Cost of s-fold Decisions in Exact Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac Statistics

TL;DR

This paper examines the exact entropy functions for Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics without Stirling approximation, analyzing the energy cost of s-fold decisions and resolving quantization paradoxes.

Contribution

It provides exact entropy forms and quantifies the energy cost of s-fold decisions, extending previous work by removing Stirling approximation and clarifying quantization effects.

Findings

Exact entropy functions derived without Stirling approximation

Energy cost of s-fold decisions depends on system knowledge

Quantization paradox resolved using Laplace-Jaynes interpretation

Abstract

The exact forms of the degenerate Maxwell-Boltzmann (MB), Bose-Einstein (BE) and Fermi-Dirac (FD) entropy functions, derived by Boltzmann's principle without the Stirling approximation (Niven, Physics Letters A, 342(4) (2005) 286), are further examined. Firstly, an apparent paradox in quantisation effects is resolved using the Laplace-Jaynes interpretation of probability. The energy cost of learning that a system, distributed over s equiprobable states, is in one such state (an s-fold decision) is then calculated for each statistic. The analysis confirms that the cost depends on one's knowledge of the number of entities N and (for BE and FD statistics) the degeneracy, extending the findings of Niven (2005).