Reinterpreting Boltzmann's H-theorem in the light of Information Theory

TL;DR

This paper reinterprets Boltzmann's H-theorem through information theory, emphasizing the distinction between actual and limiting distributions in entropy calculations, supported by computer simulations of a hard-sphere fluid.

Contribution

It introduces a new perspective on the H-theorem by relating it to information theory and the concept of limiting distributions for entropy.

Findings

The Maxwellian distribution is a limiting distribution for large systems.

Entropy of a single particle is well-defined and independent of microstates.

Computer simulations support the interpretation of the H-theorem in terms of limiting distributions.

Abstract

Prompted by the realisation that the statistical entropy of an ideal gas in the micro-canonical ensemble should not fluctuate or change over time, the meaning of the H-theorem is re-interpreted from the perspective of information theory in which entropy is a measure of uncertainty. We propose that the Maxwellian velocity distribution should more properly be regarded as a limiting distribution which is identical with the distribution across particles in the asymptotic limit of large numbers. In smaller systems, the distribution across particles differs from the limiting distribution and fluctuates. Therefore the entropy can be calculated either from the actual distribution across the particles or from the limiting distribution. The former fluctuates with the distribution but the latter does not. However, only the latter represents uncertainty in the sense implied by information theory by accounting for all possible microstates. We also argue that the Maxwellian probability distribution for the velocity of a single particle should be regarded as a limiting distribution. Therefore the entropy of a single particle is well defined, as is the entropy of an N-particle system, regardless of the microstate. We argue that the meaning of the H-theorem is to reveal the underlying distribution in the limit of large numbers. Computer simulations of a hard-sphere fluid are used to demonstrate the ideas.