Boltzmann Entropy for Dense Fluids Not in Local Equilibrium

Using computer simulations, we investigate the time evolution of the (Boltzmann) entropy of a dense fluid not in local equilibrium. The macrovariables M describing the system are the (empirical) particle density f={f(x̲ ,v̲ )} and the total energy E. We find that S(f_t,E) is a monotone increasing in time even when its kinetic part is decreasing. We argue that for isolated Hamiltonian systems monotonicity of S(M_t)=S(M_Xt) should hold generally for “typical” (the overwhelming majority of) initial microstates (phase points) X₀ belonging to the initial macrostate M₀, satisfying M_X0=M₀. This is a consequence of Liouville’s theorem when M_t evolves according to an autonomous deterministic law.