Mayer expansion

In statistical mechanics, the cluster expansion (also called the high temperature expansion or hopping expansion) is a power series expansion of the partition function of a statistical field theory around a model that is a union of non-interacting 0-dimensional field theories. Cluster expansions originated in the work of Mayer & Montroll (1941). Unlike the usual perturbation expansion, it converges in some non-trivial regions, in particular when the interaction is small.

In statistical mechanics, the properties of a system of noninteracting particles are described using the partition function. For N noninteracting particles, the system is described by the Hamiltonian

and the partition function can be calculated (for the classical case) as

From the partition function, one can calculate the Helmholtz free energy ${\displaystyle {\big .}F_{0}=-k_{B}T\ln Z_{0}}$ and, from that, all the thermodynamic properties of the system, like the entropy, the internal energy, the chemical potential etc.

When the particles of the system interact, an exact calculation of the partition function is usually not possible. For low density, the interactions can be approximated with a sum of two-particle potentials:

For this interaction potential, the partition function can be written as

and the free energy is

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