Illuminating non-equilibrium multi-step reaction dynamics with stochastic Marcus state model

Modeling the stochastic reaction dynamics is a significant task to explain the modern measurements of non-equilibrium processes at mesoscopic scales. Marcus's transition-state theory describes the reaction rate of a single-step reaction event, but how it can enlighten a multi-step stochastic reaction process in a continuous chemical-state space remains elusive. In this paper, we develop a stochastic Marcus state model with continuation methods for different reaction systems. The time-resolved evolutions of the probability density functions are expressed via Fokker-Planck equations where the drift and diffusion coefficients are determined by the free-energy functions and reorganization energy. In a system with infinitesimal-reaction transitions, the model allows a scale-invariant transform that preserves its generic form, and the equation of motion describes the over-damped Langevin dynamics space that follows the fluctuation-dissipation theorem in the chemical-state. We also prove that the Onsager reciprocal relation can be retrieved as long as the free energy obeys Schwarz's theorem, which reveals its generality in classical closed near-equilibrium systems.