Mayer and Virial Series at Low Temperature

Abstract

We analyze the Mayer and virial series (pressure as a function of the activity resp. the density) for a classical system of particles in continuous configuration space at low temperature. Particles interact via a finite range potential with an attractive tail. We propose physical interpretations of the Mayer and virial series’ radii of convergence, valid independently of the question of phase transition: the Mayer radius corresponds to a fast increase from very small to finite density, and the virial radius corresponds to a cross-over from monatomic to polyatomic gas. Our results are consistent with the Lee-Yang theorem for lattice gases and with the continuum Widom-Rowlinson model.

Appendices

Appendix A: Two Auxiliary Variational Problems

Throughout this section we assume that v is a stable pair potential with attractive tail. Consider the following two variational problems

The first variational problem appears in Theorem 3.6, and minimizers k(μ) correspond to the favored size of molecules in the gas phase as β→∞ at fixed μ. The second problem appears in Eq. (17) and, as shown in [10], minimizers k(ν) correspond to favored cluster or molecule sizes as β→∞ and ρ→0 along ρ=exp(−βν), at fixed ν.

Recall that for potentials with an attractive tail, ν ^∗:=inf _k (E _k −ke _∞)>0.

Lemma A.2

(Concavity, monotonicity and equivalence)

Let v be a stable pair interaction with attractive tail. Then:

• 1. The function μ↦ν(μ) is strictly decreasing, piecewise affine and concave in μ∈(−∞,e _∞]. The function ν↦μ(ν) is decreasing, piecewise affine and concave in ν∈[0,∞); it is strictly decreasing in ν∈[ν ^∗,∞) and equals μ(ν)=e _∞ for ν≤ν ^∗.

• 2. For μ≤e _∞ and ν≥ν ^∗, ν=ν(μ) if and only if μ(ν)=ν.

The reciprocity of μ(ν) and ν(μ) is analogous to the equivalence of the grand-canonical and the constant pressure ensembles. Indeed, the pressure and the Gibbs energy (per particle) are both obtained as Legendre transforms of the free energy, one with respect to the density, the other with respect to the volume per particle,

$$ p(\beta,\mu) = \sup_{\rho} \bigl( \mu \rho - f(\beta,\rho) \bigr),\qquad g(\beta,p) = \inf_{v} \bigl( \tilde{f}(\beta,v) + p v \bigr), $$

with $\tilde{f}(\beta,v) = v f(\beta,v^{-1})$ the free energy per particle. Equivalence of ensembles here means that p(β,⋅) and g(β,⋅) are reciprocal: the Gibbs energy is the same as the chemical potential.

Similarly, μ(ν) looks like a Legendre transform of k↦E _k with respect to k, while ν(μ) looks like a Legendre transform of E _k /k with respect to 1/k, which should be compared with the relations v=1/ρ, $\tilde{f}(\beta,v) = f(\beta,\rho) /\rho$.

Proof of Lemma A.1

1. The statement for the function μ(ν) was proven in [5, 10]. For ν(μ), we note that it is the infimum of a family of decreasing, affine functions and therefore concave and decreasing. Moreover it is almost everywhere differentiable, with derivative −k(μ), the minimizer of E _k −kμ. In particular k(μ)≥1, hence ν(μ) is strictly decreasing.

2. We prove “⇒”. The proof of the converse is similar. Thus let μ≤e _∞ and ν=inf _k (E _k −kμ). Clearly, ν≤inf _k (E _k −ke _∞)=ν ^∗, and for every k∈ℕ,

$$ \nu \leq E_k - k\mu \quad \Rightarrow\quad \mu \leq \frac{E_k - \nu}{k}, $$

whence μ≤μ(ν). On the other hand, if μ<e _∞, then E _k −kμ≥ν ^∗+k(e _∞−μ)→∞ as k→∞, so there must be a finite k such that ν=E _k −kμ. It follows that μ=(E _k −ν)/k≥μ(ν), whence μ=μ(ν). If μ=e _∞, then ν=ν ^∗ and the claim follows from the general inequality μ(ν)≤e _∞. □

Lemma A.3

(Comparison of thresholds)

Let

$$ \mu_1:= \inf_{k \geq 2} \frac{E_k}{k-1}, \qquad \nu_1 := - \mu_1. $$

Then

• either μ ₁=e _∞ and ν ^∗=−e _∞=ν ₁,

• or μ ₁<e _∞ and ν ^∗<−e _∞<ν ₁.

Proof

Lemma A.1 implies the general bounds μ ₁≤e _∞ and ν ₁≥ν ^∗. Moreover, by definition, ν ^∗≤E ₁−e _∞=−e _∞ and

$$ \nu_1 = \sup_k \frac{E_k}{1-k} \geq \lim_{k \to \infty} \frac{E_k}{1-k} = - e_\infty $$

so that ν ^∗≤−e _∞≤ν ₁. If in addition μ ₁=e _∞, then ν ₁=−e _∞ and for all k∈ℕ, E _k ≥(k−1)e _∞ from which we get ν ^∗=inf _k (E _k −ke _∞)≥−e _∞. Since in any case ν ^∗≤e _∞, we get ν ^∗=e _∞.

If μ ₁<e _∞, then ν ₁>−e _∞ and there is a p∈ℕ such that μ ₁=E _p /(p−1)<e _∞. It follows that

$$ \nu^* \leq E_p - pe_\infty = (p-1) (\mu_1 - e_\infty) - e_\infty < - e_\infty. $$

 □

Lemma A.4

(“Phase” diagram)

• 1. For μ<μ ₁, E _k −kμ has the unique minimizer k(μ)=1. Similarly, for ν>ν ₁, (E _k −ν)/k has the unique minimizer k(ν)=1.

• 2. For μ ₁<μ<e _∞, every minimizer is finite and larger or equal to 2; similarly for ν ^∗<ν<ν ₁.

• 3. For ν<ν ^∗, (E _k −ν)/k has no finite minimizer.

Proof

1. By definition, μ<μ ₁ if and only if for all k≥2, (E ₁−μ)/1=−μ<E _k −kμ. Thus for μ<μ ₁, E _k −kμ has the unique minimizer k(μ)=1. The statement on (E _k −ν)/k is proven in an analogous way.

2. For μ<e _∞, E _k −kμ≥k(e _∞−μ)→∞ as k→∞, thus (E _k −kμ) _k∈ℕ reaches its minimum at finite values of k. If k=1 was a minimizer, we would have −μ≤E _k −kμ for all k≥2, whence μ≤μ ₁. Therefore when μ>μ ₁, every minimizer k(μ) is larger or equal to two. The proof for the statement on (E _k −ν)/k is similar.

3. By definition, if ν<ν ^∗, then ν<E _k −ke _∞ for all k∈ℕ, thus (E _k −ν)/k>e _∞ for all k. It follows that inf _k (E _k −ν)/k=e _∞ and there is no finite minimizer. □

Appendix B: Free Energy at Low Temperature and Low Density

Here we give a sketch of the proof of (17). The primary aim is to show that ρ ₀ can be chosen indeed of the order of the preferred ground state density, ρ ₀≈1/a ^d. For the sake of completeness, we also make a remark on how Eq. (17) should be modified for potentials without attractive tail.

2.1 B.1 Potentials with Attractive Tail

Let $Z_{k}^{\mathrm {cl}}(\beta)$ be the cluster partition function from (29) above. Then [10, Lemma 3.1]

$$ Z_\varLambda(\beta,N) \leq \sum_{\sum_1^N {kN_k = N}}\ \prod _{k=1}^N \frac{ (|\varLambda| Z_k^\mathrm {cl}(\beta))^{N_k}}{N_k!}. $$

The sum is over integers N ₁,…,N _N ∈ℕ₀ such that ∑ _k kN _k =N. The integers describe a partition of the N particles into clusters, i.e., groups of particles close in space. Using that for suitable c>0 and all β>0 and k∈ℕ,

$$ Z_k^\mathrm {cl}(\beta) \leq \exp( - \beta E_k) \exp( c k), $$

[10, Lemma 4.3] we deduce that −βf(β,ρ) is upper bounded by the supremum of

$$ c \rho - \beta \biggl( \sum_{k\in \mathbb {N}} \rho_k E_k + \biggl(\rho-\sum_{k\in \mathbb {N}} k \rho_k\biggr) e_\infty \biggr) + \sum _{k\in \mathbb {N}} \rho_k (1 - \log \rho_k) $$

over all (ρ _k ) _k∈ℕ∈[0,∞)^ℕ such that $\sum_{1}^{\infty}k \rho_{k} \leq \rho$ (think ρ _k =N _k /|Λ|). Next, we observe that the mixing entropy can be bounded as ∑ _k ρ _k log(ρ _k /ρ)≥−2ρ, for all ρ>0 and all admissible (ρ _k ), see [10, Lemma 4.2]. Therefore we obtain

whence

$$ f(\beta,\rho) \geq - (c+3) \beta^{-1} \rho + \rho \inf_{k\in \mathbb {N}} \frac{E_k + \beta^{-1} \log \rho}{k}, $$

(42)

for all β>0 and all ρ>0.

It remains to obtain an upper bound for the free energy, or a lower bound for the partition function. Consider first the case ν ^∗>0 and ρ≥exp(−βν ^∗). In this case the infimum in Eq. (42) equals e _∞. We lower bound the partition function by integrating only over a small neighborhood of the N-particle ground state, and deduce −f(β,ρ)≥e _∞−Cβ ⁻¹logβ for suitable C and sufficiently large β. Note that this is possible if ρ is smaller than the density of the ground state, thus ρ<1/a ^d is sufficient.

Next, consider the case ρ=exp(−βν)<exp(−βν ^∗). In this case (E _k −ν)/k has a finite minimizer k=k(ν)∈ℕ. We lower bound the partition function for a cube Λ=[0,L]^d and N∈kℕ particles as follows: we split the cube into M small cubes (“cells”) with side length of the order ak ^1/d and mutual distance R. Here R is of the order of the potential range, and ak ^1/d is large enough so that a k-particle ground state fits into the small cube, as in Assumption 2. We can place approximately M=|Λ|/(ak ^1/d+R)^d small cubes in that way. We consider configurations in which particles form clusters of size k close to their ground state, such that each cluster fits completely into a small cube, and there is at most one cluster per cell.

We refer the reader to [10] for the details and content ourselves with the following remark: the procedure works provided the number M of available cells is larger than N/k. This gives the condition

$$ \rho < \bigl( a+ R k^{-1/d}\bigr)^{-d}. $$

Hence if we choose β large enough so that exp(−βν ^∗)≤(a+R)^−d, the condition is certainly fulfilled for every ρ≤exp(−βν ^∗).

Remember that for ρ≥exp(−βν ^∗), we are in the first case considered above and we only need ρ≤1/a ^d≈ ground state density. Therefore, in the end, all we need is the condition ρ<ρ ₀ with ρ ₀ of the order of 1/a ^d.

2.2 B.2 Potentials Without Attractive Tail

If v has no attractive tail, we might have ν ^∗=0, and ground states are not necessarily connected. Set $E_{1}^{\mathrm {cl}}= E_{1} = 0$ and

$$ E_k^\mathrm {cl}:= \inf \bigl\lbrace U(x_1, \ldots,x_k)\bigm| (x_1,\ldots,x_k) \in \bigl( \mathbb {R}^d\bigr)^k\ R\mbox{-connected} \bigr\rbrace \geq E_k. $$

The lower bound (42) is still true, but can be improved by replacing E _k by $E_{k}^{\mathrm {cl}}$. In fact, indices k with $E_{k}^{\mathrm {cl}}>E_{k}$ can be dropped altogether: if $E_{k}^{\mathrm {cl}}>E_{k}$, then for suitable r≥2, k ₁+⋯+k _r =k,

$$ E_k^\mathrm {cl}>E_k= E_{k_1}^\mathrm {cl}+ \cdots + E_{k_r}^\mathrm {cl}. $$

(43)

Suppose that for some ν>0, $(E_{k_{i}}^{\mathrm {cl}}- \nu ) /k_{i} \geq (E_{k}^{\mathrm {cl}}- \nu)/k$ for all i. Then

$$ \sum_1^r E_{k_i}^\mathrm {cl}\geq \sum_1^r \biggl( \nu + k_i \frac{E_k^\mathrm {cl}- \nu}{k} \biggr) = (r-1) \nu + E_k^\mathrm {cl}> E_k^\mathrm {cl}, $$

contradicting (43). Thus

$$ \inf_{k\in \mathbb {N}} \frac{E_k^\mathrm {cl}- \nu}{k} = \inf \biggl\lbrace \frac{E_k - \nu}{k} \Bigm| k \in \mathbb {N},\ E_k = E_k^\mathrm {cl}\biggr\rbrace $$

is the appropriate auxiliary variational problem to be substituted into Eq. (17). The density ρ ₀ can be chosen of the order of (a+R)^−1/d.

2.3 B.3 Non-negative Potentials

When v≥0, the situation becomes particularly simple: we have ν ^∗=0 and for all ν>0,

$$ \inf_{k\in \mathbb {N}} \frac{E_k - \nu}{k} = \inf_{k\in \mathbb {N}} \frac{E_k^\mathrm {cl}- \nu}{k} = - \nu $$

and k(ν)=1 is the unique minimizer. Equation (17) is replaced by the following: For sufficiently low temperature and density ρ smaller than or of the order of 1/R ^d,

$$ \bigl| f(\beta,\rho) - \beta^{-1} \rho \log \rho \bigr| \leq C \rho \beta^{-1} \log \beta $$

and we recognize the free energy β ⁻¹ ρ(logρ−1) of an ideal gas.

Appendix C: Cluster Expansion in the Canonical Ensemble

The virial expansion (5) can be derived directly with the help of a cluster expansion in the canonical ensemble; this was recently done in [16]. The aim of this appendix is to complement Sect. 5 and explain how Eq. (33) (without the exact formula for a(m)) is obtained with the approach from [16].

The starting point is an expression of the canonical partition function as a sum over set partitions {X ₁,…,X _r }, r∈ℕ, of the particle label set {1,…,N}:

$$ Z_\varLambda(\beta,N) = \frac{|\varLambda|^N}{N!} \sum _{\{X_1,\ldots,X_r\}} \zeta_\varLambda(X_1) \cdots \zeta_\varLambda(X_r). $$

Monomers (|X|=1) have activity 1, sets with higher cardinality have activity

$$ |X|= k \geq 2{:}\quad \zeta_\varLambda(X) = \frac{1}{|\varLambda|^k} \sum _{\gamma \in \mathcal{G}_\mathrm{c}(k)} \int_{\varLambda^k} \prod _{(ij) \in \gamma} f_{ij}(\boldsymbol {x}) \mathrm {d}x_1 \cdots \mathrm {d}x_k, $$

with $\mathcal{G}_{\mathrm{c}}(k)$ the set of connected graphs with vertices 1,…,k. Note that as |Λ|→∞, for every fixed k and β, the activity is related to the Mayer coefficients as follows:

$$ |X| =k{:}\quad \zeta_\varLambda(X) \sim \frac{k!}{|\varLambda|^{k-1}} b_k( \beta) =: \frac{B_k(\beta)}{|\varLambda|^{k-1}}. $$

The formalism of cluster expansions for polymer partition functions gives

$$ \log Z_\varLambda(\beta,N) = \log \frac{|\varLambda|^N}{N!} + \sum _{r \geq 1} \frac{1}{r!} \sum_{\stackrel{X = (X_1,\ldots,X_r)}{\mathrm {conn},\ X_i \in \Gamma_N}} n(X) \zeta_\varLambda(X_1) \cdots \zeta_\varLambda(X_r). $$

Here Γ _N is the collection of subsets of {1,…,N} of cardinality at least 2, and connectedness and n(X) are defined as follows: With $X = (X_{1},\ldots,X_{r}) \in \Gamma_{N}^{r}$ we associate the graph G(X) with vertices 1,…,r and edges {i,j}, i≠j, X _i ∩X _j ≠∅. The polymer X is called connected if the graph of overlaps G(X) is, and n(X)∈ℕ₀ is the index of G(X), i.e., n(X)=n ₊(X)−n ₋(X) with n _±(X) the number of connected subgraphs of G(X) with an even (odd) number of edges.

The N-dependence in the summation index is slightly inconvenient. We remove it by exploiting the invariance with respect to particle relabeling,

In the thermodynamic limit N,|Λ|→∞, for each cluster X=(X ₁,…,X _r ) in the sum,

This goes to zero unless $1+ \sum_{1}^{r}(k_{i} - 1) = n$ (note that “≥” for every cluster X). When this condition is satisfied, the components are necessarily distinct, X _i ≠X _j , and the overlap graph G(X) is necessarily a Husimi graph, i.e., a graph whose doubly connected components are complete graphs. Using the fact that the index of the complete graph on v vertices is (−1)^v−1(v−1)!, one finds that

$$ n(X) = (-1)^{r-1} \prod_{i=1}^j (v_i-1)! $$

with j the number of doubly connected components of G(X) and v ₁,…,v _j their respective sizes; thus v ₁+⋯+v _j =r.

Assuming we can exchange summation and thermodynamic limits, we obtain

(44)

The sum ∑⁽ⁿ⁾ is over collections of subsets {X ₁,…,X _r } with (X ₁,…,X _r ) connected, |X _i |≥2 for all i, and such that $\bigcup_{1}^{r} X_{i} = \{1,\ldots,n\}$ and $n = 1 + \sum_{1}^{r} (|X_{i}|- 1)$. Setting

$$ m_j:= \bigl| \bigl\{ i\bigm| 1 \leq i \leq r,\ |X_i|=j \bigr\}\bigr|, $$

we have

$$ n-1 = \sum_{i=1}^r \bigl(|X_i|-1\bigr) = \sum_{j=2}^n (j-1) m_j, \quad r = \sum_{j=2}^n m_j, $$

(45)

and we obtain the expansion (33) with the information a(m)>0. Additional combinatorial steps would be needed to obtain the formula for the a(m)’s, but the information that they are strictly positive is all that is needed for proofs of Propositions 3.10 and 3.11.