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 LS ’ 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 S,,(t) tends to its new equilibrium value in general faster than the expectation values of the macroscopic variables &o their new equilibrium values. We then investigate the dependence of S&t) on the size of the phase cells. For fixed t, we find a lower bound on L&(t) by going to the limit of infinite accuracy of the measuring instruments. In the limit t--f co, this lower bound on SC,(t) also converges to the equilibrium entropy of a. These properties strongly support the opinion that S,,(t) is a proper microscopic expression for the entropy for equilibrium and nonequilibrium. Finally, explicit calculations of f&(t) for the model of a point particle en-closed in a one-dimensional box are presented which confirm the general results. 1