Instanton rate theory for transitions far from equilibrium

In contrast to systems in equilibrium, the probability of a driven system going between two points in space does not only depend on the initial and final positions but on the whole path. Hence, a path-space picture is commonly adopted to capture these inherently dynamical transitions far from equilibrium.

In this work, we derive a general rate theory for rare transitions between nonequilibrium steady states of driven systems as an asymptotic approximation to the exact path-integral expression in the small-noise limit. The key component of the theory is not an optimal configuration, like the transitions state in equilibrium, but an optimal path, the instanton, connecting initial and final states. This instanton pathway minimizes the stochastic Onsager–Machlup action and provides intuitive insight into the transition mechanism. We go beyond previous work, in which several efficient algorithms for optimizing these paths have been devised, by including not only the exponential dependence of the rate on the instanton action but also the prefactor arising from harmonic fluctuations around the optimal path. We demonstrate excellent agreement of our new method with numerically exact results at small noise strength and apply it to both explicit-particle models and stochastic phase-field models, such as in the calculation of nucleation rates under external driving.