Statistical mechanics, structure of matter Definition

Gibbs entropy

Builds on Statistical ensemble Probability distribution

Definition

Let a statistical ensemble be a probability distribution $(p_i)_{i \in I}$ on a countable set $I$ of microstates. Let $k_B$ be the Boltzmann constant. The Gibbs entropy of the ensemble is $S = -k_B \sum_{i \in I} p_i \ln p_i,$ with the convention $0 \ln 0 = 0$ ; the sum lies in $[0, \infty]$ . For a classical ensemble with density $\rho$ on phase space, one sets $S = -k_B \int \rho \ln (h^{3N} \rho) \, d\Gamma$ , where $h^{3N}$ is a fixed unit of phase-space volume.

Remarks

Example: the uniform distribution on $W$ states has $p_i = 1/W$ , so $S = -k_B \sum_{i=1}^{W} \frac{1}{W} \ln \frac{1}{W} = k_B \ln W$ . A distribution concentrated on one state has $S = 0$ . Non-example: the weights $q_i = 1/2$ on three states sum to $3/2$ , so they are not a probability distribution and their Gibbs entropy is not defined. For a quantum ensemble with density operator $\rho$ , the analogue is the von Neumann entropy $-k_B \tr(\rho \ln \rho)$ . The continuous version depends on the choice of the unit $h$ and can be negative, unlike the discrete one.

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Tag	`XZTM94H0BE`
URI	`m2://XZTM94H0BE`
Kind	definition
Writing	W0 stub
MSC	82B03, 94A17, 82B30
Defines	Gibbs entropy
Uses	Statistical ensemble, Probability distribution
Changed	3 November 2026

Gibbs entropy

Definition

Remarks

Reading path

Further reading

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