Microscopic and macroscopic entropy.

Abstract

We consider physical systems in which microstates can be represented as symbol sequences. The internal correlations in a microstate give restrictions to its ``randomness,'' quantified by the measure entropy of that microstate. It is shown that, in the thermodynamic limit, the measure entropy for the microstates has an ensemble average equal to the thermodynamic entropy. A simple explanation is that the correlations necessary to generate the whole ensemble can be found in almost any one of its microstates. If microscopic phases are present, there are subensembles, each given by the statistics in almost any one of its microstates. The typical microstate in an equilibrium ensemble can be found by maximization of the measure entropy under energy constraints and general constraints on probability distributions for symbol sequences. The typical microstate does not contain information in correlations of lengths greater than interaction distance, a fact which simplifies the calculation of the entropy for systems with finite interaction length. We exemplify this by deriving the entropy for a monatomic ideal gas and the one-dimensional Ising model with an external field.