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Mean-field limit and large deviation for chemical reaction kinetics from Hamiltonian viewpoint and upwind scheme

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FKTW05 - Frontiers in numerical analysis of kinetic equations

Chemical reactions can be modeled by random time-changed Poisson processes. To characterize macroscopic behaviors in the large volume limit, the law of large numbers in the path space determines a mean-field limit nonlinear reaction rate equation describing the dynamics of the concentration of species, while the WKB expansion for the chemical master equation yields a Hamilton-Jacobi equation (HJE) and the Lagrangian gives the good rate function in the large deviation principle. By regarding chemical master equation as an upwind scheme, whose structure is preserved in HJE , we give another proof for the mean-field limit reaction rate equation.  We decompose the mean-field reaction rate equation into a conservative part and a dissipative part in terms of the stationary solution to HJE . This stationary solution is used to determine the energy landscape and thermodynamics for general chemical reactions. The associated energy dissipation law at both the mesoscopic and macroscopic levels is proved together with a passage from the mesoscopic to macroscopic one. A non-convex energy landscape emerges from the convex mesoscopic relative entropy functional in the large volume limit, which picks up the non-equilibrium features.  Furthermore, we use a reversible Hamiltonian to study a class of non-equilibrium enzyme reactions, which reduces the conservative-dissipative decomposition to an Onsager-type strong gradient flow, and a modified time reversed least action path serves as the transition paths between multiple non-equilibrium steady states with associated path affinities. In chemical reaction detailed balance case,  the comparison principle for HJE prevents a class of initial distribution converging to the equilibrium. This is a joint work with Yuan Gao at Purdue University.

This talk is part of the Isaac Newton Institute Seminar Series series.

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