Einstein-smoluchowski Equation
Magdalena Liendo
In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected[clarification needed] connection revealed independently by William Sutherland in 1904,[1][2][3] Albert Einstein in 1905,[4] and by Marian Smoluchowski in 1906[5] in their works on Brownian motion. The more general form of the equation in the classical case is[6]
A relation between the diffusion coefficient D and the distance λ that a particle can jump when diffusing in a time τ. The Einstein-Smoluchowski equation, which is D = λ2/2τ, gives a connection between the microscopic details of particle diffusion and the macroscopic quantities associated with the diffusion, such as the viscosity. The equation is derived by assuming that the particles undergo a random walk. The quantities in the equation can be related to quantities in the kinetic theory of gases, with λ/τ taken to be the mean speed of the particles and λ their mean free path. The Einstein-Smoluchowski equation was derived by Albert Einstein and the Polish physicist Marian Ritter von Smolan-Smoluchowski.
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Abstract: We study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy 17, 3205 (2015)]. We first neglect fluctuations and provide a macroscopic description of the system based on the deterministic mean field Smoluchowski equation. We then take fluctuations into account and provide a mesoscopic description of the system based on the stochastic mean field Smoluchowski equation. We establish the main properties of this equation and derive the Kramers escape rate formula, giving the lifetime of a metastable state, from the theory of instantons. We relate the properties of the generalized stochastic Smoluchowski equation to a principle of maximum dissipation of free energy. We also discuss the connection with the dynamical density functional theory of simple liquids. Keywords: Fokker-Planck equations; free energy; generalized thermodynamics; instantons; dynamical density functional theory
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