The variational formulation of the Fokker-Planck equation
SIAM Journal on Mathematical Analysis
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We study the limit of high activation energy of a special Fokker-Planck equation known as the Kramers-Smoluchowski equation (KS). This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential $H/\varepsilon$. We choose $H$ having two wells corresponding to two chemical states $A$ and $B$. We prove that after a suitable rescaling the solution to the KS converges, in the limit of high activation energy ($\varepsilon\to0$), to the solution of a simpler system modeling the spatial diffusion of $A$ and $B$ combined with the reaction $A\rightleftharpoons B$. With this result we give a rigorous proof of Kramers's formal derivation, and we show how chemical reactions and diffusion processes can be embedded in a common framework. This allows one to derive a chemical reaction as a singular limit of a diffusion process, thus establishing a connection between two worlds often regarded as separate. The proof rests on two main ingredients. One is the formulation of the two disparate equations as evolution equations for measures. The second is a variational formulation of both equations that allows us to use the tools of variational calculus and, specifically, $\Gamma$-convergence.