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Mayer cluster integrals

Connected-graph coefficients in the activity expansion of the grand potential (log grand partition function) for a classical gas; generate virial coefficients and low-density thermodynamics.
Mayer cluster integrals

Mayer’s cluster expansion rewrites the interacting Boltzmann weight in terms of “ff-bonds” and organizes the resulting series by graphs. The coefficients of the connected graphs are the Mayer cluster integrals.

They are the standard bridge between microscopic interactions and macroscopic expansions such as .

Grand partition function and the Mayer ff-bond

For a classical gas in a volume VV at inverse temperature β\beta, with activity zz and pair potential u(⋅)u(\cdot), the grand partition function is

Ξ(z,V,β)  =  ∑N=0∞zNN!∫VNexp⁡ ⁣(−βUN(x1,…,xN)) dx1⋯dxN, \Xi(z,V,\beta) \;=\; \sum_{N=0}^{\infty}\frac{z^N}{N!} \int_{V^N} \exp\!\bigl(-\beta U_N(x_1,\dots,x_N)\bigr)\, dx_1\cdots dx_N,

where for pair interactions

UN(x1,…,xN)  =  ∑1≤i<j≤Nu(xi−xj). U_N(x_1,\dots,x_N) \;=\; \sum_{1\le i<j\le N} u(x_i-x_j).

Define the Mayer ff-bond

fij  =  e−βu(xi−xj)−1. f_{ij} \;=\; e^{-\beta u(x_i-x_j)} - 1.

Then

e−βUN  =  ∏i<je−βu(xi−xj)  =  ∏i<j(1+fij), e^{-\beta U_N} \;=\; \prod_{i<j} e^{-\beta u(x_i-x_j)} \;=\; \prod_{i<j}\bigl(1+f_{ij}\bigr),

and expanding the product generates a sum over graphs on {1,…,N}\{1,\dots,N\}, with an edge {i,j}\{i,j\} contributing a factor fijf_{ij}.

Connected graphs and log⁡Ξ\log \Xi

The logarithm of the grand partition function selects connected contributions. A standard form of Mayer’s identity is

log⁡Ξ(z,V,β)=∑n≥1znn!∫Vn(∑G∈Cn∏{i,j}∈E(G)fij) dx1⋯dxn, \log \Xi(z,V,\beta) ={} \sum_{n\ge 1}\frac{z^n}{n!} \int_{V^n} \left( \sum_{G\in \mathcal{C}_n} \prod_{\{i,j\}\in E(G)} f_{ij} \right)\, dx_1\cdots dx_n,

where:

  • Cn\mathcal{C}_n is the set of connected graphs on nn labeled vertices,
  • E(G)E(G) is the edge set of the graph GG.

This is the origin of the “cluster” terminology: connected clusters of particles contribute to log⁡Ξ\log \Xi.

Definition of Mayer cluster integrals

In the thermodynamic limit (when it exists), one defines coefficients bn(β)b_n(\beta) by the activity expansion of the pressure:

1Vlog⁡Ξ(z,V,β)  =  ∑n≥1bn(β) zn. \frac{1}{V}\log \Xi(z,V,\beta) \;=\; \sum_{n\ge 1} b_n(\beta)\, z^n.

These bnb_n are the Mayer cluster integrals (up to common convention-dependent prefactors, e.g. thermal wavelength factors in continuum gases).

In this normalization, the pressure and density follow from derivatives of log⁡Ξ\log \Xi:

βp  =  1Vlog⁡Ξ  =  ∑n≥1bnzn, \beta p \;=\; \frac{1}{V}\log \Xi \;=\; \sum_{n\ge 1} b_n z^n,

and

ρ  =  z ∂z ⁣(1Vlog⁡Ξ)  =  ∑n≥1n bnzn. \rho \;=\; z\,\partial_z\!\left(\frac{1}{V}\log \Xi\right) \;=\; \sum_{n\ge 1} n\, b_n z^n.

This “log-partition per volume” viewpoint is the same object used in .

From cluster integrals to virial coefficients

Eliminating zz between βp(z)\beta p(z) and ρ(z)\rho(z) yields the virial expansion

βp  =  ρ+B2ρ2+B3ρ3+⋯ \beta p \;=\; \rho + B_2\rho^2 + B_3\rho^3 + \cdots

with coefficients BnB_n that are polynomials in the bnb_n (see ).

In the common convention where the ideal term is normalized so that b1=1b_1=1, the first relations are

B2=−b2,B3=4b22−2b3. B_2 = -b_2, \qquad B_3 = 4b_2^2 - 2b_3.

Convergence and cluster expansion theorems

Rigorous results typically prove that the series for log⁡Ξ\log \Xi converges for sufficiently small ∣z∣|z| under stability and regularity assumptions on the interaction. This is the content of , and it underlies analytic control of .

Prerequisites