Statistical mechanics relies on the complete although probabilistic descrip
tion of a system in terms of all its microscopic variables. Its object is t
o derive from this microscopic description the static and dynamic propertie
s for some reduced set of variables. The elimination of the irrelevant vari
ables 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 incom
plete 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 i
n irreversible thermodynamics, the projection operator method of statistica
l mechanics, Boltzmann's H-theorem, and spin-echo experiments. (C) 1999 Ame
rican Association of Physics Teachers.