Microscopic reversibility and macroscopic irreversibility: A lattice gas model

We present coarse-grained descriptions and computations of the time evolution of a lattice gas system of indistinguishable particles, whose microscopic laws of motion are exactly reversible, in order to investigate how or what kind of macroscopically irreversible behavior may eventually arise. With increasing coarse-graining and number of particles, relative fluctuations of entropy rapidly decrease and apparently irreversible behavior unfolds. Although that behavior becomes typical in those limits and within a certain range, it is never absolutely irreversible for any individual system with specific initial conditions. Irreversible behavior may arise in various ways. We illustrate one possibility by replacing detailed integer occupation numbers at lattice sites with particle probability densities that evolve diffusively.