Incomplete descriptions and relevant entropies

Abstract

Statistical mechanics relies on the complete although probabilistic description of a system in terms of all its microscopic variables. Its object is to derive from this microscopic description the static and dynamic properties for some reduced set of variables. The elimination of the irrelevant variables is guided by the maximum entropy criterion, which produces the least biased probability law consistent with the available information about the relevant variables. This approach defines relevant entropies which measure the missing information associated with the variables retained in the incomplete description. The relevant entropies depend not only on the state, but also on the coarseness of the reduced description of the system. Their use sheds light on questions such as the second law, both in equilibrium and in irreversible thermodynamics, the projection operator method of statistical mechanics, Boltzmann’s H-theorem, and spin-echo experiments.