# Bogolyubov inequality

in statistical mechanics

Bogolyubov's inequality for the free-energy functional is an inequality that gives rise to a variational principle of statistical mechanics. The following inequality is valid for any Hermitian operators $U _ {1}$ and $U _ {2}$:

$$\tag{* } \frac{1}{N} \langle U _ {1} - U _ {2} \rangle _ {U _ {1} } \leq \ f[U _ {1} ]- f[U _ {2} ] \leq$$

$$\leq \ \frac{1}{N} \langle U _ {1} -U _ {2} \rangle _ {U _ {2} } ,$$

where

$$f[U] \equiv - { \frac \Theta {N} } \mathop{\rm ln} \mathop{\rm Tr} e ^ {-U/ \Theta } .$$

This expression has the meaning of the free-energy density for a system with Hamiltonian $U$; the extensive parameter $N$ is the number of particles or the volume, depending on the system; $\Theta$ is the absolute temperature in energy units, and

$$\langle \dots \rangle _ {U} \equiv \ \frac{ \mathop{\rm Tr} ( \dots e ^ {-U / \Theta } ) }{ \mathop{\rm Tr} e ^ {-U/ \Theta } }$$

denotes the thermodynamic average with respect to the Hamiltonian $U$.

The Bogolyubov inequality (*) is used to obtain the exact thermodynamic limit solutions of model problems in statistical quantum physics , , in studies using the method of molecular fields , in proving the existence of the thermodynamic limit, and also in order to obtain physically important estimates for the free energies of various multi-particle systems . There exist generalizations of the Bogolyubov inequality (*) to the case of von Neumann algebras with a "trace"  and general von Neumann algebras .

How to Cite This Entry:
Bogolyubov inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bogolyubov_inequality&oldid=46092
This article was adapted from an original article by A.M. Kurbatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article