  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 $\rho^*$ is determined by the
stationary equation $\delta F[\rho]/\delta \rho|_{\rho=\rho^*}=\mu$ 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.
