On the time and cell dependence of the coarse-grained entropy. I

We consider a finite, thermally isolated, classical system which passes from an equilibrium state A by the removal of an internal constraint to another equilibrium state B after an empirical relaxation time. In the phase space of the system, cells are introduced according to the set of measuring instruments used and their experimental inaccuracies. It is shown that the coarse-grained entropy Scg(t) tends to its new equilibrium value in general faster than the expectation values of the macroscopic variables to their new equilibrium values. We then investigate the dependence of Scg(t) on the size of the phase cells. For fixed t, we find a lower bound on Scg(t) by doing to the limit of infinite accuracy of the measuring instruments. In the limit t → ∞, this lower bound on Scg(t) also converges to the equilibrium entropy of B. These properties strongly support the opinion that Scg(t) is a proper microscopic expression for the entropy for equilibrium and nonequilibrium. Finally, explicit calculations of Scg(t) for the model of a point particle enclosed in a one-dimensional box are presented which confirm the general results.