Ideal Mixture Approximation of Cluster Size Distributions at Low Density

Abstract

We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture. The present paper improves our earlier results by taking into account the mixing entropy.

Appendix: The Idealised Model in the Saha Regime

In this section, we provide explicit bounds for the approximation of the minima and the minimisers of the idealised rate function f ^ideal by the ones of the function g _ν that we introduced in (1.20) and analysed in [3] and [9]. We work in the Saha regime, where ρ=e^−βν for some ν∈(0,∞). Recall that the interaction potential v is always supposed to satisfy Assumption (V). Let us first recall the relevant notation.

The ground state energy E _k was defined in (1.15) and the quantities e _∞=lim _k→∞ E _k /k and ν ^∗=inf _k∈ℕ(E _k −ke _∞) were defined after (1.15). Set μ(ν):=inf _k∈ℕ[E _k −ν]/k. From [9, Lemma 1.3] we know that the map ν↦μ(ν) is piecewise affine. It is constant with value μ(ν)=e _∞ for ν∈(0,ν ^∗], and strictly decreasing in [ν ^∗,∞). The set $\mathcal {N}\subset[\nu^{*},\infty)$ of points at which μ changes its slope is bounded and either infinite, with the unique accumulation point ν ^∗, or finite. Furthermore, for $\nu\in(\nu^{*},\infty)\setminus\mathcal{N}$, we have $\mu(\nu) = [E_{k_{\nu}} - \nu]/k_{\nu}$ for a unique k _ν ∈ℕ, and

$$ \varDelta (\nu):= \inf\biggl\lbrace\frac{E_k - \nu }{k} - \mu(\nu) \Bigm| k \in \mathbb {N},\ k \neq k_\nu\biggr\rbrace $$

(A.28)

is strictly positive [9, Theorem 1.8].

A first quick consistency check concerns the comparison of the critical line ρ=exp(−βν ^∗) from [3, 9] with the saturation density of the ideal mixture; this strengthens Lemma 3.4.

Proposition A.5

(Saturation density)

Suppose that v satisfies Hypotheses 1, 2 and 4 and d≥2. Then, as β→∞, $\rho_{{\mathrm {sat}}}^{\mathrm {ideal}}(\beta) = \exp( - \beta\nu^{*} + O(\log\beta))$.

Next, we investigate the low-temperature asymptotics of f ^ideal(β,ρ). Recall that the free energy is a sum of two terms, see (1.19). We analyse them separately and shall see that the dominant contribution comes from the term ρμ ^ideal(β,ρ), which behaves like ρμ(ν). Observe that ρμ(ν) is precisely the approximation to the free energy f(β,ρ) proven in [9].

Proposition A.6

(Chemical potential)

Suppose that v satisfies Hypotheses 1 and 2. Let ν>0 and put ρ=exp(−βν). Then, as β→∞,

• if $\nu\in(\nu^{*},\infty)\setminus\mathcal{N}$,

  

  (A.29)

• if ν<ν ^∗ and v also satisfies Hypothesis 4, and d≥2, then

  $$ \mu^{\mathrm {ideal}}(\beta,\rho) = f_\infty^{\mathrm {cl}}( \beta) = e_\infty+ O \biggl(\frac{\log\beta}{\beta} \biggr) = \mu(\nu) + O \biggl(\frac{\log\beta}{\beta} \biggr). $$

  (A.30)

Next we state the behaviour of $m^{\mathrm {ideal}}(\beta,\rho) = \sum_{k\in \mathbb {N}} \rho_{k}^{\mathrm {ideal}}(\beta,\rho)$, the number of clusters per unit volume. Note that for an ideal mixture, this is essentially the same as the pressure, βp ^ideal(β,ρ)=m ^ideal(β,ρ) [7].

Proposition A.7

(Number of clusters (pressure))

Suppose that v satisfies Hypotheses 1 and 2. Fix ν>0 and put ρ=exp(−βν). Then, as β→∞,

• if $\nu\in(\nu^{*},\infty)\setminus\mathcal{N}$, m ^ideal(β,ρ)=(1+O(e^{−βΔ(ν)/2}))ρ/k _ν ,

• if ν<ν ^∗ and in addition v satisfies Hypothesis 4, and d≥2, then m ^ideal(β,ρ)=exp(−βν ^∗+O(logβ))=o(ρ).

Finally, we analyse the behaviour of the minimiser of f ^ideal(β,ρ,⋅).

Proposition A.8

(Cluster size distribution)

Suppose that v satisfies Hypotheses 1 and 2. Fix ν>0 and put ρ=exp(−βν). Then, as β→∞,

• if $\nu\in(\nu^{*},\infty)\setminus\mathcal{N}$,

  $$ \frac{k_\nu\rho_{k_\nu}^{\mathrm {ideal}}(\beta,\rho )}{\rho} = 1 + O\bigl(\mathrm {e}^{-\beta \varDelta (\nu)/2}\bigr), $$

  (A.31)

• if ν<ν ^∗ and in addition v satisfies Hypothesis 4, and d≥2,

  $$ \sum_{k =1}^{\infty} \frac{k \rho_{k}^{\mathrm {ideal}}(\beta,\rho)}{\rho} = O\bigl( \mathrm {e}^{ - \beta(\nu ^*-\nu) +O(\log\beta)}\bigr). $$

  (A.32)

The interpretation of (A.31) is that all but an exponentially small fraction of particles are in clusters of size k _ν , while the one of (A.32) is that the fraction of particle in finite-size clusters goes to 0 exponentially fast.

Proof of Proposition A.1

Because of Lemma 3.2, for suitable c>0 and all sufficiently large K∈ℕ and sufficiently large β,

$$ \sum_{k=1}^K k Z_k^{\mathrm {cl}}( \beta) \mathrm {e}^{\beta k f_\infty^{\mathrm {cl}}(\beta)} \leq\rho_{\mathrm {sat}}^{\mathrm {ideal}}(\beta) \leq\sum _{k=1}^K k Z_k^{\mathrm {cl}}( \beta) \mathrm {e}^{\beta k f_\infty^{\mathrm {cl}}(\beta)} + \rho_{\mathrm {sat}}^{\mathrm {ideal}}(\beta) \mathrm {e}^{- \beta c K^{1-1/d}}, $$

whence we see that

$$ \rho_{\mathrm {sat}}^{\mathrm {ideal}}(\beta) = \bigl(1+ O\bigl(\mathrm {e}^{- \beta c K^{1-1/d}} \bigr) \bigr) \sum_{k=1}^K k Z_k^{\mathrm {cl}}(\beta) \mathrm {e}^{\beta k f_\infty^{\mathrm {cl}}(\beta)}. $$

The proof is concluded by choosing K large enough so that every minimiser of E _k −ke _∞ is smaller or equal to K, since for such a K, the sum on the right-hand side of the previous equation is exp(−βν ^∗+O(logβ)). □

Proof of Proposition A.2

Consider first the case $\nu\in(\nu^{*},\infty)\setminus\mathcal {N}$. Hence, $\mu(\nu) = (E_{k_{\nu}}- \nu)/k_{\nu}$ for a unique k _ν ∈ℕ and (E _k −ν)/k−μ(ν)≥Δ(ν)>0 for all k≠k _ν . For sufficiently large β, we will have $\rho< \rho_{{\mathrm {sat}}}^{\mathrm {ideal}} (\beta)$ and therefore the chemical potential is strictly smaller than $f_{\infty}^{{\mathrm {cl}}}(\beta)$ and is given by the unique solution of Eq. (1.17) which we rewrite as

$$ 1 = \sum_{k=1}^\infty k \,z^k\, \exp\biggl(- \beta k \biggl[ f_k^{\mathrm {cl}}( \beta)- \frac{\nu}{k} - f_{k_\nu}^{\mathrm {cl}}(\beta)+ \frac{\nu}{k_\nu} \biggr] \biggr) $$

(A.33)

with the auxiliary variable

$$ z = z(\beta,\rho,\nu) := \exp\bigl(\beta\mu ^{\mathrm {ideal}}(\beta, \rho)\bigr) \exp\biggl( - \beta\biggl[ f_{k_\nu}^{\mathrm {cl}}(\beta) - \frac{\nu}{k_\nu} \biggr] \biggr). $$

(A.34)

We bound the sum in Eq. (A.33) from below by the summand for k=k _ν . This gives $1 \geq k_{\nu} z^{k_{\nu}}$ and thus z≤1. Next, we choose β ₀ such that for all β≥β ₀ and all k≠k _ν , the term in square brackets in (A.33) is larger than βΔ(ν)/2. Then

$$ 1 \leq k_\nu z^{k_\nu} + \sum_{k \neq k_\nu} k e^{- \beta k \varDelta (\nu)/2} \leq k_\nu z^{k_\nu} + \frac{\exp(- \beta \varDelta (\nu)/2)}{(1-\exp(- \beta \varDelta (\nu)/2))^2}. $$

Thus we get $k_{\nu}z^{k_{\nu}} = 1+ O(\exp(- \beta \varDelta (\nu)/2))$ and (A.29) follows from (A.34).

Now let us come to the case ν<ν ^∗. Because of Proposition A.1, for sufficiently large β, we will have $\rho> \rho_{{\mathrm {sat}}}^{\mathrm {ideal}} (\beta)$ and hence by definition $\mu^{\mathrm {ideal}}(\beta,\rho) = f_{\infty}^{{\mathrm {cl}}}(\beta)$. Equation (A.30) is then a consequence of Lemma 3.1. □

Proof of Proposition A.3

First we consider the case $\nu\in(\nu^{*},\infty)\setminus\mathcal{N}$. With z=z(β,ρ,ν) from (A.34), by an argument similar to the proof of Proposition A.2, $m^{\mathrm {ideal}}(\beta,\rho)/\rho= z^{k_{\nu}} + O(\exp( -\beta \varDelta (\nu))$. Since we saw that $k_{\nu}z^{k_{\nu}} = 1 + O(\exp( - \beta \varDelta (\nu )))$, we are done.

For the case ν<ν ^∗, we note that for sufficiently large β, $\rho>\rho_{{\mathrm {sat}}}^{\mathrm {ideal}}(\beta)$, hence $m^{\mathrm {ideal}}(\beta,\rho) = \sum _{k=1}^{\infty}Z_{k}^{{\mathrm {cl}}}(\beta) \exp( - \beta k f_{k}^{{\mathrm {cl}}}(\beta))$ and the claim follows by an argument similar to the proof of Proposition A.1. □

Proof of Proposition A.4

The case $\nu\in(\nu^{*},\infty)\setminus\mathcal{N}$ is a consequence of the identity $k_{\nu}\rho_{k_{\nu}}^{\mathrm {ideal}}(\beta,\rho) /\rho= k_{\nu}z^{k_{\nu}}$ and the argument in the proof of Proposition A.2.

In the case ν<ν ^∗ we just remark that for sufficiently large β, $\rho> \rho_{{\mathrm {sat}}}^{\mathrm {ideal}}(\beta)$ hence

$$ \sum_{k=1}^\infty\frac{k\rho_k^{\mathrm {ideal}}(\beta,\rho)}{\rho} = \frac{\rho_{\mathrm {sat}}^{\mathrm {ideal}}(\beta)}{\rho} $$

and the proof is concluded by applying Proposition A.1. □