Boltzmann H-theorem

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One of the consequences of the kinetic Boltzmann equation, according to which the function

$$H(t)\equiv\int h(t,\mathbf r)\,d\mathbf r=\int f\ln f\,d\mathbf p\,d\mathbf r,$$

is a non-increasing function of the time $t$. Here $f=f(t,\mathbf r,\mathbf p)$ is the dimensionless classical single-particle distribution function of the coordinates $\mathbf r$ and the momenta $\mathbf p$, which satisfies the Boltzmann equation. The evolution of the density $h(t,\mathbf r)$ with time is determined by the relaxational nature of the evolution of the function $f$ towards the local Maxwell distribution, while the limit value of the $H$-function as $t\to\infty$ is equal to the entropy of the ideal gas, calculated according to Gibbs, with the opposite sign. If the increments of $t$ under consideration are much larger than the time needed for the local Maxwell distribution to be established, the quantity $(-h(t,\mathbf r))$ may be identified with the entropy density, while $(-H(t))$ may be identified with the non-equilibrium entropy of an ideal gas.

From the point of view of statistical mechanics, the principal significance of the Boltzmann $H$-theorem consists in the mathematical expression of the fundamental assumptions of macroscopic thermodynamics, according to which, for example, an isolated system spontaneously tends to the state of thermodynamic equilibrium, the process being accompanied by an increase in entropy.

$H$-theorems are statements analogous to the original Boltzmann $H$-theorem, but formulated for statistical systems of a different or a more general type, including the case of non-ideal and quantified systems.

The theory was presented by L. Boltzmann in 1872.


[1] A. Sommerfeld, "Thermodynamics and statistical mechanics" , Acad. Press (1956) (Translated from German)
[2] G.E. Uhlenbeck, G.V. Ford, "Lectures in statistical mechanics" , Amer. Math. Soc. (1963)
How to Cite This Entry:
Boltzmann H-theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.A. Kvasnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article