exact discrete variable representation (DVR) of the Fokker--Planck operator. For each $\epsilon$, the results are given relative to the DVR equilibrium rate ($\nu = 0$).
• Figure 3: Instanton results for the active field theory defined in the text with parameters $\kappa = 0.01$, $u = 1/4$, $a = -1/2$, $\mu=1$. (a) Instanton at $h=0.1$ and $\nu=0.5$, where the color map shows the configuration of the field $\phi$ as a function of space and time along the path. The phase transition proceeds from the uniform reactant state ($\phi\approx -0.71$) to the uniform product state ($\phi\approx 1.32$) via a non-uniform TS, whose location is indicated by the dashed line. At the TS, part of the field extends towards the product, forming a critical nucleus. (b) Instanton rate constants over a range of applied fields $h$ and driving strengths $\nu$ with $\epsilon=5\cdot 10^{-3}$. The inset shows the same rate constants as a function of the difference between the instanton actions of the forward and the backward processes, $\Delta S$.
• Figure 4: Plot of the path length as a function of time along an instanton in the long-time limit. The comparatively fast transitions between the fixed points of the force are separated by long dwell phases at the reactant, product and TS configurations, giving rise to the typical double kink shape. The activation path reaches the TS at time $t_\text{f}$. The red dashed paths indicate equivalent activation paths related through time translation.
• Figure 5: KLT (solid lines) and instanton (dots) rate constants for the potential defined in Eq. (\ref{['equ:potentialSI']}) over a range of biases, $b$, and diffusion anisotropies with $\beta = \epsilon = 1$.