Cluster Expansion
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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 be ...
, the cluster expansion (also called the high temperature expansion or hopping expansion) is a
power series expansion In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a cons ...
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 In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
, 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 : \big. H_0=\sum_i^N \frac, and the partition function can be calculated (for the classical case) as :\big. Z_0 =\frac\int \prod_i d\vec_i\;d\vec_i \exp\left\ =\frac\left( \frac \right)^. 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 In thermodynamics, an isotherma ...
\big. F_0=-k_BT\ln Z_0 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 thermodynam ...
, 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 species ...
, 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: :\big. U\left( \ \right) = \sum_^N u_2(, \vec_i-\vec_j, ) = \sum_^N u_2(r_). For this interaction potential, the partition function can be written as :\big. Z =Z_0 \ Q , and the free energy is :F=F_0 - k_BT\!\ln\left( Q \right) , where Q is the
configuration integral In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregat ...
: : Q=\frac\int \prod_i d\vec_i\exp\left\.


Calculation of the configuration integral

The configuration integral Q cannot be calculated analytically for a general pair potential u_2(r). 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 Q can be written as a product of the form : \exp\left\=\prod_\exp\left\ . Next, define the
Mayer function The Mayer f-function is an auxiliary function that often appears in the series expansion of thermodynamic quantities related to classical many-particle systems.Donald Allan McQuarrie, ''Statistical Mechanics'' (HarperCollins, 1976), page 228 It is ...
f_ by \exp\left\=1+f_. After substitution, the equation for the configuration integral becomes: :\big. Q=\frac\int \prod_i d\vec_i \prod_ \left(1+f_\right) 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 f_, and the process continues until all the higher order terms are calculated. : \prod_ \left(1+f_\right)= 1+ \sum_\; f_ +\sum_^N \;f_\;f_+\cdots 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 Q: :\big. Q=1+\frac\alpha_1 + \frac\alpha_2+\cdots. 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 internal ...
for the system of interacting particles. The equation will have the form : PV=Nk_BT\left( 1 + \fracB_2(T) + \fracB_3(T) + \fracB_4(T)+ \cdots \right) , which is known as the
virial equation 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 internal ...
, and the components B_i(T) are the
virial coefficients Virial coefficients B_i appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potenti ...
. Each of the virial coefficients corresponds to one term from the cluster expansion (B_2(T) is the two-particle interaction term, B_3(T) 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 In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for ...
. 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