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.
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General theory
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
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
where Q is the configuration integral:
Calculation of the configuration integral
The configuration integral
Next, define the Mayer function
The calculation of the product in the above equation leads into a series of terms; the first is equal to one, the second term is equal to the sum over i and j of the terms
With this expansion it is possible to find terms of different order, in terms of the number of particles that are involved. The first term is the single-particle term, the second term corresponds to the two-particle interactions, the third to the three-particle interactions, and so on. This physical interpretation is the reason this expansion is called the cluster expansion: each term represents the interactions within clusters of a certain number of particles.
Substituting the expansion of the product back into the expression for the configuration integral results in a series expansion for
Substituting in the equation for the free energy, it is possible to derive the equation of state for the system of interacting particles. The equation will have the form
which is known as the Virial equation, and the components
This can be applied further to mixtures of gases and liquid solutions.