Sabine Jansen & Wolfgang König

Transcription

1 Ideal Mixture Approximation of Cluster Size Distributions at Low Density Sabine Jansen & Wolfgang König Journal of Statistical Physics 1 ISSN Volume 147 Number 5 J Stat Phys 2012) 147: DOI /s

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3 J Stat Phys 2012) 147: DOI /s Ideal Mixture Approximation of Cluster Size Distributions at Low Density Sabine Jansen Wolfgang König Received: 20 December 2011 / Accepted: 13 May 2012 / Published online: 31 May 2012 Springer Science+Business Media, LLC 2012 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 lie 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 nown 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 taing into account the mixing entropy. Keywords Classical particle system Canonical ensemble Equilibrium statistical mechanics Dilute system Large deviations 1 Introduction and Main Results The idea to treat an interacting particle gas as an approximately ideal mixture of droplets clusters) is classical and of wide-spread use in statistical mechanics, thermodynamics, and physical chemistry. It goes sometimes under the name of Frenel-Band theory of association equilibrium, see the textboo [7]. From a more mathematical perspective, this droplet picture has been used in investigations of percolation properties for lattice and continuum systems [6, 10, 11]. This article s main concern is to mae this droplet picture rigorous. To the best of our nowledge, all existing results wor in the grand-canonical ensemble at sufficiently negative S. Jansen ) W. König Weierstrass Institute Berlin, Mohrenstr. 39, Berlin, Germany jansen@wias-berlin.de W. König Technische Universität Berlin, Str. des 17. Juni 136, Berlin, Germany oenig@wias-berlin.de

4 964 S. Jansen, W. König values of the chemical potential, for which one expects that all clusters are finite. In contrast, we wor in the canonical ensemble. This allows us to give results that hold also in a regime where there might be infinite clusters. In this wor, we consider the free energies, the constrained free energies with fixed cluster size distributions, and the optimal distribution itself and derive bounds for the deviation from the ideal model. These bounds decay exponentially fast as a function of the inverse temperature at low densities, respectively they decay as a power of the density. 1.1 The Model We consider a system x = x 1,...,x N ) of N particles in a box Λ =[0,L] d with interaction given by U N x) = U N x 1,...,x N ) = v x i x j ), 1.1) 1 i<j N where v: [0, ) R { }is a Lennard-Jones-type potential. Our precise assumptions, which are the same as in [9], are as follows. Assumption V) The function v: [0, ) R { }satisfies the following. 1) v is finite except possibly for a hard core: there is a r hc 0 such that r <r hc : vr) =, r >r hc : vr) R. 2) v is stable. 3) The support of v is compact, more precisely, b := sup supp v<. 4) v has an attractive tail: there is a δ>0 such that vr) < 0 for all r b δ,b). 5) v is continuous in [r hc, ) 0, ). Throughout this paper, Assumption V) will be in force without further mentioning. We consider the thermodynamic limit N,L such that Λ =L d = N/ for some particle density 0, ) at positive and finite inverse temperature 0, ). The existence of the free energy per unit volume is well-nown: there is a cp 0, ], theclose-pacing density, such that for all 0, cp ), the limit 1 f,):= 1 lim N,L,L d =N/ 1 Λ log N! Λ N e U N x) dx ), 0, ) 1.2) exists in R. In the following, we always assume that 0, ) and 0, cp ). Our main concern is the cluster size distribution that is induced by the Gibbs measure as a random sequence. Fix R b, ). For a given configuration x = x 1,...,x N ) Λ N, connect two points x i and x j if their distance x i x j is R. In this way, the configuration splits into connected components maximal connected subsets), which we call clusters. Let N x) be the number of -clusters, i.e., components with particles, and let,λ x) := N x) Λ be the number of -clusters per unit volume which is zero for any sufficiently large, of course). We consider the cluster size distribution Λ =,Λ ) N as a random variable in

5 Ideal Mixture Approximation of Cluster Size Distributions 965 the set of all sequences = ) N [0, ) N, which is naturally conceived as a product probability space. One of the main results of [9] is the existence of a rate function f,, ): [0, ) N R { }such that, in the above thermodynamic limit, Λ log e U N x) 1 { ) N! Λ x) } dx inf f,,), 1.3) Λ N in the sense of a large-deviation principle. That is, the limit in 1.3) holds in the wea sense, and the level sets of the rate function f,, ) are compact; in particular, f,, ) is lower semi-continuous. Furthermore, f,, ) is convex. Moreover, if f,,) is finite, then necessarily =1. At low density, the converse is also true, i.e., f,,)< for any such that =1. The relation with the free energy given in 1.2)is { f,)= min f,,) [0, ) N, }. N Hence, the minimisers) of f,, ) play an important role as the optimal configurations) of the system. Note that the rate function and all other related objects studied in the present paper depend on the parameter R, but we will not elaborate on this dependence. Note that this dependence on R vanishes in the low temperature, low density approximation that we studied in [9], but not in the one presented here. 1.2 The Ideal-Mixture Model We now introduce the main object in terms of which we will approximate the above model. The cluster partition function of a -cluster is introduced as Z cl ) := 1 e U 0,x 2,...,x ) 1 { {0,x 2,...,x }connected } dx 2 dx, 1.4)! R d ) 1 and the associated cluster free energy per particle is f cl 1 ) := log Zcl ). It is nown that its thermodynamic limit, f cl ) = lim ), 1.5) f cl exists in R. Indeed, this was proved in [4], given that f cl)) N is bounded, and the boundedness of f cl)) N was derived in [4] indimensiond = 2andd = 3andin[9, Lemmas 4.3 and 4.5] in all dimensions. For = ) N,let f ideal,, ) := f cl ) + ) f cl ) N N + 1 log 1). 1.6) N This rate function describes the large deviations of the cluster size distribution in an idealised model that neglects the excluded-volume effect: the first term describes the internal free energy coming from the clusters of finite size, the second term the analogous contribution from clusters of infinite size, and the last term describes the entropy of placing all these clusters into the volume, not taing care of being separated from each other. See Sect. 1.5 for an integer partition model that has f ideal,, ) as a rate function. We are going to compare f,,) with f ideal,, ) for small densities and low temperatures 1/. That these two should be close to each other is intuitively clear, since low

6 966 S. Jansen, W. König temperature should ensure that clusters assume a compact shape, and low density should give enough space to place the clusters at positive mutual distance. The main purpose of this paper is to mae this reasoning rigorous. 1.3 Our Hypotheses We need further assumptions about ground states and the cluster partition functions. Roughly speaing, we need to assume some Hölder continuity of the energy U ) close to the ground states and that the relevant clusters at zero and low temperature, respectively, have a compact shape, i.e., occupy at most a box with volume of order of the number of particles. These hypotheses are believed to be true for many potentials of the type in Assumption V), and they can be seen to be satisfied for the ground states. However, for positive temperatures, their rigorous understanding has not yet been completed. The purpose of the hypothesis of bounded density is the following. As we indicated above, the fundamental idea is to split the configuration into its clusters and to collect the internal free energies of all the clusters. However, one also needs to describe the entropy of a configuration, that is, the combinatorial complexity for the placement of all the clusters into some cube. This tas is very hard without further information. In the low-density approximation, we will solve this tas by neglecting the excluded-volume effect, which maes it much easier. For this, we need to now that the clusters do not require a large diameter. Our second hypothesis ensure this for the ground states, and the last two hypotheses ensure this for positive low temperature. First we formulate our hypothesis about uniform Hölder continuity of the energy around the ground states and a strong form of stability. Recall that the pair potential v is called stable if 1 inf x R d ) U x) is bounded from below in, which means that the ground states do not clump too strongly. Hypothesis 1 There is a r min r hc, ) such that v is uniformly Hölder continuous in [r min, ) and, for all N, every minimiser x 1,...,x ) R d ) of the energy U has interparticle distance lower bounded as x i x j r min for any i j. This hypothesis can be seen to be satisfied under some mild additional assumptions on v relating the negative part of v to its behaviour at zero; see [3, Proof of Lemma 3.1] or [13, Lemma 2.2]. The Hölder continuity allows us to give low-temperature estimates of the form ) log 1 log 1! R d ) e U x) dx = 1 inf x R d ) U x) + O as, uniformly in N. Our next hypothesis says that the minimising configurations the ground states) do not occupy more space than a box with volume of order of the number of particles. Hypothesis 2 Ground states have a compact shape) There is a constant c>0 such that for all N every minimiser x 1,...,x ) R d ) of the energy U has interparticle distance upper bounded by x i x j c 1/d for any i j. This hypothesis is nown to be satisfied for some classes of potentials having a large intersection with those satisfying Assumption V), however, only in dimension one and two. See [12] and[1].

7 Ideal Mixture Approximation of Cluster Size Distributions 967 Now we proceed with two more restrictive hypotheses, whose validity has actually not been clarified for all interesting potentials of the type that we consider. They concern, at positive sufficiently low temperature, the diameter of the relevant clusters. An important object is the internal cluster energy coming from a box of volume A d : Z cl,a ) := 1 e Ux) 1{x connected} dx!a d 1 dx. 1.7) [0,A] d ) The reader may chec that lim A Z cl,a ) = Z cl ) holds for every fixed and.thecorresponding free energy is defined as f cl,a ) := 1 log Zcl,A ). It is tempting to believe and this is the content of our next hypothesis) that, at least at sufficiently low temperature, a box of volume of order should capture almost all the internal free energy of a cluster: Hypothesis 3 Clusters have a compact shape) For some c 0, ) and every sufficiently large, 1 lim 1 log Zcl,c1/d ) = lim log Zcl ). 1.8) However, this hypothesis has not even been proved for the relatively simple case of the two-dimensional Ising model, see the discussion in [5]. It is commonly believed that 1.8) is true for low temperature and wrong for high temperature, since the cluster is believed to assume a tree-lie structure and to occupy therefore a much larger portion of space. This phenomenon is often called a collapse transition: as the temperature decreases below a critical value, the volume per particle collapses to some finite value. The following hypothesis is in the spirit of Hypothesis 3 and goes much beyond it: if, for large, the relevant configurations for f cl ) have a compact shape, then the number of particles that have not the optimal number of neighbours should be of surface order. Therefore the correction to the large- asymptotics should be of surface order of a ball with volume : Hypothesis 4 For some C>0and all sufficiently large, f cl cl ) f, N. 1.9) To the best of our nowledge, such a deep statement has not been proved for any interesting potential satisfying Assumption V). Actually for our proofs we only need a lower bound against C ε for some ε>0 instead of C 1 1/d. 1.4 Our Results Our first main result applies to all cluster size distributions, not only minimisers of the rate functions, and is therefore possibly of interest for non-equilibrium thermodynamic models. Theorem 1.1 Comparison of f,, ) and f ideal,, )) Let the pair potential v satisfy Hypotheses 1 and 2. Then there are, and C>0 such that for all [, ), 0, ), and K N with K</3) 1/d+1), and all = ) N [0, ) N satisfying N, f ideal,, ) f,,) f ideal,, ) + C ε K,, ), 1.10) where ε K,, ) = d+2)/d+1) K + K ) log m >K log m >K 1.11)

8 968 S. Jansen, W. König and we abbreviated K := K =1 and m >K := =K+1. If in addition Hypothesis 3 holds, then in 1.11) we can replace K ) with =K+1. Theorem 1.1 is proved in Sect. 2. Next, we compare the minimum and the minimisers under the two stronger hypotheses on the compact shape of the relevant clusters at positive temperature and the finite size correction of the cluster free energy. Let { f ideal, ) := inf f ideal,, ) [0, ) N, }. 1.12) N It is not difficult to see that there is a unique minimiser ideal, ) = ideal, )) N.We set m ideal, ) :=, ) [0,]. =1 ideal Theorem 1.2 Suppose that Hypotheses 1, 3 and 4 hold, and assume that d 2. Then there are,,c,c > 0 such that, for all [, ) and 0, ), the following holds. 1) Free energy: 0 f,) f ideal, ) C mideal, ) 1/d+1). 1.13) 2) Let =,) = ) N be a minimiser of f,, ), and put m := N. Then m m ideal, ) 1 2 ) C 1/d+1) 1 and 2 H m ; ideal, ) C 1/d+1). m ideal, ) 1.14) Here Ha; b) = N b a + a log a ) b is the relative entropy between two finite measures a and b on N, and we recall Pinser s inequality: when a and b are probability measures, then 1 2 Ha; b) a b 2 var. The proof of Theorem 1.2 is in Sect As we explained in more detail in [9, Section 1.6], our results do not imply a transition from non-existence to existence of unbounded clusters at any fixed temperature, but only indicate the occurrence of such a transition in a certain limiting sense as the temperature and the particle density both vanish. If we do not assume that Hypotheses 3 and 4 are true, our rigorous bounds hold in the temperature-density plane only in a region away from the critical line given by = exp ν ) with ν 0, ) defined as follows. Introduce the ground-state energy, E := inf U x). 1.15) x R d ) Then e := lim E / exists in, 0),andν := inf N E e ) is positive [3, 9]; note that Assumption V)4) is needed for proving the positivity of ν. Theorem 1.3 Let v satisfy Hypotheses 1 and 2. Then, for any ε>0 there are ε,c ε,c ε > 0 such that for all [ ε, ) and 0, ) satisfying 1 log >ν + ε, 1.13) and 1.14) hold with C and C replaced by C ε and C ε, respectively. The proof of Theorem 1.3 is in Sect We would lie to note that our condition R>b is needed in the preparatorial Sect. 3.1.

9 Ideal Mixture Approximation of Cluster Size Distributions Discussion Let us explain in more detail the significance of Theorems 1.2 and 1.3, and in which way they improve results of [9]. We start by recalling the properties of the idealised problem; see also [2, Sect. 4]. As mentioned above, for all and, f ideal,, ) has a unique minimiser ideal, )) N, which can be characterised as follows. Let sat ideal ) := e N [f cl ) f cl )] 0, ] 1.16) be the saturation density of the ideal mixture. If Hypothesis 4 holds actually also under much weaer bounds than the one in 1.9)), at low temperature, sat ideal ) is finite. In general, however, it can be infinite. For <sat ideal ), letμ ideal, ),f cl )) be the unique solution of e [μideal,) f cl)] =, 1.17) =1 and for sat ideal ), letμ ideal, ) := f cl ). Then, as follows from 3.27) below,the minimiser ideal, )) is given by ideal, ) = e [μideal,) f cl)], 1.18) and the ideal free energy from 1.12)isgivenby f ideal, ) = μ ideal, ) 1 mideal, ). 1.19) Moreover, f ideal, ) is analytic and strictly convex in 0,sat ideal )), and linear with slope f cl ) in [ideal sat ), ). In particular, the ideal mixture undergoes a phase transition as the density is varied if and only if the saturation density of the ideal mixture is finite. The transition is from a gas phase where all particles are in finite-size clusters, to a condensed phase where a positive fraction goes into unboundedly large clusters: for all <sat ideal, ), we have =1 ideal, ) =, while for >sat ideal ), =1 ideal ) = sat ideal )<. Armed with this nowledge, we can compare our results with those of [9]. In [9], we approximated the rate function f,, ), more precisely the function q = q ) N 1 f,,q /) N ), with E ν g ν q) = 1 q )e + q, ν := 1 log. 1.20) =1 =1 This function is easier to formulate, but involves more approximations, and has some rather unphysical properties. This approximation was proved in the so-called Saha regime, where large and small are coupled with each other via the equation = e ν for some parameter ν 0, ), and the limiting rate function g ν turned out to be piecewise linear with at least one in, but also possibly more. Each in represents a phase transition, and the minimiser in the ins is not unique. This is in strong contrast with the approximate rate function f ideal,, ) studied in this paper, which possesses only one minimiser and only at most) one phase transition. As we mae explicit in the next paragraph, f ideal,, ) can itself be approximated by g ν, in particular by neglecting an entropy term. It is the smoothing effect of this term that gets lost in that approximation, and possibly a lot of new ins appear in this way. We now that these additional ins correspond to cross-overs inside the gas

10 970 S. Jansen, W. König phase, but not to sharp phase transitions see [8] for a discussion of this). Hence, the full ideal mixture captures the behaviour of the physical system much better than the function studied in [9]. We note that both f ideal,, ) and g ν considered in [3, 9] appear as exact large deviations rate functions for simple random partitions models. We consider vectors N ) 1 N N N 0 with N =1 N = N as integer partitions, and loo at the not normalised) measures on partitions given by μ ideal,n,λ{ N1,...,N N ) }) := = N Λ Z cl))n N! ) M Λ M N 1,...,N N M! =1 μ CKMS,N,Λ {N1,...,N N } ) := Λ M M! N ) e E N, =1 N Z cl )) N, =1 1.21) where M := N =1 N. In the thermodynamic limit N, Λ such that N/ Λ, under μ ideal,n,λ, the vector N / Λ ) N satisfies a large deviations principle with speed Λ and rate function f ideal,, ). On the other hand, under μ CKMS,N,Λ, the vector N /N) N satisfies a large deviations principle with speed N and rate function q g ν q) 1 q N, which differs from the approximate functional g ν from [3, 9] only by a vanishing term. Hence, from 1.21) we see that g ν ) arises from f ideal,, ) by two simplifications: Z cl) exp E ) and the omission of the multinomial coefficient, that is, the cluster free energy is approximated by the ground state energy, and the mixing entropy is neglected. The second main difference with [9] is that our error bounds are much better. In [9], the free energy per particle f, )/ is approximated up to errors of the order log )/. In contrast, when = e ν for fixed ν>0, as, the error in 1.13) vanishes exponentially fast in. Moreover, 1.13) may be written as f,)= μ ideal, ) 1 mideal, ) 1 + O 1/d+1))). This is interesting because for >sat ideal ), wehavem ideal, ) sat ideal ). Thus the free energy equals the ideal free energy plus an error which is small compared to the smallest of the two terms in 1.19), 1 mideal. For completeness and for the reader s convenience, in the Appendix, we provide approximations of the idealised mixture model in terms of g ν in the Saha regime with exponentially small errors. 2 Proof of Theorem 1.1 In this section, we prove Theorem 1.1. We will use a convexity argument and split an arbitrary sequence into its components on the first K entries and the ones on the remainder. Hence, we consider these two parts separately. 2.1 Case 1: All Clusters Have Size K Here we consider satisfying = 0for>Kwith some K N. Recall from Assumption V) that b is the interaction range and R b, ) determines the notion of connectedness.

11 Ideal Mixture Approximation of Cluster Size Distributions 971 Lemma 2.1 Let >0, N. Set C := sup x 2/3 x 1 log1 x). Then i) For all A>0, f cl cl,a ) f ). ii) For all A>3R, f cl,a ) f cl CdR ) + A. Proof i) For all,,a,wehavez cl,a ) Z cl). Indeed, for any x 1 R d, we obviously have 1 e U x 1,x 2,...,x ) 1{x 1,x 2,...,x ) connected} dx 2 dx Z cl! [0,A] d ) 1 ). Integrating over x 1 [0,a] d and dividing by a d shows that Z cl,a ) Z cl ). This implies that f cl,a ) f cl). ii) Let N and A>2R. Letx 1 [0,A] d have distance R to the boundary of the cube. Then, writing x = x 1,...,x ), 1 e Ux) 1{x connected} dx 2 dx! [0,A] d ) 1 = 1 e Ux) 1{x connected} dx 2 dx! R d ) 1 = Z cl ). Thus, integrating over x 1 and recalling the definition of Z cl,a ) in 1.7), Z cl,a ) A 2R)d Z cl A d ). Therefore, when A>3R,wetae 1 log and deduce [ f cl,a ) f cl ) 1 log 1 2R A ) d ] f cl CdR ) + A. Lemma 2.2 Fix, > 0. Let K N and A>0 such that A + R) d < 1. Then for all such that K =1 =, f,,) K =1 f cl,a ) + 1 K log 1) =1 + 1 K K ) log 1 A + R) d =1 =1 ) + log 1 + R ) ) d. A 2.22) Proof Let Λ =[0,L] d.for2 K, setn := Λ.SetN 1 := N K =2 N,and M := K =1 N.DivideΛ into cubes of side-length A, at mutual distance R. We call these cubes cells. The number D of cubes that can be placed in this way is D L/A + R) d. Let Z Λ,N,N 1,...,N K ) := 1 e U N x) 1 { } {1,...,K}: N x) = N dx. N! Λ N We can lower bound this constrained partition function by integrating only over configurations such that: a) there is at most one cluster per cell, and b) each cluster is contained in

12 972 S. Jansen, W. König one of the cells. The number of partitions of the particle label set {1,...,N} into N 1 sets of size 1, N 2 sets of size 2, etc., is equal to N!/ K =1!N!). For a given set partition, the number of ways to assign distinct cells to the sets is DD 1) D M + 1). Thus we find K A d Z cl,a )) N Z Λ,N,N 1,...,N K ) DD 1) D M + 1). N =1! We observe that DD 1) D M + 1) A d) M ) L d ) M A + R 1 M ) A d M ) A dm = Λ M 1 A + R ) d ) MA+ M R)d. A + R L Λ It follows that in the limit N,L, N/L d 1, lim inf Λ log Z Λ,N,N 1,...,N K ) is not larger than the right-hand side of 2.22). The proof of the lemma is then concluded as in the proof of [9, Proposition 3.2]. Proposition 2.3 Fix >0, 2 d+1 /3) 1+1/d, and K N with K< 1/d+1). Then, for suitable C > 0 and all with K =1 = and = 0 for K + 1, f,,) f ideal,, ) + C 1/d+1). Proof Let C>0beasin Lemma 2.1 and set m := K =1. Then by Lemmas 2.1 and 2.2, f,,) f ideal,, ) CdR A + Cm A + R) d m + dr ), A provided A>3KR and A + R) d m 2/3. Set A = 1/d+1) /3R). ThenA>3KR because by assumption K< 1/d+1),andA + R) d m 2 d A d = 2 d d/d+1) 2/3 because we have assumed 2 d+1 /3) 1+1/d. Thus we have f,,) f ideal,, ) C 1/d+1), 2.2 Case 2: All Clusters Have Size K where C d ) := C 3R) + d 6dR2. Proposition 2.4 Suppose that v satisfies Hypotheses 1 and 2. Then there are,,c > 0 such that for all [, ), 0, ), for all K N 0 and all such that =K+1 and = 0 for K, f,,) f ideal,, ) + C log m log m e. 2.23) If in addition Hypothesis 3 holds, we can replace 1 log with =K+1 ) 1 log. Proof By [9, Theorem 1.8], there are,,c > 0 such that for all and, f,,) E + log ) ) + e + C log. 2.24) =K+1 =K+1

13 Ideal Mixture Approximation of Cluster Size Distributions 973 Because of [9, Lemma 4.3], we can further increase C and so that for all N and, E f cl C ) + log and e f cl C ) + log. Moreover, setting m := =K+1, =K+1 log 1 Here we have used that =K+1 =K+1 log 1) m log m m 1 + log 2 + log m + log =K+1 ). m =K+1 see [9, Lemma 4.1]. The inequality 2.23) follows. If in addition Hypothesis 3 holds we remar, first, that for all 1/c and all sufficiently large, with c as in Hypothesis 3, f,,0) f cl) = f ideal,, 0). Because of the convexity of f,, ), =1 f,,) f,, ) + =K+1 f,,0), =1 := ) 1. m ), We deduce that in 2.24) we can replace e with f cl ), and the claim follows. 2.3 General Case Proof of Theorem 1.1 We already now that f,,) f ideal,, ) [9, Lemma 3.1], so we need only prove the upper bound for f,,). LetK N and such that =1. The cases =1 = and = 0 were treated in Propositions 2.3 and 2.4, thus we may assume 0 < K =1 <.Set K := K =1.Let be as in Proposition 2.4 and suppose that. Set and := K 1 K / { small /1 K /) if 1 K, := 0, if K + 1, { 0 if 1 K, large := /[ K /], if K + 1. It was shown in [9, Section 2.5] that the map, ) f,,) is a supremum of convex functions and hence is itself convex. Thus we can write f,,) 1 ) K f,, small) + K f,, large). 2.25) and

14 974 S. Jansen, W. König Propositions 2.3 and 2.4 yield f,,) f ideal,, ) C K K =1 small ) 1/d+1) m K log 1 ) K + C K) log 2 m >K log m >K e, provided, 2 d+1 /3) 1+1/d,andK< 1/d+1). The conditions on and K are certainly satisfied if we assume that 2 3 min1, ) and K /3) 1/d+1).Ifthisisthe case, we can further bound the right-hand side of the inequality above as C 3 1/d+1) + C K ) d+2)/d+1) + C K) log 2 m >K log m >K e. Since m >K 2/3, we can upper bound 1 + log D)log m >K for some suitable constant C) > 0. We obtain the bound of Theorem 1.1. The improved bound under Hypothesis 3 is deduced from the corresponding statement in Proposition Proof of Theorems 1.2 and 1.3 We prove Theorems 1.2 and 1.3 in Sects. 3.2 and 3.3 below, respectively, after providing some preparations in Sect Tail Estimates The main idea for the proof of Theorems 1.2 and 1.3 is to apply Theorem 1.1 to the minimiser ideal, ) of the ideal rate function. Therefore we will need estimates on the tails of the ideal minimiser, see Lemmas 3.2 and 3.3 below. First we recall a statement about the low-temperature behaviour of the cluster free energy. Recall the ground-state energy E defined in 1.15). Lemma 3.1 Cluster free energy at low temperature) There are 0 > 0 and C>0 such that for 0, N : f cl ) E C cl, and f ) e C. Moreover, for each fixed N, lim f cl) = E /. If in addition the pair potential satisfies Hypothesis 1, then for 0, N : f cl + C log, and cl f + C log and lim f cl). Proof The limit statement for f cl ) as follows by a standard argument for exponential integrals. The lower bounds for f cl cl ) and f ) have been proven in [9, Lemma 4.3]. The upper bounds follow from [9, Lemma 4.5] and the observation that for all,a > 0and N, f cl ) f cl,a ), see Lemma 2.1. Note that our condition R>bis needed in the proof of [9, Lemma 4.5] and hence in the above proof.

15 Ideal Mixture Approximation of Cluster Size Distributions 975 In the following, we omit the arguments, ) of the objects ideal and m ideal for brevity. Recall that ideal is given in 1.18)andthatm ideal = N ideal and m ideal >K = =K+1 ideal. Lemma 3.2 Suppose that Hypothesis 4 is true, and assume that d 2. Then there are C,c, >0, K 0 N such that for all K K 0, [, ), and >0, m ideal >K =K+1 C, and ideal e ck1 1/d. 3.26) m ideal m ideal N ideal N ideal Proof According to Hypothesis 4, we can choose c>0, K 0 N, and such that for and K K 0, exp [ f cl cl ) f )]) e ck1 1/d. =K+1 Let ν ) be such that E ν ) ν )e = ν. Recall from 1.18) that ideal, ) exp[f cl cl ) f )]). Then, for K max{k 0,ν )}, we have, also using Lemma 3.1, =K+1 ideal =1 ideal e ck1 1/d ideal ν ) = e ck1 1/d e ν +O 1 log )) e ck1 1/d 2ν ), where the last inequality holds for sufficiently large. ForK sufficiently large, this last expression is smaller than e ck1 1/d /2. Writing c/2 instead of c, this shows the second assertionin3.26). Furthermore, for these K, N ideal m ideal K =1 ideal K =1 ideal + e ck1 1/d /2 K + 1. Using this for K = K 0, this also shows the first assertion in 3.26). If we do not assume that Hypothesis 4 is true, we have an analogue of Lemma 3.2 for densities much smaller than exp ν ), valid for all d N. Lemma 3.3 Let v satisfy Hypothesis 1. Fix ε>0. Then there are ε,k ε,δ ε > 0 such that for all [ ε, ), exp ν + ε)) and K K ε, m ideal >K =K+1 C m ideal ε, and ideal e Kδε. m ideal m ideal For the proof, we give first a lower bound for the saturation density of the ideal mixture, which is of interest in itself. Under additional hypotheses, a stronger statement holds, see Proposition A.1. Lemma 3.4 lim inf 1 log ideal sat ) ν. Proof Let N. We pic, as a lower bound, only the -th summand in 1.16). Then, as, according to Lemma 3.1, 1 log ideal sat ) 1 log + f cl cl ) f )) = E e ) + o1).

16 976 S. Jansen, W. König Letting and taing the supremum over of the right-hand side yields the desired result. Proof of Lemma 3.3 Fix ε>0. By Lemma 3.4, thereisa 0 > 0 such that for all 0, e ν +ε) <sat ideal ). Thus for 0 and e ν +ε),wehave<sat ideal, ), and μ ideal, ) solves 1.17). Let μ ε := inf N E ν ε)/. Thenμ ε <e. Indeed, by definition of ν,thereisa N such that E e ν +ε/2, and for this, μ ε E ν ε)/ e ε/2) < e. Choosing 0 N such that μ ε = E 0 ν ε)/ 0,wefind μ ideal, ) f cl 0 ) + 1 log E 0 ν + ε) + O 1 log ) 0 0 = μ ε + O 1 log ). Since μ ε <e and [E ν + ε)]/ e as,thereare ε N, Δ ε > 0 such that ε E ν + ε) μ ν + ε ) + Δ ε. It follows that for ε, ideal, ) exp Δ ε + O 1 log ))). Choose ε 0 large enough so that the O 1 log ) term is Δ ε /2, then we find for ε and ε, ideal, ) exp Δ ε /2). Noting that for z<1, as K, K z = Oz K ), we deduce that for sufficiently large K, =K+1 ideal, ) exp Kδ ε ). Now fix K 1 K ε large enough so that exp ε δ ε K 1 ) 1/2. Then K 1 = ideal + =1 =K 1 +1 ideal, ) K 1 m ideal + /2 whence 2K 1 m ideal and, for sufficiently large K, =K+1 ideal 2K m ideal 1 exp Kδ ε ). 3.2 Proof of Theorem Free Energy f,) By Theorem 1.1, forall and, f,)= inf f,,) inf f ideal,, ) = f ideal, ). Thus we need only prove the upper bound to f,). To this aim we note that

17 Ideal Mixture Approximation of Cluster Size Distributions 977 f,) f,, ideal, ) ) = f ideal, ) + f,, ideal, ) ) f ideal,, ideal, ) )), where we recall that ideal, ) is the unique minimiser of f ideal,, ). To lighten notation, we will drop the, )-dependence in the notation and write ideal, m ideal, instead of ideal, ), m ideal, ), etc. Theorem 1.1 yields f,, ideal) f ideal,, ideal) C ) ) ideal d+2)/d+1) K + ideal log m ideal >K log mideal >K, =K+1 provided, <, andk</3) 1/d+1). By Lemma 3.2, if we assume 0 and K K 0, the term in the big parenthesis is bounded above by a constant times m ideal ) 1/d+1) + e ck1 1/d log + ck 1 1/d e ck1 1/d. Choosing K as a constant times 1/d+1), we see that the second and third summands are bounded by the first. This gives the desired bound on the free energy Minimisers Every minimiser of f,, ) satisfies f,)= f,,) f ideal,, ). Therefore, according to part 1), 0 f ideal,, ) f ideal, ) f,) f ideal, ) C mideal 1/d+1). Now by an explicit computation, f ideal,, ) f ideal, ) = H ; ideal) + f cl ) μideal), 3.27) where := =1.Letp := /m and p ideal := ideal /m ideal.then H ; ideal) ) m = m ideal g + mh p; p ideal), m ideal where gx) := 1 x + x log x. Notethatgx) 0forallx>0. Summarising the last three displays, we find, ) m m ideal g + mh p; p ideal ) + f cl ) μideal) Cm ideal 1/d+1). m ideal Since each of the three terms on the left-hand side is nonnegative, we obtain that each of them is not larger than the right-hand side, and this implies ) m g C 1/d+1) and H p; p ideal) C mideal m ideal m 1/d+1). Since gx) 0 implies x 1andgx) 1 x) 2 /2asx 1, we deduce that for sufficiently small and some suitable constant C > 0, 1 m/m ideal C 1/2d+2),andthe corresponding bound for the relative entropy easily follows. 3.3 Proof of Theorem 1.3 The argument is exactly the same as for Theorem 1.2. In Theorem 1.1 applied to ideal, we note that for sat ideal,wehave 1 ideal = ; this observation replaces the use of Hypothesis 3. Later we use Lemma 3.3 instead of Lemma 3.2.

18 978 S. Jansen, W. König 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 wor in the Saha regime, where = e ν 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 was defined in 1.15) and the quantities e = lim E / and ν = inf N E e ) were defined after 1.15). Set μν) := inf N [E ν]/.from [9, Lemma 1.3] we now that the map ν μν) is piecewise affine. It is constant with value μν) = e for ν 0,ν ], and strictly decreasing in [ν, ).ThesetN [ν, ) of points at which μ changes its slope is bounded and either infinite, with the unique accumulation point ν, or finite. Furthermore, for ν ν, ) \ N,wehaveμν) =[E ν ν]/ ν for a unique ν N, and { E ν } Δν) := inf μν) N, ν A.28) is strictly positive [9, Theorem 1.8]. A first quic consistency chec concerns the comparison of the critical line = exp ν ) from [3, 9] with the saturation density of the ideal mixture; this strengthens Lemma 3.4. Proposition A.1 Saturation density) Suppose that v satisfies Hypotheses 1, 2 and 4 and d 2. Then, as, sat ideal ) = exp ν + Olog )). Next, we investigate the low-temperature asymptotics of f ideal, ). 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 μ ideal, ), which behaves lie μν). Observe that μν) is precisely the approximation to the free energy f,)provenin[9]. Proposition A.2 Chemical potential) Suppose that v satisfies Hypotheses 1 and 2. Let ν>0and put = exp ν). Then, as, if ν ν, ) \ N, μ ideal, ) = f cl ν ) ν log ν + O 1 e Δν)/2) ν ) log = μν) + O, A.29) if ν<ν and v also satisfies Hypothesis 4, and d 2, then ) log μ ideal, ) = f cl ) = e + O = μν) + O log ). A.30) Next we state the behaviour of m ideal, ) = N ideal, ), the number of clusters per unit volume. Note that for an ideal mixture, this is essentially the same as the pressure, p ideal, ) = m ideal, ) [7]. Proposition A.3 Number of clusters pressure)) Suppose that v satisfies Hypotheses 1 and 2. Fix ν>0 and put = exp ν). Then, as,

19 Ideal Mixture Approximation of Cluster Size Distributions 979 if ν ν, ) \ N, m ideal, ) = 1 + Oe Δν)/2 ))/ ν, if ν<ν and in addition v satisfies Hypothesis 4, and d 2, then m ideal, ) = exp ν + Olog )) = o). Finally, we analyse the behaviour of the minimiser of f ideal,, ). Proposition A.4 Cluster size distribution) Supposethat v satisfies Hypotheses 1 and2. Fix ν>0 and put = exp ν). Then, as, if ν ν, ) \ N, ν ideal ν, ) = 1 + O e Δν)/2), A.31) if ν<ν and in addition v satisfies Hypothesis 4, and d 2, =1 ideal, ) = O e ) ν ν)+olog ). A.32) The interpretation of A.31) is that all but an exponentially small fraction of particles are in clusters of size 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, K K Z cl )efcl ) sat ideal ) Z cl )efcl ) + sat ideal )e ck1 1/d, =1 whence we see that ideal sat =1 ) = 1 + O e ck1 1/d )) K =1 Z cl )efcl ). The proof is concluded by choosing K large enough so that every minimiser of E e is smaller or equal to K, sinceforsuchak, the sum on the right-hand side of the previous equation is exp ν + Olog )). Proof of Proposition A.2 Consider first the case ν ν, ) \ N. Hence, μν) = E ν ν)/ ν for a unique ν N and E ν)/ μν) Δν) > 0forall ν. For sufficiently large, wewillhave<sat ideal ) and therefore the chemical potential is strictly smaller than f cl ) and is given by the unique solution of Eq. 1.17) which we rewrite as [ 1 = z exp f cl ) ν f cl ν ) + ν ]) A.33) ν =1 with the auxiliary variable z = z,, ν) := exp μ ideal, ) ) [ exp f cl ν ) ν ]). A.34) ν

20 980 S. Jansen, W. König We bound the sum in Eq. A.33) from below by the summand for = ν.thisgives1 ν z ν and thus z 1. Next, we choose 0 such that for all 0 and all ν, the term in square bracets in A.33) is larger than Δν)/2. Then 1 ν z ν + e Δν)/2 exp Δν)/2) ν z ν + 1 exp Δν)/2)). 2 ν Thus we get ν z ν = 1 + Oexp Δν)/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, we will have >sat ideal ) and hence by definition μ ideal, ) = f cl ). Equation A.30)is then a consequence of Lemma 3.1. Proof of Proposition A.3 First we consider the case ν ν, ) \ N. With z = z,, ν) from A.34), by an argument similar to the proof of Proposition A.2, m ideal, )/ = z ν + Oexp Δν)). Sincewesawthat ν z ν = 1 + Oexp Δν))), we are done. For the case ν<ν, we note that for sufficiently large, >sat ideal ), hence m ideal, ) = =1 Zcl cl ) exp f )) and the claim follows by an argument similar to the proof of Proposition A.1. Proof of Proposition A.4 The case ν ν, ) \ N is a consequence of the identity ν ideal ν, )/ = ν z ν and the argument in the proof of Proposition A.2. In the case ν<ν we just remar that for sufficiently large, >sat ideal ) hence ideal, ) = ideal sat ) =1 and the proof is concluded by applying Proposition A.1. Acnowledgements We gratefully acnowledge financial support by the DFG-Forschergruppe FOR718 Analysis and stochastics in complex physical systems. References 1. Au Yeung, Y., Friesece, G., Schmidt, B.: Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. Calc. Var. Partial Differ. Equ. 44, ) 2. Ball, J.M., Carr, J., Penrose, O.: The Becer-Döring cluster equations: basic properties and asymptotic behaviour of solutions. Commun. Math. Phys. 104, ) 3. Collevecchio, A., König, W., Mörters, P., Sidorova, N.: Phase transitions for dilute particle systems with Lennard-Jones potential. Commun. Math. Phys. 299, ) 4. Dicman, R., Schieve, W.C.: Proof of the existence of the cluster free energy. J. Stat. Phys. 36, ) 5. Dicman, R., Schieve, W.C.: Collapse transition and asymptotic scaling behavior of lattice animals: Low-temperature expansion. J. Stat. Phys. 44, ) 6. Georgii, H.-O., Häggström, O., Maes, C.: In: The Random Geometry of Equilibrium Phases, Phase Transitions and Critical Phenomena, vol. 18, pp Academic Press, San Diego 2001) 7. Hill, T.L.: Statistical Mechanics: Principles and Selected Applications. McGraw-Hill, New Yor 1956) 8. Jansen, S.: Mayer and virial series at low temperature. J. Stat. Phys. 2012). doi: / s Jansen, S., König, W., Metzger, B.: Large Deviations for Cluster Size Distributions in a Continuous Classical Many-Body System 2011, preprint) 10. Lebowitz, J.L., Penrose, O.: Cluster and percolation inequalities for lattice systems with interactions. J. Stat. Phys. 16, ) 11. Mürmann, M.G.: Equilibrium properties of physical clusters. Commun. Math. Phys. 45, ). 12. Radin, C.: The ground state for soft diss. J. Stat. Phys. 26, ). 13. Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262, )