Fast-slow chemical reactions: convergence of Hamilton-Jacobi equation and variational representation

TL;DR

This paper investigates the limiting behavior of Hamilton-Jacobi equations and variational representations in fast-slow chemical reactions, demonstrating how microscopic stochastic models converge to macroscopic deterministic dynamics on a slow manifold.

Contribution

It provides a rigorous analysis of the fast-slow limit for viscosity solutions of Hamilton-Jacobi equations with boundary conditions, linking microscopic stochastic models to macroscopic reaction rate equations.

Findings

Convergence of viscosity solutions to an effective Hamilton-Jacobi equation.

Identification of the effective Hamilton-Jacobi equation on the slow manifold.

Establishment of a variational representation for the limiting solutions.

Abstract

Microscopic behaviors of chemical reactions can be described by a random time-changed Poisson process, whose large-volume limit determines the macroscopic behaviors of species concentrations, including both typical and non-typical trajectories. When the reaction intensities (or fluxes) exhibit a separation of fast-slow scales, the macroscopic typical trajectory is governed by a system of $\varepsilon$-dependent nonlinear reaction rate equations (RRE), while the non-typical trajectories deviating from the typical ones are characterized by an $\varepsilon$-dependent exponentially nonlinear Hamilton-Jacobi equation (HJE). In this paper, for general chemical reactions, we study the fast-slow limit as $\varepsilon\to 0$ for the viscosity solutions of the associated HJE with a state-constrained boundary condition. We identify the limiting effective HJE on a slow manifold, along with an effective variational representation for the solution. Through the uniform convergence of the viscosity solutions and the $\Gamma$-convergence of the variational solution representations, we rigorously show that all non-typical (and also typical) trajectories are concentrated on the slow manifold and the effective macroscopic dynamics are described by the coarse-grained RRE and HJE, respectively. This approach for studying the fast-slow limit is applicable to, but not limited to, reversible chemical reactions described by gradient flows.