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Convergence of the Virial Expansion

Sufficient conditions (via cluster/Mayer expansions) guaranteeing analyticity of the pressure and a convergent virial series at low density (or small activity) for a classical gas.
Convergence of the Virial Expansion

Prerequisites

Setting and expansions

Consider a classical continuum gas in a bounded region Λ⊂Rd\Lambda\subset\mathbb{R}^d at inverse temperature β>0\beta>0 with a translation-invariant pair potential ϕ(x)\phi(x). The grand-canonical partition function is

ΞΛ(z,β)=∑N=0∞zNN!∫ΛNexp⁡ ⁣(−β∑1≤i<j≤Nϕ(xi−xj)) dx1⋯dxN, \Xi_\Lambda(z,\beta) ={} \sum_{N=0}^\infty \frac{z^N}{N!} \int_{\Lambda^N} \exp\!\Big( -\beta \sum_{1\le i<j\le N}\phi(x_i-x_j) \Big)\,dx_1\cdots dx_N,

where zz is the activity.

The finite-volume pressure (in units of β\beta) is

βpΛ(z)=1∣Λ∣log⁡ΞΛ(z,β), \beta p_\Lambda(z)=\frac{1}{|\Lambda|}\log \Xi_\Lambda(z,\beta),

and the thermodynamic pressure is the limit defining .

Introduce the Mayer function

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

Under suitable assumptions, one has an absolutely convergent cluster (Mayer) expansion

βp(z)=∑n≥1bn(β) zn,ρ(z)=zddz(βp(z))=∑n≥1n bn(β) zn, \beta p(z)=\sum_{n\ge 1} b_n(\beta)\, z^n, \qquad \rho(z)= z\frac{d}{dz}\big(\beta p(z)\big)=\sum_{n\ge 1} n\,b_n(\beta)\,z^n,

where ρ(z)\rho(z) is the particle density.

The virial expansion is the equation of state as a power series in ρ\rho:

βp(ρ)=ρ+∑n≥2Bn(β) ρn, \beta p(\rho)=\rho+\sum_{n\ge 2} B_n(\beta)\,\rho^n,

with coefficients BnB_n (the ) determined combinatorially from the cluster integrals bnb_n (see ).

Theorem (Convergence of Mayer and virial expansions)

Assume the pair potential ϕ\phi is:

  1. Stable: there exists B≥0B\ge 0 such that for every NN and every configuration (x1,…,xN)(x_1,\dots,x_N), ∑1≤i<j≤Nϕ(xi−xj)≥−B N. \sum_{1\le i<j\le N}\phi(x_i-x_j)\ge -B\,N.
  2. Tempered / integrable Mayer function: the quantity C(β)=∫Rd∣e−βϕ(x)−1∣ dx C(\beta)=\int_{\mathbb{R}^d} \big|e^{-\beta\phi(x)}-1\big|\,dx is finite.

Then there exists z0(β)>0z_0(\beta)>0 such that:

  • The series ∑n≥1bnzn\sum_{n\ge1} b_n z^n and ∑n≥1nbnzn\sum_{n\ge1} n b_n z^n converge absolutely for ∣z∣<z0(β)|z|<z_0(\beta).

  • The functions p(z)p(z) and ρ(z)\rho(z) are analytic on ∣z∣<z0(β)|z|<z_0(\beta), with ρ(0)=0\rho(0)=0 and ρ′(0)=1\rho'(0)=1.

  • Consequently, there exists ρ0(β)>0\rho_0(\beta)>0 such that z↦ρ(z)z\mapsto \rho(z) is invertible and analytic for ∣ρ∣<ρ0(β)|\rho|<\rho_0(\beta), and the virial series

    βp(ρ)=ρ+∑n≥2Bn(β)ρn \beta p(\rho)=\rho+\sum_{n\ge2} B_n(\beta)\rho^n

    converges absolutely for ∣ρ∣<ρ0(β)|\rho|<\rho_0(\beta).

One explicit sufficient small-activity condition commonly used in practice is

e2βB ∣z∣ C(β)<1e, e^{2\beta B}\,|z|\,C(\beta) < \frac{1}{e},

which guarantees convergence of the cluster expansion (hence analyticity of pp and ρ\rho) and yields a nontrivial analyticity neighborhood for the virial expansion.

Low-order relations between bnb_n and BnB_n

With the normalization b1=1b_1=1 (so that ρ(z)=z+O(z2)\rho(z)=z+O(z^2)), the first virial coefficients can be expressed in terms of the cluster coefficients as

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

and higher BnB_n are polynomials in b2,…,bnb_2,\dots,b_n.

Consequences and interpretation

  • Analytic equation of state at low density: convergence implies p(ρ)p(\rho) is analytic for sufficiently small ρ\rho; in particular, there is no thermodynamic singularity in that regime.
  • Uniform control via : the same estimates typically imply exponential decay of correlations and uniqueness of the Gibbs state in the corresponding parameter region (high temperature / low density).