Gibbs entropy

In classical statistical mechanics, the entropy function earlier introduced by Rudolf Clausius is interpreted as statistical entropy using probability theory. The statistical entropy perspective was introduced in 1870 with the work of physicist Ludwig Boltzmann.

The macroscopic state of a system is defined by a distribution on the microstates. The entropy of this distribution is given by the Gibbs entropy formula, named after J. Willard Gibbs. For a classical system (i.e., a collection of classical particles) with a discrete set of microstates, if ${\displaystyle E_{i}}$ is the energy of microstate i, and ${\displaystyle p_{i}}$ is the probability that it occurs during the system's fluctuations, then the entropy of the system is

Entropy changes for systems in a canonical state

A system with a well-defined temperature, i.e., one in thermal equilibrium with a thermal reservoir, has a probability of being in a microstate i given by Boltzmann's distribution.

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