Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classical Resolution of the Gibbs Paradox from the Equal Probability Principle: An Informational Perspective

Zheng Zhang

TL;DR

The paper tackles the Gibbs paradox in classical statistical mechanics by showing that, under the equal probability principle, entropy changes during gas mixing can be understood as information rather than a paradox. It identifies the Gibbs entropy with Shannon entropy $S=-k\int \rho \ln \rho \, dpdq$ and decomposes total information into partition ignorance $S_d$ and kinetic-state ignorance $S_A,S_B$, revealing why additivity fails. For identical gases, partition ignorance cancels upon mixing, yielding $\Delta S=0$, while for different gases a finite $\Delta S$ arises, consistent with thermodynamics, all without invoking a $1/N!$ correction. The approach ties information to extractable work via $W \le -kT\Delta I$ and positions information as a physical resource, suggesting a paradigm shift in statistical mechanics where entropy and information are deeply linked.

Abstract

The Gibbs paradox is a conventional paradox in classical statistical mechanics, typically resolved by invoking quantum indistinguishability through the 1/N! correction. In this letter, we present a resolution within classical ensemble theory, which relies solely on the equal probability principle and does not invoke the 1/N! correction. Our resolution can be naturally interpretated from a purely informational perspective, where the Gibbs entropy is explicitly regarded as the Shannon entropy, quantifying ignorance rather than disorder. From this informational perspective, we also clarify the connection between information and extractable work in the gas mixing processes. Our work opens a new avenue to reconsider the role of information in statistical mechanics.

Classical Resolution of the Gibbs Paradox from the Equal Probability Principle: An Informational Perspective

TL;DR

The paper tackles the Gibbs paradox in classical statistical mechanics by showing that, under the equal probability principle, entropy changes during gas mixing can be understood as information rather than a paradox. It identifies the Gibbs entropy with Shannon entropy and decomposes total information into partition ignorance and kinetic-state ignorance , revealing why additivity fails. For identical gases, partition ignorance cancels upon mixing, yielding , while for different gases a finite arises, consistent with thermodynamics, all without invoking a correction. The approach ties information to extractable work via and positions information as a physical resource, suggesting a paradigm shift in statistical mechanics where entropy and information are deeply linked.

Abstract

The Gibbs paradox is a conventional paradox in classical statistical mechanics, typically resolved by invoking quantum indistinguishability through the 1/N! correction. In this letter, we present a resolution within classical ensemble theory, which relies solely on the equal probability principle and does not invoke the 1/N! correction. Our resolution can be naturally interpretated from a purely informational perspective, where the Gibbs entropy is explicitly regarded as the Shannon entropy, quantifying ignorance rather than disorder. From this informational perspective, we also clarify the connection between information and extractable work in the gas mixing processes. Our work opens a new avenue to reconsider the role of information in statistical mechanics.
Paper Structure (8 sections, 35 equations, 1 figure)

This paper contains 8 sections, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of the Gibbs paradox
  3. The Gibbs paradox
  4. Traditional resolution
  5. Resolution based on the equal probability principle
  6. Informational perspective interpretation
  7. Information and work
  8. Discussion and conclusion

Figures (1)

  • Figure 1: Information differences among four scenarios in the Gibbs paradox. Colored circles means different types of molecules. Compared to scenario (c), scenario (a) has additional $2N\ln2$ nats information regarding the side to which each particle belongs. From (a) to (b), the wall is removed, the $2N\ln2$ nats information is lost, resulting a mixing entropy $2kN\ln2$. Scenarios (b)(c) and (d) has no information difference (or entropy difference), since we are equally ignorant about the positions of particles in these scenarios.