Stochastic dynamics and thermodynamics around a metastable state based on the linear Dean-Kawasaki equation

Abstract

The Dean-Kawasaki equation forms the basis of the stochastic density functional theory (DFT). Here it is demonstrated that the Dean-Kawasaki equation can be directly linearized in the first approximation of the driving force due to the free energy functional $F[\rho ]$ of an instantaneous density distribution $\rho$, when we consider small density fluctuations around a metastable state whose density distribution ${\rho }^{\ast }$ is determined by the stationary equation $\delta F[\rho ]/\delta \rho {|}_{\rho ={\rho }^{\ast }}=\mu$ with $\mu$ denoting the chemical potential. Our main results regarding the linear Dean-Kawasaki equation are threefold. First, (i) the corresponding stochastic thermodynamics has been formulated, showing that the heat dissipated into the reservoir is negligible on average. Next, (ii) we have developed a field theoretic treatment combined with the equilibrium DFT, giving an approximate form of $F[\rho ]$ that is related to the equilibrium free energy functional. Accordingly, (iii) the linear Dean-Kawasaki equation, which has been reduced to a tractable form expressed by the direct correlation function, allows us to compare the stochastic dynamics around metastable and equilibrium states, particularly in the Percus-Yevick hard sphere fluids; we have found that the metastable density is larger and the effective diffusion constant in the metastable state is smaller than the equilibrium ones in repulsive fluids.

accepted for publication in J. Phys. A

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