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

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 $\delta F[\rho ]\mathrm{/}\delta \rho {\mathrm{\mid }}_{\rho ={\rho }^{\ast }}=\mu$ 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.