In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
, 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 . 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.
Classical case
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
In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz en ...
and, from that, all the thermodynamic properties of the system, like the
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
, the internal energy, the
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...
, 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:
:
Calculation of 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 by
. After substitution, the equation for the configuration integral becomes:
:
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
, and the process continues until all the higher order terms are calculated.
:
Each term must appear only once. 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 non-interaction term (corresponding to no interactions amongst particles), the second term corresponds to the two-particle interactions, the third to the two-particle interactions amongst 4 (not necessarily distinct) particles, and so on. 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.
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
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...
for the system of interacting particles. The equation will have the form
:
,
which is known as the
virial equation, and the components
are the
virial coefficients.
Each of the virial coefficients corresponds to one term from the cluster expansion (
is the two-particle interaction term,
is the three-particle interaction term and so on).
Keeping only the two-particle interaction term, it can be shown that the cluster expansion, with some approximations, gives the
Van der Waals equation.
This can be applied further to mixtures of gases and liquid solutions.
References
*
*
*
*
* , chapter 9.
*
*
*{{cite book , last1=Friedli , first=S. , last2=Velenik , first2=Y. , title=Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction , publisher=Cambridge University Press , year=2017 , isbn=9781107184824 , url=http://www.unige.ch/math/folks/velenik/smbook/index.html
Statistical mechanics