Virial decomposition

From Encyclopedia of Mathematics
Revision as of 17:17, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

virial series

The series on the right-hand side of the equation of state of a gas:

where is the pressure, is the temperature, is the specific volume, and is the Boltzmann constant. The term of the series which contains the -th virial coefficient describes the deviation of the gas from ideal behaviour due to the interaction in groups of molecules. can be expressed in terms of irreducible repeated integrals :

summed over all natural numbers , , subject to the condition

In particular,


is the volume of the gas, the integration extends over the total volume occupied by the gas, and is the interaction potential. There is a rule for writing down for any in terms of . The expression obtained after simplification is:

In practice, only the first few virial coefficients can be calculated.

Power series in , with coefficients expressed in terms of , can be used to represent equilibrium correlation functions for particles; a corollary of this fact is that the equation of state can be obtained in a simple manner [3].

There exists a quantum-mechanical analogue of the virial decomposition.


[1] J.E. Mayer, M. Goeppert-Mayer, "Statistical mechanics" , Wiley (1940)
[2] R. Feynman, "Statistical mechanics" , M.I.T. (1972)
[3] N.N. Bogolyubov, "Problems of a dynamical theory in statistical physics" , North-Holland (1962) (Translated from Russian)
[4] G.E. Uhlenbeck, G.V. Ford, "Lectures in statistical mechanics" , Amer. Math. Soc. (1963)
How to Cite This Entry:
Virial decomposition. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.P. Pavlotskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article