Distribution of cracks in a chain of atoms at low temperature

We consider a one-dimensional classical many-body system with interaction potential of Lennard-Jones type in the thermodynamic limit at low temperature $1/\beta\in(0,\infty)$. The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with $N$ particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of $N\exp(- \beta e_\mathrm{surf}/2)$ with $e_\mathrm{surf}>0$ a surface energy. For the proof, the system is mapped to an effective model, which is a low-density lattice gas of defects. The results require conditions on the interactions between defects. We succeed in verifying these conditions for next-nearest neighbor interactions, applying recently derived uniform estimates of correlations.


Introduction
A fundamental problem in statistical and solid mechanics is to gain insight into the structure of matter and to derive material properties from basic atomistic interaction models. A complete theoretic understanding of why atoms at low energy arrange in (almost) periodic patterns and how defects form appears to be out of reach in full generality to date. In view of this state of affairs, recent years have witnessed remarkable progress under simplifying assumptions and shed light on a number of important model cases.
A basic, yet non-trivial model problem is given by one-dimensional chains of atoms. Assuming that particles interact via a classical pair interaction potential such as the Lennard-Jones potential, their crystallization in ground states at zero temperature has been well understood since the pioneering contributions [Ven78,GR79,Rad84,RS83]. Even results in a purely quantum mechanical framework have been obtained more recently in [BLB02]. Allowing for configurations whose energy is slightly larger while keeping the temperature to be zero, one is led to considering chains of atoms that may undergo fracture. Also this regime is well examined by now, in particular for systems with nearest neighbor (NN) and next-to-nearest neighbor (NNN) interactions, see [BC07,SSZ11,Hud13]. In contrast to the case of pure nearest neighbor interactions such as [Tru96,BDMG99,BG02], such models show a non-trivial competition between NN bonds lying in the convex region and NNN bonds occupying the concave region of the interaction potential. This, in particular, leads to nontrivial surface corrections and boundary layers. Extension to more general finite range interactions are considered in [BLO06,SS18], a detailed analysis of boundary layers and surface energies is provided in [JKST20].
We also mention that, by way of contrast, much less is known in higher dimensions, still within the zero temperature regime. We refer to [HR80,Rad81,SmH05,The06,FT15] for crystallization results for specific pair potentials and to [AFG00,FS14,FS15a,FS15b] for partial results relating atomistic models to a corresponding variational continuum Griffith functional.
Still in a one-dimensional setting, in our recent contribution [JKST20] we proved that at small but non-zero temperature a chain of atoms under positive pressure is well approximated by the corresponding ground state configuration. In the present article we show that the picture is rather different for a chain of atoms in thermal equilibrium at small non-zero temperature and vanishing pressure. Indeed, at positive temperature, no matter how small, there is no global crystallization in one-dimensional systems for typical interaction potentials. Yet we will see that an alternating pattern of crystalline clusters and cracks emerges whose statistics can be precisely described in terms of an associated surface/defect energy.
We assume that atoms interact via a Lennard-Jones type potential, energy minimizers have their interatomic spacing (in the bulk) equal to the minimizer a > 0 of a Cauchy-Born energy density. Thermal equilibrium is investigated within the framework of classical equilibrium statistical mechanics [Rue69,Pre09]. This means that we study families of probability measures indexed by the number N of atoms, the length L of the chain, and a positive parameter β > 0 called inverse temperature. Each configuration has probability weight proportional to exp(−βU ), with U the energy of the configuration. Our results are formulated as asymptotic estimates for finite N, L, β but they should be read with the following limits in mind: thermodynamic limit L, N → ∞ at fixed ℓ = L/N first, low-temperature limit β → ∞ second.
Our main result (Theorem 2.3) roughly says that for elongated chains i.e. L/N = ℓ > a, and large β, the chain of atoms typically fills space by alternating approximately crystalline domains with empty regions of space. By approximately crystalline domain we mean a cluster of consecutive atoms with spacing approximately equal to the optimum ground state spacing a, except at both ends of the cluster where boundary layers may form. Empty regions of space occur when two consecutive atoms x j ≤ x j+1 are separated by a large spacing z j = x j+1 − x j , which we call gaps, voids, cracks, or broken bonds. We identify a surface free energy e (R) surf (β) for a threshold parameter R which corresponds to a critical length beyond which atomic bonds are considered broken and whose R-dependence vanishes for β → ∞ and show that: (i) The number of cracks (hence also the number of crystalline clusters) is of the order of N √ ℓ − a exp(−βe (R) surf (β)/2), (ii) the typical length of a crack is of the order of √ ℓ − a exp(βe (R) surf (β)/2), and (iii) the typical number of atoms in a crystalline cluster is of the order of exp(βe (R) surf (β)/2)/ √ ℓ − a. A posteriori we will see that this asymptotic behavior is in fact essentially independent of the choice of R for large β.
In particular, the number of cracks is not bounded but instead proportional to the number of atoms in the chain, moreover each crack is of microscopic length even though the length is exponentially large in β. This behavior is similar to one-dimensional Ising chains with nearest neighbor interaction at low temperature [ST90] or with Kac interactions and small Kac parameter [COP93,CMP18]. For the Ising model intervals of positive magnetization and negative magnetization play a role analogous to our crystalline clusters and empty domains.
Let us briefly explain what makes our result demanding. One-dimensional Gibbs measures with finite-range interactions are easily treated with transfer operators and infinite-dimensional versions of Perron-Frobenius theory; absence of phase transitions, analyticity of thermodynamic potentials, and decay of correlations follow right away [Rue69]. The challenge taken up here is to characterize how the objects whose existence is trivially guaranteed by Perron-Frobenius theory depend on the inverse temperature β. In principle it should be possible to do so by studying the β-dependency of the transfer operator. However, as pointed out by Cassandro, Merola, and Presutti already in the context of the Ising model, "to carry out the whole program along these lines looks maybe possible but not easy at all" [CMP18]. Therefore we follow a different route and instead map the chain of atoms to an effective model which is a weakly interacting lattice gas of defects, see Sections 3.2 and 5.1. Sites j of the lattice correspond to labels of nearest-neighbor spacings. A particle or defect is present in the effective model if the original bond is broken, i.e., the gap is large. The presence of a defect comes with a small weight q β > 0, the effective activity, which is related to the free energy of formation of the defect (see Remark 2). Defects separated by k particles have an effective interaction V β (k) which can be written in terms of the energy of the particles enclosed between cracks (Eqs. (3.6) and (5.3)).
A principal difficulty is to show that the effective interaction between defects can be neglected. This is achieved with Theorem 5.7, which is our main technical result, for chains of atoms with NN and NNN interactions. It crucially relies on bounds on the decay of correlations. Such estimates are highly nontrivial for interactions beyond nearest neighbors since, as alluded to above, boundary layers will form and give rise to non-trivial surface energy contributions. In the case of NN and NNN interactions, a sufficiently strong result has recently been established in [JKST20, Theorem 2.11]. With a view to the interesting question if our conclusions extend to more general finite range interactions, we formulate our results in a way that allows for such an adaption subject to sufficiently good decay of correlations estimates becoming available.
The article is organized as follows. Section 2 describes the model and main results. Section 3 explains key proof ingredients, namely equivalence of ensembles and, on a heuristic level, the effective lattice gas of defects. In addition it proposes an alternative scenario replacing Theorem 2.3 when some of our assumptions fail. Section 4 analyzes in detail the corresponding class of effective models. These general results are applied in Section 5 to the constant-pressure ensemble, the results for the canonical ensemble are deduced in Section 6.

Model and main results
2.1. Model. Consider N particles on a line, with positions 0 = x 1 < x 2 < · · · < x N = L and spacings z j = x j+1 − x j . Neighboring particles and next-to-nearest neighbors interact via a pair potential v : [0, ∞) → R ∪ {∞} which is repulsive for short distances (in fact we shall assume the existence of a hard core) and attractive for spacings larger than a unique energy minimizing bond length. The precise assumptions are collected in Section 2.2 below. The total energy of a configuration (x 1 , . . . , x N ) is Since our analysis extends in a straightforward way to more general interactions involving a finite number of m ∈ N particles subject to improved estimates on correlations being available, cf. the discussion in Section 1, we more generally consider We allow for m ∈ N ∪ {∞} and specify explicitly whenever m < ∞ or m = 2 is exploited, but will sometimes drop the superscript m so as to lighten notation. The canonical partition function at inverse temperature β > 0, with one particle pinned at x 1 = 0 and another at x N = L is where we put z N −1 = L − N −2 j=1 z j . The canonical Gibbs measure is the probability measure P (m,β)

N,L
on Ω N,L := {z ∈ R N −1 Fix an average spacing ℓ > 0. The Helmholtz free energy per particle is The existence of the limit (2.1) and some basic properties are well-known [Rue69, Chapter 3]. Moreover ℓ → f (m) (β, ℓ) is convex and continuously differentiable [DM67], see also [Rue69,Chapter 3.4.8]. For one-dimensional systems and the pair potentials under consideration, there is no phase transition, and ℓ → f (m) (β, ℓ) is strictly convex and analytic [GMS70,Dob73,Dob74,CO81]. The above references refer to the model with m = ∞; for finite m, such results are easily proven using transfer operators [Rue69,Chapter 5.6]. The pressure is defined as In addition to the free energy and pressure, we provide results on the distribution of interparticle spacings. We investigate the following objects. Let R > 0 be some large truncation parameter. We refer to spacings z j ≥ R as cracks and to groups of particles enclosed between consecutive cracks as clusters. Let be the number of clusters. For M N = n + 1 let 1 ≤ i 1 < · · · < i n ≤ N − 1 be the indices i for which z i ≥ R. We also set i 0 = 0 and i n+1 = N . Let be the empirical distributions of the cluster cardinalities and of the crack lengths (minus R); note that they are probability measures.

2.2.
Assumptions. In this section we introduce and discuss the four assumptions on the pair potential under which we will be working throughout this article. Their role is threefold. First of all, they ensure that standard results from statistical mechanics concerning the existence of the thermodynamic limit, continuity of the pressure and absence of phase transitions in dimension one apply. Second, they ensure the periodicity of ground states and allow for a transfer of the low-temperature asymptotics from our previous article [JKST20]. Third, for average spacings ℓ larger than the ground state periodicity a, they allow us to estimate interactions across cracks and to show that cracks do not aggregate.
(i) Shape of v: z max is the unique minimizer of v and satisfies v(z max ) < 0. Furthermore, v is decreasing on (0, z max ) and increasing and non-positive on (z max , ∞).
v ′′ is decreasing on [z min , z max ] and increasing and non-positive on [2z min , ∞).
Assumptions 1(i)-(v) are rather generic conditions on a pair potential with hard core. They are imported from [JKST20] and we refer to [JKST20] for a thorough discussion of these assumptions. They also allow us to estimate interactions across cracks: Indeed, under Assumption 1(i)-(v), there exists a constant C ≥ 0 such that, for any N ∈ N, R ≥ z max , m ∈ N ∪ {∞}, and z 1 , . . . , z N −1 satisfying We leave the elementary proof to the reader. Assumption 1(vi), which relates the hard core radius to the full repulsive zone of v, is a mild technical assumption which enters in the proof of Theorems 5.7 and 5.9 below. In view of the fact that typical next-to-nearest neighbor bonds are attractive it might be achieved upon enlarging r hc while keeping the essential properties of the model. For compactly supported potentials, interactions across cracks vanish if R is longer than the interaction range. Our next assumption ensures this, more generally, it ensures that the entropic push for large crack lengths wins over the attractive part of the interaction; it enters in Lemma 5.4 below.
Assumption 2 (on the truncation parameter). The truncation parameter R ≥ z max is so large that it satisfies • C/R s−2 < e surf /2 otherwise, with C > 0 as in Eq. (2.4) and e surf > 0 as in Eq. (2.5) below.
In fact, as our analysis in the second case does not make use of an infinite interaction range, the second condition is sufficient for compactly supported potentials as well. We distinguish the case R ≥ sup supp(v) as it allows for an approximation with an ideal lattice gas. We close this subsection commenting on the role of the interaction parameter m. A restriction to finite range m < ∞ is natural for compactly supported potentials with a hard core. Indeed, if v(r) = ∞ for r ≤ r hc for some r hc > 0 and v(r) = 0 for r ≥ R * , then any configuration z = (z 1 , . . . , z N −1 ) with finite energy satisfies v(z i + · · · + z j−i ) = 0 whenever |j − i|r hc > R * . Hence, U For v with unbounded range a restriction to finite (and in fact small m) is quite common in atomistic models of solid state physics and indeed less restrictive than a truncation of the potential v itself as leading order contributions to crack energies are still kept. As our main theorems are proven for NN and NNN interactions, we introduce the following assumption.
While most auxiliary results apply to m ≥ 3 as well, Assumption 3 enters in the proof of Theorem 5.7 below, where we need a good control on the β-dependence of correlations [JKST20, Theorem 2.11]. Improved correlation bounds for general interaction range may help get rid of the restriction, but this is beyond this article's scope.
2.3. Results. In this section, we formulate our two main results on the large-N behavior of the Nparticle system at low temperature: Theorem 2.2 on the free energy and the pressure, and Theorem 2.3 on the statistics of cluster sizes, crack lengths, and number of clusters.
First, we need to recall results from [JKST20]. Let be the N -particle ground state energy. The following limits exist: The ground state energy per particle e 0 is characterized with the help of the Cauchy-Born density as e 0 = inf r>0 W (r) = W (a) with a ∈ (z min , z max ) the unique global minimizer of W [JKST20, Section 2.1]. The surface energy e surf accounts for boundary layers at the end of long chains. The reader may also think of e surf as the energy of a defect consisting of a large spacing z j , see Remark 2 below. For positive temperature, analogous quantities and assertions are collected in the following proposition. The truncated partition function appearing on the left-hand side of (2.6) will play an important role in the present article.
Proposition 2.1 ( [JKST20]). Under Assumptions 1(i)-(v), for every β > 0 and 0 ≤ p < |v(z max )|/z max , there are uniquely defined quantities g (R) surf (β, p), g (R) (β, p) such that, as N → ∞, (2.7) In particular, the R-dependence vanishes in the zero-temperature limit. Some technical remarks are in order. Indeed, Sections 2.2 and 2.3 in [JKST20] assume a fixed positive pressure constant p > 0. Those results extend to p = 0 or temperature-dependent pressures p = p β → 0 if the integration is restricted to compact intervals z j ∈ [0, R]. Indeed the positivity of p is only needed to ensure exponential tightness, see [JKST20, Lemma 5.1 and 5.3]. But exponential tightness comes for free in compact spaces, the condition p > 0 is no longer needed.
Our first result concerns the asymptotics of the free energy and the pressure as β → ∞ at fixed elongation ℓ.
The theorem is proven in Section 6.1. The leading-order asymptotic behavior of the free energy density in both (a) and (b) is written more succinctly with the convex hull W * * of W as We remark that in view of our general assumptions on v we cannot expect the Cauchy-Born rule to hold near r hc , so that ℓ * > r hc in general. Our second result describes in detail the distribution of cracks for elongated chains ℓ > a. The case ℓ < a corresponds to positive pressure and was already analysed in detail in [JKST20]. Define For simplicity we suppress the R-dependence from the notation for q β,ℓ . We let Geom(p) denote the probability measure on N with probability weights p(1 − p) k−1 , and || · || TV the total variation distance between two probability measures, i.e., ||µ − ν|| TV = sup A |µ(A) − ν(A)|. Note that both p(β, ℓ) and q β,ℓ behave as exp(−βe surf /2 + o(β)).
The theorem is proven in Section 6.2. These estimates imply three laws of large numbers for N → ∞ under the distribution P (β) N,ℓN at sufficiently low temperature with exponentially fast decay of the deviation from the mean by some threshold that is vanishingly small when β is large. In particular, the number of clusters, M N , behaves like N q β,ℓ (1 + O(δ (1) β )) with a probability converging to 1 exponentially fast. Furthermore, the number of clusters of size k behaves like N q β,ℓ (1 − q β,ℓ ) k−1 (1 + O(δ (1) β )) for every k ∈ N, and hence the average cluster cardinality is about 1/q β,ℓ . Moreover, the distribution of a typical crack length (distance between neighboring clusters) is approximately an exponential variable with parameter βp(β, ℓ) and hence on average of size 1/βp(β, ℓ).
Theorem 2.3 makes no statement about the spacings inside the clusters (however, see [JKST20] for more precise assertions in the constant-pressure model), but Lemma 5.11(a) implies that the average spacing is ≈ a. Hence, the N particles in the interval [0, ℓN ] are, with high probability, organized into N q β,ℓ clusters that cover each an interval of length a/q β,ℓ and the same number of gaps in between, each with a size ≈ 1/(βp(β, ℓ)). Since a + q β,ℓ /(βp(β, ℓ)) ≈ ℓ, which follows from a comparison of Theorem 2.2(b) with (2.8), this explains how the N particles fill up the system of length ℓN .
We finally remark that our asymptotic estimates are essentially independent of the choice of R. To leading order this is a consequence of Eq. (2.7). It also follows a posteriori from Theorems 2.2 and 2.3 as the crack length, for a fixed R, is exponentially distributed with parameter βp(β, ℓ) = e −βe (R) which is itself exponentially small in 1/β. For any other R ′ > R the probablity of finding spacings which are larger than R but not larger than R ′ thus becomes negligible at large β exponentially fast in 1/β. Remark 1. In the elementary case of nearest neighbor models [Tak42] (i.e., m = 1) and smooth v, one has a = argmin v(r), e (R) surf (β) = −e (R) 0 (β) and In particular, the R-dependence is explicitly seen to enter in exponentially small correction terms only. Harmonic approximations in case of more general pair potentials v would require to replace v ′′ (a) by more complicated terms from Hessians or WKB expansions [Hel02,SL17], see also [JKST20, Section 2.3]. (For related techniques in the context of computational approximation schemes for the simulation of atomistic materials see [BLBLP10, BLPV15, SL17, BDO20].) We do not pursue this here.

Proof ingredients and heuristics
One-dimensional systems are best treated in the constant-pressure ensemble, also called isothermalisobaric or NpT ensemble, which does not fix the length of the N -particle chain but instead fixes the external pressure. We formulate and prove all the results analogous to Theorems 2.2 and 2.3 for the constant-pressure ensemble in Section 5 (Theorems 5.9 and 5.10) and derive Theorems 2.2 and 2.3 from them in Section 6.
In the present section, we introduce the constant-pressure ensemble in Section 3.1 and give in Section 3.2 extensive heuristics about what properties are to be expected and how the various quantities behave and how they are related to each other. We also introduce and explain the effective model to which we will compare the ensemble when we carry out the proofs in Sections 4-5. In Section 3.3 we give a modification of the heuristics in a case that we are not considering rigorously in the present article; it leads to a slightly different picture.
3.1. Equivalence of ensembles and pressure-density (stress-strain) relation. The partition function of the constant-pressure ensemble at pressure p is defined as (3.1) We write Q (β,p) N for the corresponding probability measure on R N −1 . The Gibbs free energy (also called free enthalpy) per particle is The existence of the limit is well-known, moreover p → g(β, p) is concave and it is related to the Helmholtz free energy by the relations [Rue69, Chapter 5.6.6] which formulate the equivalence of the ensembles at the level of thermodynamic potentials. By standard results on Legendre transforms, as f (β, ·) is strictly convex and continuously differentiable, g(β, ·) is strictly concave and continuously differentiable, moreover Explicit computations on the equivalence of ensembles and the stress-strain (or force-elongation) relation for one-dimensional systems with nearest or next-nearest neighbor interactions, in a context closer to applications to materials modelling, are given by Legoll and Lelièvre [LL12, Section 2], see also [BLBLP10].

Effective gas of defects. An important quantity is the truncated constant-pressure partition function
which restricts to small gaps and describes a cluster of cardinality k. Let us give heuristics about its behavior for large β and how it is used for a description of the entire constant-pressure ensemble in terms of a decomposition in its clusters and the gaps in between. We assume that β → ∞ and βp β → 0. We look at a realization of the N -particle ensemble with n ∈ {0, . . . , N − 1} cracks.
be the collection of configurations (chains) that have large gaps (the cracks) precisely at the places i 1 , . . . , i n . Suppose that interaction across cracks can be neglected. It is plausible that Q (R) k (β, p β ) ≈ e −βE k . Then, on the event B N (i), the entire chain decomposes into n cracks and n + 1 clusters: Thus setting We recognize the partition function for an effective lattice gas on {1, 2, . . . , N − 1} with activity q and interaction potential (i, j) → V (j − i). Each site j = 1, . . . , N − 1 corresponds to a bond z j between neighboring particles, and a defect is present at j if z j ≥ R is a crack. If V was neglected, then the lattice gas would be ideal, and the right hand side of (3.7) would be equal to e −β(N e 0 +e surf ) (1 + q) N −1 .
Remark 2. The reader may also think of − 1 β log q as the Gibbs free energy of formation of a defect. Computing free energies of defect formation is a non-trivial task, see e.g. [BDO20] and the references therein. The Gibbs free energy of defect formation is a sum of two contributions: an energetic contribution e surf that accounts for missing interactions across the crack, and an entropic contribution 1 β log(βp β ) that comes from integrating over different possible lengths of the crack z j ≥ R. At fixed pressure only the energetic contribution would survive in the zero-temperature limit, however in our context the pressure is exponentially small in β (see Eq. (3.8) below) and both the energetic and the entropic contributions are relevant.
From the definition (2.5) of e 0 and e surf we know that V (k) → 0 as k → ∞. Hence, we work in a perturbative regime and need to control that V is small enough in an appropriate sense. Criteria for this are well-known. Indeed, according to [Rue69,Theorem 4.2.3], if the quantity qC(β) is small, where then the effect of interactions is negligible, and we may approximate the effective model by the ideal lattice gas.
In this approximation, under the assumption that qC(β) is small, we get a number of consequent crucial approximations. Indeed, the collection of bonds (effective lattice sites) is approximately independent, and the probability that site j is occupied (z j ≥ R) approaches the (tiny) number q/(1 + q) (and with the remaining probability 1/(1 + q), it is not). As a consequence, the number of particles in successive clusters becomes geometric with this parameter. Furthermore, the length z j − R of a crack minus R is approximately exponentially distributed with small parameter βp β and expected length 1/βp β . Additional arguments that analyse the energy term show that any length of a spacing inside a cluster approaches the ground state spacing a. As a consequence, any spacing is ≈ a with probability 1/(1 + q) and ≈ R + 1/βp β otherwise. In particular, the average length of a spacing then is Assuming that q is very small (low density of defects because of large β), the only way that a length ℓ ∈ (a, ∞) can be achieved is that which yields Hence, the smallness of qC(β) would lead to a complete picture of the behavior of the chain, which is the one that we describe in Theorems 2.2 and 2.3. The number C(β) is a common measure in statistical mechanics for the overall strength of the interactions, see [Rue69, Definition 4.1.2]. However, there is a priori no reason that it be small. In general, it can go to infinity exponentially fast as β → ∞. Indeed, . Under our assumptions on the pair potential, we have E k ≥ (k − 1)e 0 for all k ∈ N [JKST20, Lemma 3.2] and hence e surf + e 0 ≥ 0. (In particular, V (k) ≥ 0, which justifies the first inequality in the above estimate.) As soon as the inequality is strict, we find that C(β) → ∞ exponentially fast. Hence, our plan works only if q = q β vanishes as β → ∞ fast enough. If the pressure p β goes to zero not too fast so that βp β ≫ exp(−βe surf ) → 0-for example, by choosing p β as in (3.8)-we see that q = q β → 0. A necessary condition for q β C(β) → 0, when q β is as in (3.8), is certainly that e 0 + e surf /2 ≤ 0. This is indeed the case in which we are working in the present article, see Lemma 5.5 and Theorem 5.7.
3.3. An alternative scenario. Let us present a modified heuristics in the case where e surf /2 > |e 0 |, which we do not handle rigorously in this article. We still assume that inf k∈N Let us make one more approximation step on the right-hand side of (3.7). We introduce the solution and introduce an N-valued random variable T β which assumes the value k ∈ N with probability q β u k β e −βV (k) . Then independent copies of T β play the role of the cardinalities of the clusters. (Notice that the geometric distribution from Theorem 2.3 is recovered with the approximation V (k) = 0, under which u β = 1/(1 + q β ).) The right-hand side of (3.7) can be further transformed using these variables, which we carry out in Section 4.
With the ansatz u β ≈ exp(−βt β ) where t β = exp(−βe surf /2), and with the help of (3.8), Eq. (3.9) becomes (3.10) Further approximations yield (splitting the sum at k = 1) where we used in the last step that |e 0 | < 1 2 e surf . (Here is the point at which the heuristics deviates from the situation considered in this article.) In order to find the expectation of T β , we approximate, again splitting the sum at k = 1, Hence, using that T β assumes each k ∈ N with probability qu k β e −βV (k) and recalling that V (k) = E k − ke 0 − e surf , we see that the average cardinality of a given cluster is Accordingly, the average number of clusters is ≈ N/E[T β ] ≈ N exp(−β|e 0 |). We expect the chain of atoms to have a length given by the number of clusters times the sum of average cluster length and average crack length Since our container has length N ℓ, this suggests βt β ≈ √ ℓ − a exp(−βe surf /2), in agreement with our ansatz for t β . This leads us altogether to a picture that is slightly different from Theorem 2.3: • The cluster size T β is no longer approximately geometric anymore because the dominant contribution to the infinite sum (3.10) comes from bounded k. Put differently, defects tend to gather at finite mutual distance.
On the other side, the following features are the same in both heuristics: • E[T β ] → ∞, and the size-biased law (the cardinality of the cluster containing a given particle) P( T β = k) = kP(T β = k)/E[T β ] is still comparable to a size-biased geometric law with parameter exp(−βt β ). • The crack length has an exponential law with parameter βp β and hence an average length ≈ 1/βp β (but with value ≈ exp(β|e 0 |), see above).
This heuristics provide intuition also in less restrictive situations than under our precise assumptions of Section 2.2. In particular, Eq. (3.10) is applicable with e surf = lim inf k→∞ (E k − ke 0 ) when E k − ke 0 is not convergent, which can happen for non-convex interactions where parity plays a role [BC07]. The comparison of e surf and e 0 as well as the evaluation of inf k∈N (E k − ke 0 ), which in general need not be equal to |e 0 |, are in turn closely related to the location of fracture in zero-temperature models [BC07,SSZ11].

Weakly interacting lattice gas
In this section we analyze an abstract lattice gas model motivated by Eq. (3.7). For the reader's orientation, it is helpful to recall the heuristics of Sections 3.2 and 3.3 until (3.11). We will have no parameter β.
In Section 4.1 we introduce the model and find some first properties of its free energy in terms of standard renewal theory. In Section 4.2 we introduce the random variable T that plays the role of the number of clusters and derive precise estimates about its distance to the geometric distribution. Large-deviation principles and the relevant estimates are derived in Section 4.3.
4.1. Effective free energy. It is convenient to work with f (k) = e −βV (k) − 1 rather than the interaction itself. Thus we assume that a number q > 0 and a map f : N → [−1, ∞) are given such that (4.1) Put differently, we assume that C := ∞ k=1 |f (k)| is finite, and q = ε/C for 0 ≤ ε < 1. We think of q and ε as small numbers, whereas f (k) can be large, at least for small k. Consider the partition function It can be studied either directly, using standard tools of statistical mechanics such as cluster expansions, or with the help of standard renewal theory from probability theory; see [Fel71,Chapter XI].
We are going to use the latter. Let u ∈ (0, 1) be the unique solution of and let T, T 1 , T 2 , T 3 , . . . be independent identically distributed random variables with law (4.4) (Then T i plays the role of the cardinality of the i-th cluster.) The partition function (4.2) of the defect gas is related to the random variables T i by = u −N P(∃n ∈ N : T 1 + · · · + T n = N ).
(Recall that the integers i k correspond to locations of cracks and the variables T k count cluster sizes, i.e., the number of points enclosed between two successive cracks.) It follows from standard renewal theory that u N Z N (q) → 1/E[T ] as N → ∞, hence the effective free energy is given by It is actually close to log(1 + q), as we have that (4.6) Indeed, by a straightforward computation, Eq. (4.3) is equivalent to Then (4.6) follows from the fact that u ∈ (0, 1) and the monotonicity of x → x/(1 − x) in (−1, 1).
In addition to the formula (4.5) for the free energy, renewal theory also yields an explicit description of the thermodynamic limit: as N → ∞, the bulk behavior is given by a stationary renewal process. In our setup, this means in particular that the probability that a nearest neighbor bond (i, i + 1) is broken has probability 1/µ where µ = E[T ], and given that the bond is broken, the particle i + 1 belongs to a k-cluster with probability P(T = k) (same statement for particle i).

4.2.
Approximately geometric variables. We continue our analysis of the abstract gas model introduced in Section 4.1. For non-interacting defects, that is, V (k) = 0, i.e., f ≡ 0, we have u = 1/(1 + q) and the random variable T (standing for the cluster size) has precisely a geometric law. For weak interactions and small q, we may expect approximately a geometric law. In this section we provide explicit estimates.
Lemma 4.1. For every r ∈ N and τ > 0, The function g is analytic on the open half-plane {t ∈ C : ℜt < 0} and bounded by ε on the closed half-plane {t ∈ C : ℜt ≤ 0}. Let t = −τ q ∈ (−∞, 0) and r ∈ N. By Cauchy's formula, Let G be a geometric random variable with law P(G = k) = q(1 + q) −k for k ∈ N. We compare the laws L(T ) and L(G) of T and G. Let T and G be the size-biased variables associated with T and G, i.e., P( T = k) = kP(T = k)/E[T ] and P( G = k) = kq 2 (1 + q) −k−1 . Recall the total variation norm µ TV = k∈N |µ(k)| of a signed measure µ on N.

The latter term is equal to
It follows that T − G TV is of order O(ε). The size-biased distributions are treated in a similar way.
We also need some control of the cumulant generating function of T and its Legendre transform. Let (1 + f (k))u k e tk , I(x) = ϕ * (x) = sup t∈R (tx − ϕ(t)). . In view of the geometric approximation, we expect var[T ] ≈ 1/q 2 , and that the quadratic approximation to I(x) for x ≈ µ becomes I(x) ≈ 1 2 q 2 (x − µ) 2 . The next lemma provides a corresponding lower bound with some uniformity as q, ε → 0.

Large deviations.
The system that we wish to investigate can be expressed exactly in terms of a lattice gas of defects as in Section 4.1 only when interactions across cracks vanish, i.e., for compactly supported potentials, see Assumption 4 in Section 2.2. In the general case, we estimate the contribution of interactions across cracks by some small number λ times the number of cracks, see Lemma 5.1 below. In order to quantify the effect of this small contribution we use large deviations theory. Providing this is the purpose of the present section. We keep all the notation from Sections 4.1 and 4.2.
For the reader's convenience, we briefly repeat what a large deviations principle (LDP) is, see [DZ98] for more about this theory. We say that a sequence of random variables X N with values in a Polish space X satisfies an LDP with speed N and with lower semi-continuous rate function I : if for every open set G ⊂ X and every closed subset F ⊂ X , The intuitive idea behind this is that P(X N ≈ x) ≈ e −N I(x) for x ∈ X . Below, we will be working with X chosen as N and the set of probability measures on N and the product of the two. Let (M N ) N ∈N be a sequence of N-valued random variables with law where λ = −ϕ(t). It follows that This proves Eq. (4.12). Now ϕ is a smooth, strictly convex, monotone increasing bijection from (−∞, − log u) onto R. It follows that ψ : R → (log u, ∞) is a monotone increasing bijection and strictly convex as well. The Gärtner-Ellis theorem shows that (M N /N ) N ∈N satisfies a large deviations principle with speed N and rate function J = ψ * . The explicit expression for the Legendre transform ψ * follows from I = ϕ * and [HUML03, Theorem 4].
Next, let be the empirical distribution of T 1 , . . . , T M N . Then ν N is a random variable with values in the space of probability measures on N equipped with the topology of weak convergence. Note that on this space weak convergence is equivalent to pointwise convergence.
Lemma 4.6. There exist δ > 0 and c, C > 0 such that the following holds for all ε, q, λ ∈ (0, δ). With for y > 0 and J(y, ν) = ∞ otherwise. Let δ, λ, C, A be as in Lemma 4.5. Thus A c = {y : with the help of Varadhan's lemma. On A c × B, we have Here Lemmas 4.5 and 4.6 are formulated in terms of the variable T only. Combined with the information that T is close to the geometric variable G introduced in Section 4.2, we obtain the following.

Constant-pressure ensemble
In this section we formulate and prove the analogs of Theorems 2.2 and 2.3 in the constant-pressure ensemble. Our main results in this section are Theorems 5.9 and 5.10. Motivated by the heuristics from Section 3 we focus on β-dependent pressures p β such that as β → ∞. (5.1) Recall that we write Q (β) N for the Gibbs measure in the constant-pressure ensemble, i.e., the probability measure on R N −1 + with probability density z = (z 1 , . . . , , and the partition function Q N (β, p β ) is given by (3.1).

Effective model. As in (3.4), define the truncated partition function by
when N ≥ 2, and set Q (R) 1 (β, p) := 1. Recall that Assumption 1(i) and 2 imply that v(r) ≤ 0 for r ≥ R. As interactions across cracks are always zero or negative, discarding them decreases the Boltzmann weight exp(−βU ). Summing over the number and locations of cracks, we get the inequality with the conventions i 0 = 0 and i n+1 = N . (If v = 0 on (R, ∞), the inequality (5.2) is in fact an equality.) With g (R) (β, p β ), g (R) surf (β, p β ) as in Proposition 2.1 we define For simplicity we suppress the R-dependence from the notation for V β , f β , and q β . In this notation, Eq. (5.2) becomes (5.6) Let T, T 1 , . . . be i.i.d. random variables as in Section 4 with f = f β and q = q β . The β, R-dependence is suppressed from the notation. Think of T k = i k − i k−1 in Eq. (5.6). Then with u β = u as in Eq.  N let B N (i 1 , . . . , i n ) be the event that there are exactly n cracks, located after the particles with labels i 1 , . . . , i n , as in (3.5). As in (2.2) we denote by M N the number of clusters in a configuration. To avoid confusion we write M lg N for the lattice gas variable from Eq. (4.11). Also recall the generating function ϕ = ϕ β from Eq. (4.8).
For later purpose we formulate a similar bound for the empirical distribution ν N of the crack lengths defined in (2.3). Let Y i , i ∈ N, be i.i.d. exponential random variables with parameter βp β , defined without loss of generality on the same underlying probability space as the lattice gas variable M lg N . The Y i 's are assumed to be independent of M lg N . Define Lemma 5.2. Under Assumptions 1(i)-(v), for all β > 0, for all n ∈ N 0 , and D a measurable subset of the set of probability measures on R + , we have Proof. We use the notation of Lemma 5.1 and its proof. Refining the first inequality in (5.12), we see with the help of (2.4). The expression in the second line is rewritten with the aid of the i.i.d. exponential random variables with probability density function βp β exp(−βp β r) as Substituting this expression in the second line of (5.14) we obtain compare the third line in (5.12). We divide by Q N (β, p β ), combine with the lower bound (5.7) and obtain an inequality similar to (5.10): .
To conclude, we note that on the event M lg N = n + 1, we have 1

5.2.
Bounds on effective quantities. In order to apply the results from Section 4, we need to check that q β , λ β , and q β ∞ k=1 |f β (k)| are small. We start with q β . Proof. In our previous work [JKST20, Theorem 2.5] we investigated the asymptotic behavior of g(β, p) and g surf (β, p) at fixed p > 0 and for the full partition function Q N (β, p). The strict positivity of the pressure was needed to ensure exponential tightness of measures on R N + or R Z + as β → ∞. For restricted partition functions with spacings in [0, R], exponential tightness comes for free and the results extend to vanishing pressure p β → 0. The asymptotic relations for g (R) (β, p β ) and g (R) surf (β, p β ) follow. Together with the definition (5.5) of q β and our choice of pressure (5.1), this implies the asymptotic behavior of q β .
Next we estimate λ β defined in Eq. (5.8). The following lemma crucially needs Assumption 2 on the size of the truncation parameter R.
Next we make sure that ε β = q β ∞ k=1 |f β (k)| vanishes for β → ∞. Let us first check a necessary condition. Notice that for every fixed k In view of Lemma 5.3, in order that at least lim β→∞ q β sup k |f β (k)| = 0, it is necessary that 1 2 e surf − (E k − ke 0 ) < 0 for all k ∈ N. The following lemma implies this. It is related to bounds derived in [SSZ11], see also [BC07, Remark 2.3].
From [JKST20, Theorem 2.2], we know that the surface energy is smaller than the clamped surface energy, i.e., e surf ≤ − m k=1 kv(ka). Therefore Since m = 2, the right-hand side equals v(a), which is negative.
Remark 3. The proof shows that Lemma 5.5, and thus also the following Lemma 5.6, is valid for m ∈ N ∪ {∞} provided v satisfies the estimate v(a) < m k=3 (k − 2)v(ka). This is e.g. true for the Lennard-Jones potential v(r) = r −2s − r −s provided s ≥ 3.
Corollary 5.8. Under Assumptions 1-3, In particular, q β and f β satisfy the condition (4.1), and the results from Section 4 are applicable. As a preparation for the following proofs, we note that the shift-invariant restricted Gibbs measure µ (R) β at pressure p on [0, R] Z is given by for all f ∈ C([0, R] ℓ ) and sequences i k with i k → ∞ and k − i k → ∞. (The reader is referred to [JKST20] for details on µ (R) β .) In terms of the interaction between a left and a right part of an infinite chain (in particular, W n ((z j ) j∈Z ) = v(z n + z n+1 ) if m = 2) in [JKST20, Proposition 4.9] and its proof one finds the explicit formulae and for f ∈ C([0, R] ℓ ) whenever n ≥ m − 1 for the unrestricted quantities Q (β,p) n+1 , Q n+1 (β, p) which directly transfer to Q (R) n+1 (β, p), Q (R,β,p) n+1 . Proof of Theorem 5.7. Because of our restriction to spacings in [0, R], the results from [JKST20] extend to vanishing pressure p β → 0. This holds true in particular for [JKST20, Theorem 2.11], which together with Proposition 4.9 and its proof in [JKST20] shows the existence of some constants c, γ > 0 such that |f β (k)| ≤ e cβ e −γk , k ∈ N. To see this, we note that [JKST20, Theorem 2.11] gives The claim then follows with a possibly larger c from ) β (e βW 0 )] 2 − 1 and the shift-invariance of µ (R) β . By Lemma 5.6, we have It follows that, for all n ∈ N, by splitting the sum after the n-th summand, Theorem 5.9. Suppose Assumptions 1-3 hold true. Let p β be as in Eq. (5.1) and q β , λ β , ε β as in (5.5), (5.8) and Corollary 5.8. Set δ β := max( λ β , ε β ). Then for suitable c, C, β 0 ≥ 0 and all β ≥ β 0 , It follows in particular that as N → ∞, the probability of the event M N /N = q β (1 + O(δ β )) converges to 1.
Remark 4. The estimate on ν N actually holds true for every δ β ≥ λ β . Moreover, for compactly supported potentials v, we have λ β = 0 and for each j, the distribution of z j − R conditional on z j ≥ R is exactly an exponential law with parameter βp β .
Proof. To avoid confusion we write M lg N and ν lg N for the auxiliary lattice gas variables defined in Eqs. (4.11) and (4.14). The statements for M N /N and ν N are consequences of Lemma 5.1 and Proposition 4.7. More precisely, with A = {n ∈ N : |n − q β N | ≥ Cq β N max{q β , δ β }}, one has  For the empirical distribution of crack lengths, let Y 1 , Y 2 , . . . be i.i.d. random variables with exponential law Y i ∼ Exp(βp β ). The variables are taken independent of T 1 , T 2 , . . . and M lg N . By Lemma 5.2 with ν lg N as in (5.13), .
is the expected length of a k-cluster at inverse temperature β and pressure p while ℓ(β, p) and ℓ (R) (β, p) represent the average spacings between consecutive particles in a chain or cluster with infinitely many particles. (In Lemma 5.11 below we will see that 1 k L (R) k (β, p) → ℓ (R) (β, p) uniformly in p as k → ∞.) Recall that q β and p β are both of order exp(−βe surf /2 + o(β)).
The proof requires several lemmas.
Lemma 5.12. Under Assumption 1 and 3, we have as β → ∞, Proof. From the definition of L (R) k (β, p) and ℓ (R) (β, p) and Lemma 5.11(b), we get for some k-independent c and all sufficiently large β. Letting k → ∞ we find |g (R) surf (β, p β ) − e surf (β)| ≤ cβp β . This proves the first part of the lemma. The expression for q β follows from the definition (5.5) of q β and the fact that βp β = o(β −1 ) by our choice of p β .
In order to analyze the system length, we condition on the number M N of clusters and express the system length as a sum of (conditionally) independent random variables. Let T 1 , T 2 , . . . be i.i.d. random variables with law as in (4.4), representing the cluster cardinalities. Further let X i , Y i be random variables with the following properties: The Y i 's are i.i.d. with law Y i ∼ Exp(βp β ). They are also independent of the T i 's and the X i 's. The X i 's are i.i.d. and satisfy for all k ∈ N and measurable B ⊂ R + , and P( Then Λ n represents the system length conditional on the event {M N = n} that there are n clusters, neglecting the effect of interactions across cracks.
Proof. We have where T is the size-biased variable. Lemma 4.2 tells us that By Lemma 5.11, for every fixed k 1 ∈ N, Since q β → 0 exponentially fast, we have We let first β → ∞, then k 1 → ∞, and find altogether E[X i ] ∼ 1 q β ℓ (R) (β, p β ), and we conclude with Lemma 5.11(a).
Lemma 5.15. Under the assumptions of Theorem 5.10, there exists c > 0 such that for all sufficiently large β, Proof. We use Markov's exponential inequality as in the proof of Lemma 5.13. We have, for |t| ≪ q, Using (4.9) with c = 2 √ c τ we find that β uniformly for small q β and |t| ≤ τ q β where τ ∈ (0, 1) is fixed. It follows that With ε = c λ β and t = 4q β λ β /c the remaining part of the proof is analogous to Lemma 5.13 and is left to the reader.
To conclude, we use Lemma 5.14 for E[X 1 ] and Lemma 5.12 for q β and we obtain the second inequality in Theorem 5.10.
The statements on the empirical distributions are deduced in a similar fashion from the corresponding empirical distributions in Theorem 5.9. For the geometric distributions, the proof is easily completed with the observation ∞ k=1 q β (1 + q β ) k − q β,ℓ (1 + q β,ℓ ) k → 0.