Mayer expansion (construction)

Graph expansion of the grand partition function of a classical interacting gas in powers of activity, using the Mayer f-bond.

Mayer expansion (construction)

The Mayer expansion is a classical, explicit instance of a cluster expansion for continuum gases. It expands the grand partition function in powers of the activity $z$ and expresses $\log \Xi$ (hence the pressure) as a sum over connected graphs built from the Mayer f-function.

Setup: grand canonical gas with pair interactions

Consider a classical gas in a bounded region $\Lambda\subset\mathbb{R}^d$ at inverse temperature $\beta$ (see inverse temperature $\beta$ ), described in the grand canonical ensemble . For a pair potential $\phi(x)$ , the total interaction energy for $N$ particles at positions $x_1,\dots,x_N$ is

U_N(x_1,\dots,x_N)=\sum_{1\le i<j\le N}\phi(x_i-x_j).

The grand partition function is

\Xi_\Lambda(z,\beta) ={} \sum_{N\ge 0}\frac{z^N}{N!}\int_{\Lambda^N} \exp\!\big(-\beta\,U_N(x_1,\dots,x_N)\big)\,dx_1\cdots dx_N,

where the configuration integral is an ordinary (Lebesgue) multiple integral; the underlying measure-theoretic viewpoint is the Lebesgue integral .

Mayer f-bond and graph expansion

Define the Mayer f-function (or f-bond)

f(x)=e^{-\beta \phi(x)}-1.

Since

e^{-\beta U_N}=\prod_{1\le i<j\le N} e^{-\beta\phi(x_i-x_j)} =\prod_{1\le i<j\le N}\big(1+f(x_i-x_j)\big),

expanding the product produces a sum over graphs $G$ on vertex set $\{1,\dots,N\}$ , where each edge $(i,j)$ contributes a factor $f(x_i-x_j)$ .

The key structural fact is that $\log \Xi_\Lambda$ is obtained by summing only connected graphs. One convenient way to write it is

\log \Xi_\Lambda(z,\beta) ={} \sum_{n\ge 1}\frac{z^n}{n!}\int_{\Lambda^n} \Phi_n^T(x_1,\dots,x_n)\,dx_1\cdots dx_n,

with the connected (truncated) kernel

\Phi_n^T(x_1,\dots,x_n) ={} \sum_{\substack{G\ \text{connected}\\\text{on }\{1,\dots,n\}}} \ \prod_{(i,j)\in E(G)} f(x_i-x_j).

Pressure and cluster integrals

The pressure is recovered from the grand partition function using pressure from the partition function . Formally,

\beta\,p(z,\beta) ={} \lim_{|\Lambda|\to\infty}\frac{1}{|\Lambda|}\log \Xi_\Lambda(z,\beta) ={} \sum_{n\ge 1} b_n(\beta)\, z^n,

where $b_n(\beta)$ are the cluster integrals. The first few have simple forms under translation invariance; for example,

b_1(\beta)=1, \qquad b_2(\beta)=\frac12\int_{\mathbb{R}^d} f(x)\,dx \quad (\text{when the integral exists}).

The number density satisfies

\rho(z,\beta)= z\,\frac{\partial}{\partial z}\big(\beta p(z,\beta)\big) =\sum_{n\ge 1} n\,b_n(\beta)\,z^n,

which is the starting point for virial expansions (pressure as a series in $\rho$ ) at low density.

Convergence regime and interpretation

For physically reasonable pair potentials (typically stable and tempered) the Mayer series converges for sufficiently small activity $z$ (low density) and/or high temperature. In that regime, the connected-graph structure makes $\log\Xi_\Lambda$ extensive and compatible with the thermodynamic limit .

Physically, the Mayer expansion reorganizes the grand canonical sum into contributions from interacting clusters of particles. Disconnected groups factorize in $\Xi_\Lambda$ and cancel when taking the logarithm, leaving only genuinely collective (connected) interaction effects.