Thermodynamic limit, large deviation and global energy landscape for non-equilibrium chemical reactions

Chemical reactions can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors such as the mean-field equation and the large deviation rate function can be studied via the WKB reformulation. The WKB reformulation for the backward equation is Varadhan's discrete nonlinear semigroup and is also a monotone scheme which approximates the limiting first order Hamiltonian-Jacobi equations(HJE). The discrete Hamiltonian is a m-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well posedness of chemical master equation. The convergence from the monotone schemes to the viscosity solution of HJE is proved via constructing barriers to overcome the polynomial growth coefficient in Hamiltonian. This convergence of Varadhan's discrete nonlinear semigroup to the continuous Lax-Oleinik semigroup yields the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered. Moreover, the LDP for invariant measures can be used to construct the global energy landscape for non-equilibrium reactions. It is also proved to be a selected unique weak KAM solution to the corresponding stationary HJE. Based on this stationary solution, a decomposition into Hamiltonian flow and gradient flow for general mean-field reaction rate equation will also be discussed.