Unlike the usual perturbation expansion which usually leads to a divergent asymptotic series, the cluster expansion may converge within a non-trivial region, in particular when the interaction is small and short-ranged.
The cluster expansion coefficients are calculated by intricate combinatorial counting.
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
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 where Q is the configuration integral: The configuration integral
cannot be calculated analytically for a general pair potential
One way to calculate the potential approximately is to use the Mayer cluster expansion.
This expansion is based on the observation that the exponential in the equation for
can be written as a product of the form Next, define the Mayer function
, and the process continues until all the higher order terms are calculated.
This physical interpretation is the reason this expansion is called the cluster expansion: the sum can be rearranged so that each term represents the interactions within clusters of a certain number of particles.
Each of the virial coefficients corresponds to one term from the cluster expansion (
Keeping only the two-particle interaction term, it can be shown that the cluster expansion, with some approximations, gives the Van der Waals equation.