Statistical mechanics, structure of matter Foundations of equilibrium statistical mechanics Problem

Gibbs paradox

Builds on Microcanonical ensemble Microstate Gibbs entropy

Problem

Count the microstates of a classical ideal gas of $N$ identical particles in volume $V$ by phase-space volume alone, without the factor $1/N!$ . The resulting microcanonical entropy is not extensive: doubling $N$ , $V$ and the energy does not double it. Equivalently, removing a partition between two equal samples of the same gas at the same temperature and pressure raises this entropy by $2 N k_B \ln 2$ . Yet nothing observable changes, and reinserting the partition restores the original state. The problem is to explain this contradiction and to find the correct count of microstates.

Remarks

The standard resolution divides the phase-space volume by $N!$ , treating permutations of identical particles as the same microstate. With this factor the entropy is extensive, the mixing of identical gases produces no entropy, and the result is the Sackur–Tetrode equation. For two different gases the mixing entropy $2 N k_B \ln 2$ is real, and it does not shrink as the gases become more similar; it drops to $0$ only when they are identical. Quantum mechanics justifies the factor through the indistinguishability of identical particles, while other authors argue that it already follows from classical thermodynamic considerations; the interpretation is still discussed.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Tag	`AVZ2R5C45M`
URI	`m2://AVZ2R5C45M`
Kind	problem
Writing	W0 stub
MSC	82B05, 82B30, 82D05
Status	This copy of the site has no build data. Run `m2 status AVZ2R5C45M`.
About	Microcanonical ensemble
Uses	Microcanonical ensemble, Microstate, Gibbs entropy
Changed	3 November 2026

Gibbs paradox

Problem

Remarks

Reading path

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