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Analytical Mechanics in Stochastic Dynamics: Most Probable Path, Large-Deviation Rate Function and Hamilton-Jacobi Equation

Abstract

Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Langrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation dXt=b(Xt)dt+Ο΅dWtdX_t=b(X_t)dt+\epsilon dW_t where WtW_t is a Brownian motion. In the limit of vanishingly small Ο΅\epsilon, the solution to the stochastic differential equation other than xΛ™=b(x)\dot{x}=b(x) are all rare events. However, conditioned on an occurence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with L=βˆ₯qΛ™βˆ’b(q)βˆ₯2/4\mathcal{L}=\|\dot{q}-b(q)\|^2/4 and Hamiltonian equations with H(p,q)=βˆ₯pβˆ₯2+b(q)β‹…pH(p,q)=\|p\|^2+b(q)\cdot p. Hamiltonian conservation law implies that the most probable trajectory for a "rare" event has a uniform "excess kinetic energy" along its path. Rare events can also be characterized by the principle of large deviations which expresses the probability density function for XtX_t as f(x,t)=eβˆ’u(x,t)/Ο΅f(x,t)=e^{-u(x,t)/\epsilon}, where u(x,t)u(x,t) is called a large-deviation rate function which satisfies the corresponding Hamilton-Jacobi equation. An irreversible diffusion process with βˆ‡Γ—bβ‰ 0\nabla\times b\neq 0 corresponds to a Newtonian system with a Lorentz force qΒ¨=(βˆ‡Γ—b)Γ—qΛ™+1/2βˆ‡βˆ₯bβˆ₯2\ddot{q}=(\nabla\times b)\times \dot{q}+1/2\nabla\|b\|^2. The connection between stochastic motion and analytical mechanics can be explored in terms of various techniques of applied mathematics, for example, singular perturbations, viscosity solutions, and integrable systems.Comment: 23 pages, 2 figure

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