Classical particles in the continuum subjected to high density boundary conditions
aa r X i v : . [ m a t h - ph ] S e p Classical particles in the continuum subjectedto high density boundary conditions
Aldo Procacci ∗ and Sergio A. Yuhjtman † September 18, 2020
Abstract
We consider a continuous system of classical particles confined in a finite region Λ of R d interacting through a superstable and tempered pair potential in presence of non freeboundary conditions. We prove that the thermodynamic limit of the pressure of the systemat any fixed inverse temperature β and any fixed fugacity λ does not depend on boundaryconditions produced by particles outside Λ whose density may increase sub-linearly with thedistance from the origin at a rate which depends on how fast the pair potential decays atlarge distances. In particular, if the pair potential v ( x − y ) is of Lennard-Jones type, i.e. itdecays as C/ k x − y k d + p (with p >
0) where k x − y k is the Euclidean distance between x and y , then the existence of the thermodynamic limit of the pressure is guaranteed in presence ofboundary conditions generated by external particles which may be distributed with a densityincreasing with the distance r from the origin as ρ (1 + r q ), where ρ is any positive constant(even arbitrarily larger than the density ρ ( β, λ ) of the system evaluated with free boundaryconditions) and q ≤ min { , p } . In the area of rigorous results in statistical mechanics it is a widely accepted belief that the entropy,free energy and pressure of a many-body system are independent on the boundary conditionsimposed to the system. This is a well established fact for bounded spin systems in a lattice andsimple proofs can be found in many elementary textbooks. The situation becomes less clear whenunbounded spin systems on a lattice are analyzed. In that case the proofs of the independency of thefree energy from the boundary conditions are much more involved and in general limitations on theallowed boundary conditions are needed, see e.g. [14], [3] and [19]. The situation is even less clearwhen we consider continuous systems constituted by many classical particles confined in a box Λ ofthe d -dimensional Euclidean space R d and interacting via a pair potential (e.g. the Lennard-Jonespotential or other similar potentials). Concerning specifically these systems, the vast majority ofthe rigorous result about the properties of the thermodynamic functions (e.g. pressure, free energy,entropy) in the thermodynamic limit have been deduced (mainly in the sixties/seventies, but alsomore recently) in ensembles submitted to free boundary condition or periodic boundary conditions.We refer the reader in particular to the papers [18], [21], [22], [13], [4], [6], [5], [24], [7], to theoverlooked but relevant papers [1] and [2] and their recent revisitation [15], to the classic books[23], [8], and to [20], [16] for recent significative improvements. ∗ Departamento de Matem´atica, Universidade Federal de Minas Gerais, Belo Horizonte-MG, Brazil - [email protected] † Departamento de Matem´atica, Universidad de Buenos Aires, Buenos Aires, Argentina - [email protected] β and any fixed activity λ is independent ofboundary conditions produced by particles outside Λ whose density may increase sub-linearly withthe distance from the origin at a rate which depends on how fast the pair potential decays at largedistances. In particular, if the pair potential v ( x − y ) is of Lennard-Jones type, i.e. it decaysas C/ k x − y k d + p (with p >
0) where k x − y k is the Euclidean distance between x and y , thenthe existence of the thermodynamic limit of the pressure is guaranteed in presence of boundaryconditions generated by external particles which may be distributed with a density increasing withthe distance r from the origin as ρ (1 + r q ), where ρ is any positive constant (even hugely larger thanthe density ρ ( β, λ ) of the system evaluated with free boundary conditions) and q ≤ min { , q } . We consider a continuous system of classical particles confined in a bounded compact region Λ of R d , which we assume to be a cubic box of size 2 L centered at the origin. So from now on the symbollim Λ ↑∞ means simply that L → ∞ . We denote by x i ∈ R d the position vector of the i th particle ofthe system and by k x i k its Euclidean norm. We suppose that particles interact via a translationalinvariant pair potential v : R d → R ∪ { + ∞} and are subjected to a boundary condition ω generatedby particles in fixed positions outside Λ. The boundary condition ω is a locally finite set of pointsof R d representing the positions of fixed particles in R d . Namely, ω must be a countable set ofpoints in R d (not necessarily distinct) such that for any compact subset C ⊂ R d it holds that | ω ∩ C | < + ∞ (here | ω ∩ C | denotes the cardinality of the set ω ∩ C ). We call Ω the space of alllocally finite configurations of particles in R d and, given a cube Λ ⊂ R d , we denote by Ω Λ the setof all finite configurations of particles in Λ.As usual, we will suppose that each particle inside Λ, say at position x ∈ Λ, feels the effect of theboundary condition ω through the field generated by the particles of the configuration ω which arein Λ c = R d \ Λ. Free boundary conditions correspond to the case ω = ∅ . We are interested instudying the behavior of the system in the limit Λ ↑ ∞ with a given boundary condition ω andhow eventually this limit may be influenced by ω , having in mind that, as the volume Λ invades2 d , the fixed particles of ω entering in Λ are disregarded and only those boundary particles outsideΛ influence particles inside Λ. We will denote below by | Λ | = (2 L ) d the volume of Λ and by ∂ Λ theboundary of Λ. We define, for x ∈ Λ, d Λ x = inf y ∈ ∂ Λ k x − y k In the suite we will frequently use the following notation. Given a configuration ω ∈ Ω, a function f : R d → R ∪ { + ∞} , a cubic box Λ ⊂ R d and a point x ∈ Λ, we set E f Λ ( x, ω ) = X y ∈ ω ∩ Λ c f ( x − y )With this notation, for any fixed volume Λ and for any fixed boundary condition ω , the partitionfunction of the system in the grand canonical ensemble at inverse temperature β ≥ λ ≥ ω Λ ( β, λ ) = ∞ X n =0 λ n n ! Z Λ dx . . . Z Λ dx n e − β " P ≤ i A pair potential v ( x ) is regular if Z R d | e − βv ( x ) − | dx < + ∞ (8) regu As shown in [21] and [18] the finiteness of the integral given in (8) guarantees that the zero-freeregion of the partition function (4) around λ = 0 in the complex plane does not shrink to zero asΛ → ∞ . This implies the existence of a disc D R = { λ ∈ C : | λ | ≤ R } with R independent of Λ inwhich log Ξ ∅ Λ ( β, λ ) is analytic (see e.g. Chapter 4 in [23]). If the model has to describe a real gasthis is a minimal request: at least for small values of the fugacity the system must be a pure gas. Definition 2.3 A pair potential v is superstable if v can be written as v = v + v with v stable and v non-negative and strictly positive near the origin. The existence of the limit (7) and its continuity as a function of λ and β , when particles interactvia a superstable and regular pair potential is a well established fact since the sixties (see [4], [6],[22], [5], [24] and [23]). Much later Georgii [9, 10] showed that the limit (3) with non free boundaryconditions exists if the pair potential, beyond superstable and regular, has a hard-core or divergesin a non summable way at short distances. Its result holds for all “tempered boundary conditions ω (see (2.24) in [9] or (2.6) in [10]), which basically means that, for some finite positive constant t , ω must be such that lim sup Λ →∞ | Λ | − P ∆ δ ∈ Λ δ | ω ∩ ∆ δ | ≤ t where Λ δ is a collection of cubes ∆ δ of fixed size δ > .1.1 Assumptions on the pair potential The translational invariant pair potential v ( x ) is supposed to be Lebesgue measurable and to satisfythe following assumptions.(i) v is superstable . Namely, v can be written as the sum of two functions v = v + v with v stable with stability constant B and v non-negative and strictly positive near theorigin in a strong sense: there exist two constants a > c > v ( x ) ≥ c for all k x k ≤ a (9) deca (ii) v is tempered , namely, there exist b > η : [0 , + ∞ ) → [0 , + ∞ ) such that, Z ∞ η ( r ) r d − dr < ∞ (10) . b and | v ( x ) | ≤ η ( k x k ) for all x ∈ R d such that k x k ≥ b (11) . Note that assumption (ii) is basically equivalent to impose that v is regular according (8) (seecomment after Definition 4.1.2 in [23]). Let us define for later convenience v ± ( x ) = max { , ± v ( x ) } so that v ( x ) = v + ( x ) − v − ( x )Assumption (ii) immediately implies that v − and v + are such that v ± ( x ) ≤ η ( k x k ) for all x ∈ R d such that k x k ≥ b . Moreover, due to stability v − is bounded tout court in the whole space R d , i.e., v − ( x ) ≤ B for all x ∈ R d . Remark We will assume, without loss of generality, that η is a continuous function which isconstant in the interval [0 , b ] at the value 2 B (i.e. η ( r ) = 2 B for all r ≤ b ) so that v − ( x ) ≤ η ( k x k )for all x ∈ R d . Moreover we will choose η sufficiently well behaved in such a way that, for δ > ⊂ R d of size δ , there is a constant C δ independent on the positionof ∆ in R d such that δ d sup y ∈ ∆ δ η ( k y k ) ≤ C δ Z y ∈ ∆ δ η ( k y k ) dy (12) kade We will use inequality (12) in the following.We further define the function V : [0 , + ∞ ) → (0 , + ∞ ) with V ( r ) = Z ∞ R d \ B r (0) η ( k x k ) dx (13) er B r (0) = { x ∈ R d : k x k < r } is the d -dimensional open ball of radius r centered at the originin R d . Note that, due to (10), it holds thatlim r →∞ V ( r ) = 0 (14) vrinfty We now establish the class of boundary conditions under which the thermodynamic limit of thepressure can be proved to exist. We will suppose hereafter that R d is partitioned in elementary cubes ∆ δ of suitable size δ > R dδ the set of all these cubes and, given x ∈ R d , we will denoteby ∆ δ ( x ) the cube of R dδ to which x belongs. Moreover, given a d -dimensional cube Λ of size 2 L centered at the origin of R d , we agree to choose δ in such a way that 2 L/δ is integer and we callΛ δ the set whose elements are the elementary cubes forming Λ.Given ω ∈ Ω, we define the density of ω as the function ρ ωδ : R d → [0 , + ∞ ) : x ρ ωδ ( x ) with ρ ωδ ( x ) = δ − d | ω ∩ ∆ δ ( x ) | (hence ρ ωδ ( x ) is constant for all x ∈ ∆ d ( x )). Since ω is locally finite, ρ ωδ ( x )is everywhere finite. Definition 2.4 Given a superstable and tempered pair potential v according to the assumptions (i) and (ii) , a continuous monotonic non-decreasing function g : [0 , + ∞ ) → [0 , + ∞ ) is called admissibleif the following conditions hold. g ( α + β ) ≤ g ( α ) + g ( β ) (15) subli Z R d η ( k x k ) g ( k x k ) dx < + ∞ (16) integr An admissible function is called non-trivial if lim u →∞ g ( u ) = + ∞ (17) ntriv Let ρ be a non-negative constant and let g : [0 , + ∞ ) → [0 , + ∞ ) be admissible. We will setΩ ρ,g = { ω ∈ Ω : ρ ωδ ( x ) ≤ ρ (1 + g ( k x k )) , ∀ x ∈ R d } (18) orog and define the set of allowed configurations asΩ ∗ g = ∪ ρ ≥ Ω ρ,g (19) oog Therefore, the allowed configurations in Ω ∗ g are those whose density increases at most as ρ (1+ g ( k x k ))for some constant ρ , where g is an admissible function according to Definition 2.4. Note that if g isnon-trivial, the density ρ ωδ ( x ) of a configuration ω ∈ Ω ∗ g becomes arbitrarily large as we move awayfrom the origin. On the other hand when g is identically zero, Ω ∗ is the set of configurations withbounded density. Remark . It should be noted that although the intermediate set Ω ρ,g defined in (18) depends on δ , the set of allowed configurations Ω ∗ g defined in (19) does not depend on the choice of δ .Indeed, consider two different partitions of R d where the cubes have sizes δ and δ . We take ω ∈ Ω δ ρ,g and we will show that, for some finite ˜ ρ , ω ∈ Ω δ ˜ ρ,g . Let x ∈ R d and consider all the cubes6n the δ -partition that intersect the cube of the δ -partition containing x . Take points y , ..., y m ,one in each such cube. Then, for some constant K depending only on δ and δ , we have: ρ ωδ ( x ) ≤ m X i =1 Kρ ωδ ( y i ) ≤ m X i =1 Kρ (1 + g ( k y i k ))For k x k sufficiently large, say k x k > R , we have k y i k ≤ k x k , so using the properties of g we have g ( k y i k ) ≤ g ( k x k ). Thus, in these cases we have ρ ωδ ( x ) ≤ mKρ (1 + 2 g ( k x k )) ≤ mKρ (1 + g ( k x k )) = ρ ′ (1 + g ( k x k ))For the values of x such that k x k ≤ R we can simply pick ρ ′′ > ρ ωδ ( x ) ≤ ρ ′′ ≤ ρ ′′ (1 + g ( k x k ))Therefore if we take ˜ ρ = max { ρ ′ , ρ ′′ } we have that ρ ωδ ( x ) ≤ ˜ ρ (1 + g ( k x k )) for all x ∈ R d .We define, for later use, the function W : [0 , + ∞ ) → [0 , + ∞ ) such that for any r ≥ W ( r ) = Z R d \ B r (0) η ( k x k ) g ( k x k ) dx (20) wr Note that, by (16), we have that lim r →∞ W ( r ) = 0 (21) wrinfty We will now show that the assumptions on the pair potential and on the boundary conditionsestablished above guarantee that the grand canonical partition function defined in (1) is an analyticfunction of λ in the whole complex plane.We first show the following preliminary Lemma. Lemma 2.1 Let v be a pair potential satisfying assumptions (i) and (ii) , let g be admissible andlet ω ∈ Ω ∗ g . Then there exists a finite constant ˜ κ such that, for any x ∈ Λ E v − Λ ( x, ω ) ≤ ˜ κ (1 + g ( L )) Proof . If ω ∈ Ω ∗ g , then there exists ρ ∈ [0 , ∞ ) such that ρ ωδ ( y ) ≤ ρ (1 + g ( k y k )) for all y ∈ R d . Thengiven x ∈ R d and r ≥ E v − Λ ( x, ω ) = X y ∈ ωy ∈ Λ c v − ( x − y ) ≤ X y ∈ ω v − ( x − y ) ≤ X ∆ δ ∈ R dδ sup y ∈ ∆ δ v − ( x − y ) | ω ∩ ∆ δ |≤ δ d X ∆ δ ∈ R dδ sup y ∈ ∆ δ η ( k x − y k ) ρ ωδ ( y )Now, by inequality (12), we have δ d sup y ∈ ∆ δ η ( k x − y k ) ρ ωδ ( y ) ≤ C δ Z y ∈ ∆ δ η ( k x − y k ) ρ ωδ ( y ) dy E v − Λ ( x, ω ) ≤ C δ X ∆ δ ∈ R dδ Z y ∈ ∆ δ η ( k x − y k ) ρ ωδ ( y ) dy = C δ Z R d η ( k x − y k ) ρ ωδ ( y ) dy = C δ ρ Z R d η ( k x − y k )(1 + g ( k y k )) dy ≤ C δ ρ Z R d η ( k x − y k )(1 + g ( k x − y k + k x k )) dy Therefore, E v − Λ ( x, ω ) ≤ C δ ρ (cid:20)Z R d g ( k y k ) η ( k y k ) dy + (1 + g ( k x k )) Z R d η ( k y k ) dy (cid:21) Recalling definitions (13) and (20), and observing that, for any x ∈ Λ we have that k x k ≤ √ dL and that, by (15), g ( √ dL ) ≤ g ( dL ) ≤ dg ( L ), we can conclude that E v − Λ ( x, ω ) ≤ C δ ρ [ W (0) + (1 + dg ( L )) V (0)] ≤ ˜ κ (1 + g ( L ))with ˜ κ = C δ ρd ( W (0) + V (0)). (cid:3) Lemma 2.1 above implies straightforwardly the following Proposition. Proposition 2.1 Let v be a pair potential satisfying assumptions (i) and (ii) , let g be admissibleand let ω ∈ Ω ∗ g . Then the grand canonical partition function defined in (1) is an analytic functionof λ in the whole complex plane. Proof . By assumption (i) on the pair potential v we have X ≤ i 0, we alsohave that the finite volume pressure (2) is well defined and finite for all ( β, λ ) ∈ [0 , + ∞ ) × [0 , + ∞ )as soon as the pair potential is stable and tempered according to (i) and (ii) and ω ∈ Ω ∗ g . Of course,even with p ω Λ ( λ, β ) well defined for every finite Λ and for every ω ∈ Ω ∗ g , the problem of the existenceof the thermodynamic limit (3) and its independency on ω is another story.8 .2 Results We conclude this section by enunciating the main results of this note in form of four Theorems. Thefirst two theorems establish general conditions under which, for any boundary condition ω ∈ Ω ∗ g ,lim sup Λ ↑∞ | Λ | − log Ξ ω Λ ( β, λ ) and lim inf Λ ↑∞ | Λ | − log Ξ ω Λ ( β, λ ) are bounded from above and frombelow by lim Λ ↑∞ | Λ | log Ξ ∅ Λ ( β, λ ) respectively. Theorem 2.1 Consider a continuous system of classical particles interacting through a superstableand tempered pair potential v according to assumptions (i) and (ii) and let ω ∈ Ω ∗ g with g admissibleaccording to Definition 2.4.Let V and W be the functions defined in (13) and (20) respectively and suppose that lim R →∞ g ( R ) R R W ( s ) dsR = lim R →∞ [ g ( R )] R R V ( s ) dsR = 0 (22) diffi Then, for any λ ≥ and β ≥ it holds lim sup Λ ↑∞ | Λ | log Ξ ω Λ ( β, λ ) ≤ lim Λ ↑∞ | Λ | Ξ ∅ Λ ( β, λ ) (23) limsup Remark . Condition (22) basically imposes constraints to the possible growth of the function g depending on how rapidly the potential decays at large distances. Theorem 2.2 Consider a continuous system of classical particles interacting through a superstableand tempered pair potential v according to assumptions (i) and (ii) and let ω ∈ Ω ∗ g with g admissibleaccording to Definition 2.4.Let V be the function defined in (13) and suppose that there exists a continuous function h ( L ) suchthat lim L →∞ h ( L ) = ∞ , lim L →∞ h ( L ) /L = 0 and lim L →∞ (1 + g ( L )) V ( h ( L )) = 0 (24) exi Then, for any λ ≥ and β ≥ it holds lim inf Λ ↑∞ | Λ | log Ξ ω Λ ( β, λ ) ≥ lim Λ ↑∞ | Λ | Ξ ∅ Λ ( β, λ ) (25) liminf The proofs of Theorem 2.1 and Theorem 2.2 are given in Sections 3 and 4 respectively.The next two theorems, which follows straightforwardly from Theorem 2.1 and 2.2, produce tworelevant examples in which the thermodynamic limit of the pressure under boundary conditionsbelonging to Ω ∗ g exists and it is equal to the free boundary condition pressure. Theorem 2.3 Let v superstable and tempered according to assumptions (i) and (ii) and let ω ∈ Ω ∗ (i.e configurations with bounded density), then lim Λ ↑∞ log Ξ ω Λ ( β, λ ) = lim Λ ↑∞ | Λ | Ξ ∅ Λ ( β, λ ) (26) limlim roof . Let us first show that if g = 0 (hence we are considering boundary conditions with boundeddensity), any superstable and tempered pair potential satisfies (22) and therefore, by Theorem 2.1,inequality (23) holds. Indeed, if g = 0 then the condition (22) simply boils down tolim R →∞ R R V ( s ) dsR = 0 (27) faci Now, if lim R →∞ R R V ( s ) ds < + ∞ then equation (27) is trivially true. On the other hand, iflim R →∞ R R V ( s ) ds = ∞ , then by l’Hopital rulelim R →∞ R R V ( R ) dsR = lim R →∞ V ( R ) = 0and thus we have that inequality (23) holds.Secondly, if g = 0, we can choose h ( L ) = √ L which clearly satisfies the hypothesis of Theorem 2.2.Therefore, by Theorem 2.2, also inequality (25) holds. (cid:3) In the second example, we suppose that v is of Lennard-Jones type. In this case the densitydistribution of the boundary condition is allowed to increase sublinearly with the distance from theorigin. Theorem 2.4 Let v superstable and tempered according to assumptions (i) and (ii) and supposethat the function η is such that, for some constant C and some p > , η ( r ) ≤ Cr d + p for all r ≥ b (28) poly Let ω ∈ Ω ∗ g with g ( r ) = r q and q > such that q < 12 min { , p } (29) qsub Then (26) holds true. Proof . Let us first prove that inequality (23) holds by using Theorem 2.1. We start by showingthat the function g ( r ) = r q with 0 < q < p is admissible according to Definition 2.4. Clearlylim r →∞ r q = + ∞ and, recalling that η ( r ) has been chosen to take the value 2 B in the interval[0 , b ], Z η ( k x k ) g ( k x k ) dx ≤ BV d b q + d + S d Z ∞ b r p − q dr < + ∞ where V d and S d are the volume and the surface of the d dimensional unit sphere respectively.Moreover for any q < α, β > α + β ) q ≤ α q + β q . In conclusion g ( r ) = r q is admissible for all q such that 0 < q < min { , p } .Now let us analyze the left hand side of (22).lim R →∞ g ( R ) R R W ( s ) dsR (30) lim 10n what follows we will denote as K , K , . . . constants not depending on R and S d is the surfaceof the unit sphere in d dimensions. Observe that Z R W ( s ) ds = Z R ds Z R d \ B s (0) k x k q η ( k x k ) dx = Z b ds Z R d \ B s (0) k x k q η ( k x k ) dx + Z Rb ds Z R d \ B s (0) k x k q η ( k x k ) dx = Z b ds Z B b (0) \ B s (0) k x k q η ( k x k ) dx + Z b ds Z R d \ B b (0) k x k q η ( k x k ) dx ++ Z Rb ds Z R d \ B s (0) k x k q η ( k x k ) dx ≤ B Z b ds Z B b (0) \ B s (0) k x k q dx + Z b ds Z R d \ B b (0) C k x k d + p − q dx ++ Z Rb ds Z R d \ B s (0) C k x k d + p − q dx where in the last inequality here above we have used that, by assumption, η ( k x k ) = 2 B for k x k < b and we have bounded η ( k x k ) ≤ C k x k d + p for k x k ≥ b . Therefore we get Z R W ( s ) ds ≤ S d " B Z b ds Z bs r d − q dr + Z b ds Z ∞ b Cr p − q dr + Z Rb ds Z ∞ s Cr p − q dr ≤ S d " Bb Z b r d − q dr + Z b ds Z ∞ b Cr p − q dr + Z Rb ds Z ∞ s Cr p − q dr ≤ S d " Bb (cid:18)Z b r d − q dr + Z ∞ r Cr p − q dr (cid:19) + Z Rb ds Z ∞ s Cr p − q dr ≤ K + S d Z Rb Cs p − q ds Hence, g ( R ) R Z R W ( s ) ds ≤ K R q − + CS d R q − Z Rb s p − q ds Since q < 1, the number R q − goes to zero when R → ∞ . Thus, to show (22) we only have to dealwith R q − R R s q − p ds . Since 2 q < p , we can pick a t > q + t < p , which is the same as q − p < − q − t . This t can also be chosen so that q + t < Z R s q − p ds < Z R s − q − t ds = 11 − q − t ( R − q − t − R q − Z R s q − p ds < − q − t ( R − t − R q − ) R →∞ −→ R →∞ [ g ( R )] R R V ( s ) dsR (31) lim can be analyzed proceeding similarly. Doing so we get[ g ( R )] R Z R V ( s ) ds ≤ K R q − + K R q − Z Rb s p ds which goes to zero by imitating the above argument. This time we have to use q < / q < p/ 2. This concludes the proof of inequality (23).Let us now prove that also inequality (25) holds. If g ( L ) = L q where q < min { , p } then wecan choose e.g. h ( L ) = L / . Indeed with this choice we have clearly that lim L →∞ h ( L ) = + ∞ and lim L →∞ h ( L ) /L = 0. Moreover recalling that by hypothesis g ( L ) = L q , V ( L ) ≤ C/L p and q < min { , p } , the l.h.s. of (24) is, for any L > 1, such that(1 + g ( L )) V ( h ( L )) ≤ (1 + CL q ) L p = 1 L p + CL p − q and thus lim L →∞ (1 + g ( L )) V ( h ( L )) = 0. Therefore, by Theorem 2.2 inequality (25) holds. Inconclusion (26) holds true if the hypothesis of Theorem 2.4 stands. (cid:3) In this section we will denote shortly by ~x a generic configuration ( x , ..., x n ) ∈ Λ n so that ~x ∈ Λmeans ( x , ..., x n ) ∈ Λ n for some n ∈ N . We will use below the following shorter notations. ∞ X n =0 λ n n ! Z Λ dx . . . Z Λ dx n ( · ) . = Z Ω Λ dµ λ ( ~x )( · ) (32) poisson v ( ~x ) = X { x,y }⊂ ~x v ( x − y ) E v Λ ( ~x, ω ) = X x ∈ ~x E v Λ ( x, ω ) (33) evecx E v ± Λ ( ~x, ω ) = X x ∈ ~x E v ± Λ ( x, ω ) (34) evecpm So that Ξ ω Λ ( β, λ ) = Z Ω Λ dµ λ ( ~x ) e − β [ v ( ~x ) − E v Λ ( ~x,ω ) ] (35) short Again, we are supposing that R d is partitioned in elementary cubes ∆ δ of size δ > δ chosenin such a way that, for fixed cube Λ of size 2 L centered at the origin, | Λ δ | /δ d is integer so thatΛ δ denotes the set of elementary cubes forming Λ. We let Ω δ to denote the set of configurations12 x ∈ Λ such that in each cube ∆ δ ∈ Λ δ there is one and only one particle. We define the followingcrucial quantity. S ω Λ ( δ ) = sup ~x ∈ Ω δ E v − Λ ( ~x, ω ) (36) sl Note that, if ω ∈ Ω ∗ g then by Lemma 2.1 S ω Λ ( δ ) is well defined since it is bounded from aboveby | Λ δ | ˜ κ (1 + g ( L )). Note also that if S ω Λ ( δ ) = 0, then E v − Λ ( x, ω ) = 0 for all x ∈ Λ, therefore E v Λ ( ~x, ω ) = E v + Λ ( ~x, ω ) for all ~x ∈ Λ and henceΞ ω Λ ( β, λ ) = Z Λ d~xe − βv ( ~x ) − βE v +Λ ( ~x,ω ) ≤ Z Λ d~xe − βv ( ~x ) = Ξ ∅ Λ ( β, λ )which implies trivially (23). Therefore we may suppose without loss of generality thatlim inf Λ →∞ S ω Λ ( δ ) > sig Let us also define K ω Λ = sup x ∈ Λ E v − Λ ( x, ω )Note that S ω Λ ( δ ) > ⇐⇒ K ω Λ > S ω Λ ( δ ), we have, for any δ > 0, that S ω Λ ( δ ) ≥ K ω Λ (38) SK Moreover, via Lemma 2.1, we can bound K ω Λ ≤ ˜ κ (1 + g ( L )) (39) Kbo We will begin the proof of Theorem 2.1 by proving below, as a consequence of the assumed super-stability of the pair potential v , a key lemma (Lemma 3.1 below). Guessing that the statement ofthis lemma may sound rather technical, we anticipate, before enunciating it, its interpretation andits purpose. If we have a configuration of particles inside Λ that feels a strong negative energy fromthe outside particles (measured by the quantity pS ω Λ ( δ ) where p is an integer), then this configura-tion must be constituted by a large number of particles and thus there are many pairs of particlesat short distance. Lemma 3.1 below shows the contribution to the energy of this large number ofshort-distance pairs of particles inside Λ is strongly positive (i.e. of the order p S ω Λ ( δ ) /K ω Λ ). Thispositive energy, as will be shown later on, is more than enough to compensate the effect from theoutside particles, so that this kind of configurations will have low probability density and thus willbe under control. Lemma 3.1 Let δ ∈ (0 , a/ √ d ) . Given a potential v as in the theorem 2.1, let p ∈ N , ~x a configu-ration in a box Λ and ω ∈ Ω such that K ω Λ > and E v − Λ ( ~x, ω ) > pS ω Λ ( δ ) . Then v ( ~x ) ≥ c p ( p − S ω Λ ( δ ) K ω Λ where a and c are the constants appearing in (9). roof . Due to definition (36), if E v − Λ ( ~x, ω ) > pS ω Λ ( δ ), then there exists at least a cube ∆ δ ∈ Λ δ containing p + 1 particles. Indeed if ~x is a configuration with at most p particles in each cube then E v − Λ ( ~x, ω ) ≤ pS ω Λ in contradiction with the hypothesis. Since E v − Λ ( ~x, ω ) > pS ω Λ > ( p − S ω Λ then forthe same reason we can find a cube ∆ δ containing at least p particles of the configuration ~x and,since δ < a/ √ d , all these particles in ∆ δ are at mutual distance less the a . Choose one particleinside ∆ δ , call x its position and call ~x = ~x \ { x } . We have that v ( ~x ) = X x ∈ ~x v ( x − x ) + v ( ~x ) ≥ c ( p − 1) + v ( ~x )Remove now x from ~x so that we are left with the new configuration ~x . This new configurationis such that E v − Λ ( ~x , ω ) = E v − Λ ( ~x, ω ) − E v − Λ ( x , ω ) > pS ω Λ ( δ ) − K ω Λ ≥ pS ω Λ ( δ ) − S ω Λ ( δ ) = ( p − S ω Λ ( δ )So we could extract at least a point from the configuration ~x and yet, for the new configuration ~x , the condition E v − Λ ( ~x , ω ) > ( p − S ω Λ ( δ ) still holds. We can therefore repeat the process andextract m ≥ ~x in such way that pS ω Λ ( δ ) − mK ω Λ > ( p − S ω Λ ( δ )i.e. m must be such that 1 ≤ m < S ω Λ ( δ ) K ω Λ Namely, we can extract m = (cid:22) S ω Λ ( δ ) K ω Λ (cid:23) points from the configuration ~x in such way that for the remaining configuration ~x ′ = ~x \{ x , . . . x m } it holds E v − Λ ( ~x ′ , ω ) > ( p − S ω Λ ( δ )and v ( ~x ) ≥ c (cid:22) S ω Λ ( δ ) K ω Λ (cid:23) ( p − 1) + v ( ~x ′ ) ≥ c (cid:22) S ω Λ ( δ ) K ω Λ (cid:23) ( p − 1) + v ( ~x ′ )Now the remaining configuration ~x ′ has the property E v − Λ ( ~x ′ , ω ) > ( p − S ω Λ ( δ ). So, applying thesame process to bound v ( ~x ′ ) we get v ( ~x ) ≥ c (cid:22) S ω Λ ( δ ) K ω Λ (cid:23) ( p − 1) + c (cid:22) S ω Λ ( δ ) K ω Λ (cid:23) ( p − 2) + v ( ~x ′′ )where now ~x ′′ is such that E v − Λ ( ~x ′′ , ω ) > ( p − S ω Λ ( δ ). Iterating we get v ( ~x ) ≥ c (cid:22) S ω Λ ( δ ) K ω Λ (cid:23) p ( p − j S ω Λ ( δ ) K ω Λ k ≥ S ω Λ ( δ ) K ω Λ (because, by (38), j S ω Λ ( δ ) K ω Λ k ≥ (cid:3) Λ ↑∞ S ω Λ ( δ ) K ω Λ | Λ | = 0 (40) limsl Proof of (40) . We are supposing that Λ is a d dimensional cube centered at the origin of size2 L . We make a partition of R d in elementary cubes ∆ δ of size δ chosen in such a way that Λ isformed by an integer number of elementary cubes and also in such a way that, for some constant C δ , inequality (12) holds.Recalling definitions (34) and (36) we have S ω Λ ( δ ) = sup ~x ∈ Ω δ E v − Λ ( ~x, ω ) ≤ X ∆ δ ⊂ Λ sup x ∈ ∆ δ E v − Λ ( x, ω ) ≤ X ∆ δ ⊂ Λ sup x ∈ ∆ δ E v − d Λ x ( x, ω ) ≤ δ d X ∆ δ ⊂ Λ sup x ∈ ∆ δ X ∆ ′ δ ∈ Λ c sup y ∈ ∆ ′ δ η ( k x − y k ) ρ ωδ ( y )Now, similarly as we did in the proof of Lemma 2.1 we may use inequality (12) to bound, for someconstant C δ , δ d sup y ∈ ∆ ′ δ η ( k x − y k ) ρ ωδ ( y ) ≤ C δ Z y ∈ ∆ ′ δ η ( k x − y k ) ρ ωδ ( y ) dy Hence S ω Λ ( δ ) ≤ C δ X ∆ δ ⊂ Λ sup x ∈ ∆ δ Z y ∈ Λ c η ( k x − y k ) ρ ωδ ( y ) dy ≤ C δ X ∆ δ ⊂ Λ Z y ∈ Λ c sup x ∈ ∆ δ η ( k x − y k ) ρ ωδ ( y ) dy Now, we again bound δ d sup x ∈ ∆ δ η ( k x − y k ) by C δ R ∆ δ η ( k x − y k ) dx and we get S ω Λ ( δ ) ≤ C δ δ d X ∆ δ ⊂ Λ Z y ∈ Λ c (cid:16) Z ∆ δ η ( k x − y k ) dx (cid:17) ρ ωδ ( y ) dy = C δ δ d X ∆ δ ⊂ Λ Z ∆ δ dx Z y ∈ Λ c η ( k x − y k ) ρ ωδ ( y ) dy = C δ δ d Z Λ dx Z y ∈ Λ c η ( k x − y k ) ρ ωδ ( y ) dy I.e., setting K δ = C δ δ d , we get S ω Λ ( δ ) ≤ K δ Z x ∈ Λ dx Z y ∈ Λ c η ( k x − y k ) ρ ωδ ( y ) dy ≤ ρK δ Z x ∈ Λ dx Z y ∈ Λ c η ( k x − y k )(1 + g ( k y k )) dy ≤ ρK δ Z x ∈ Λ dx Z y ∈ Λ c η ( k x − y k ) h g ( k x − y k ) + g ( k x k ) i dy Z y ∈ Λ c η ( k x − y k ) h g ( k x − y k ) + g ( k x k ) i dy ≤ Z k y k≥ d Λ x η ( k y k ) h g ( k y k ) + g ( k x k ) i dy where recall that d Λ x is the distance of x ∈ Λ from the boundary ∂ Λ of Λ. Moreover, sincesup x ∈ Λ g ( k x k ) = g ( √ dL ) ≤ g ( dL ) ≤ dg ( L ) we can bound Z y ∈ Λ c η ( k x − y k )[1 + g ( k x − y k ) + g ( k x k )] dy ≤ Z k y k≥ d Λ x η ( k y k ) h g ( k y k ) + dg ( L ) i dy = W ( d Λ x ) + (1 + dg ( L )) V ( d Λ x )where in the last line we have used definitions (13) and (20). Therefore, setting F ( d Λ x ) = W ( d Λ x ) + (1 + dg ( L )) V ( d Λ x )we have that S ω Λ ( δ ) ≤ ρK δ Z x ∈ Λ F ( d Λ x ) dx Now, recalling that Λ is a d -dimensional hypercube of size L centered at the origin and thus0 ≤ d Λ x ≤ L/ 2, we have that Z x ∈ Λ F ( d Λ x ) dx = Z L F ( r )2 d h (cid:16) L − r (cid:17)i d − dr ≤ dL d − Z L F ( r ) dr and thus we have, for L so large that g ( L ) ≥ S ω Λ ( δ ) ≤ d ρL d − K δ Z L h W ( r ) + (1 + g ( L )) V ( r ) i dr ≤ d ρL d − K δ Z L h W ( r ) + g ( L ) V ( r ) i dr Now, by Lemma 2.1 we have, for Λ sufficiently large (so that g ( L ) > K ω Λ ≤ ˜ κg ( L )Therefore, since | Λ | = (2 L ) d ≥ L d , we have, setting κ δ ≤ d ρL d − K δ ˜ κ , S ω Λ ( δ ) K ω Λ | Λ | ≤ κ δ " g ( L ) R L W ( r ) drL + [ g ( L )] R L V ( r ) drL (41) rhs and thus, given that g satisfies (22), (40) is proved. (cid:3) We are now in the position to prove the Theorem 2.1. Let set E Λ = [ S ω Λ ( δ ) K ω Λ ] | Λ | (42) ella By (40) we have that lim Λ →∞ E Λ | Λ | = lim Λ →∞ (cid:18) S ω Λ ( δ ) K ω Λ | Λ | (cid:19) = 0 (43) flsl Λ →∞ E Λ S ω Λ ( δ ) K ω Λ = lim Λ →∞ (cid:18) | Λ | S ω Λ ( δ ) K ω Λ (cid:19) = + ∞ (44) flsl We now can writeΞ ω Λ ( β, λ ) = Z Ω Λ dµ λ ( ~x ) e − βv ( ~x ) − βE v Λ ( ~x,ω ) = Z Ω Λ : E v − Λ ( ~x,ω ) ≤ E Λ dµ λ ( ~x ) e − βv ( ~x ) − βE v Λ ( ~x,ω ) + Z Ω Λ : E v − Λ ( ~x,ω ) >E Λ dµ λ ( ~x ) e − βv ( ~x ) − βE v Λ ( ~x,ω ) ≤ e βE Λ Z Ω Λ : E v − Λ ( ~x,ω ) ≤ E Λ dµ λ ( ~x ) e − βv ( ~x ) + Z Ω Λ : E v − Λ ( ~x,ω ) >E Λ dµ λ ( ~x ) e − βv ( ~x )+ βE v − Λ ( ~x,ω ) ≤ e βE Λ Z Ω Λ dµ λ ( ~x ) e − βv ( ~x ) + Z Ω Λ : E v − Λ ( ~x,ω ) >E Λ dµ λ ( ~x ) e − β [ v ( ~x ) − E v − Λ ( ~x,ω )] Namely, we getΞ ω Λ ( β, λ ) ≤ e βE Λ Ξ ∅ Λ ( β, λ ) + Z Ω Λ : E v − Λ ( ~x,ω ) >E Λ dµ λ ( ~x ) e − β [ v ( ~x ) − E v − Λ ( ~x,ω )] (45) terms Let us consider the second term in the r.h.s. of inequality (45). By hypothesis v = v + v with v stable with stability constant equal to B . Therefore we can bound v ( ~x ) − E v − Λ ( ~x, ω ) ≥ − B | ~x | + v ( ~x ) − E v − Λ ( ~x, ω )We can now use Lemma 3.1 to bound from below v ( ~x ) − E v − Λ ( ~x, ω ). Let p be defined as thefollowing integer. p = $ E v − Λ ( ~x, ω )2 S ω Λ ( δ ) % + 2By the fact that we are considering here second term in the r.h.s. of inequality (45) where E v − Λ ( ~x, ω ) > E Λ and since (44) implies that E Λ /S ω Λ ( δ ) goes to infinity when Λ ↑ ∞ , we havethat E v − Λ ( ~x, ω ) /S ω Λ ( δ ) is surely larger than 4 for Λ large enough. Then, using that x ≥ ⌊ x ⌋ + 2 forall x ≥ 4, we have E v − Λ ( ~x, ω ) = E v − Λ ( ~x, ω ) S ω Λ ( δ ) S ω Λ ( δ ) > $ E v − Λ ( ~x, ω )2 S ω Λ ( δ ) % + 2 ! S Λ ( δ ) ω = pS ω Λ ( δ )Hence we can use Lemma 3.1 to bound v ( ~x ) − E v − Λ ( ~x, ω ) ≥ c p ( p − S ω Λ ( δ ) K ω Λ − E v − Λ ( ~x, ω )= c $ E v − Λ ( ~x, ω )2 S ω Λ ( δ ) % + 2 ! $ E v − Λ ( ~x, ω )2 S ω Λ ( δ ) % + 1 ! S ω Λ ( δ ) K ω Λ − E v − Λ ( ~x, ω )17 c E v − Λ ( ~x, ω )2 S ω Λ ( δ ) ! S ω Λ ( δ ) K ω Λ − E v − Λ ( ~x, ω )= E v − Λ ( ~x, ω ) " c E v − Λ ( ~x, ω ) S ω Λ ( δ ) K ω Λ − ≥ E Λ (cid:20) c E Λ S ω Λ ( δ ) K ω Λ − (cid:21) where in the last line we have once again considered that we are bounding the second term in r.h.s.of (45) in which the integral is over configurations ~x such that E v − Λ ( ~x, ω ) ≥ E Λ .In conclusion we have obtained that v ( ~x ) − E v − Λ ( ~x, ω ) ≥ E Λ (cid:20) c E Λ [ S ω Λ ( δ ) K ω Λ ] − (cid:21) . = G Λ Let us analyze the behaviour of the ratio G Λ / | Λ | as Λ → ∞ . Recalling (42) and (39), we get G Λ | Λ | ≥ E Λ | Λ | (cid:20) c E Λ S ω Λ ( δ ) K ω Λ − (cid:21) = c (cid:18) | Λ | S ω Λ ( δ ) K ω Λ (cid:19) − (cid:18) S ω Λ ( δ ) K ω Λ | Λ | (cid:19) and thus in force of (43) and (44) we have thatlim Λ →∞ G Λ | Λ | = + ∞ ThereforeΞ ω Λ ( β, λ ) ≤ e βE Λ Ξ ∅ Λ ( β, λ ) + Z ΩΛ Ev − Λ ( ~x,ω ) >E Λ dµ λ ( ~x ) e − β [ v ( ~x ) − E v − Λ ( ~x,ω )] ≤ e βE Λ Ξ ∅ Λ ( β, λ ) + e − βG Λ Z Λ dµ λ ( ~x ) e + βB | ~x | ≤ e βE Λ Ξ ∅ Λ ( β, λ ) + e − βG Λ e + λ | Λ | e βB ≤ e −| Λ | ( βG Λ | Λ | − λe βB ) + e βE Λ Ξ ∅ Λ ( β, λ )and thus 1 | Λ | log Ξ ω Λ ( β, λ ) ≤ | Λ | log (cid:20) e −| Λ | ( βG Λ | Λ | − λe βB ) + e βE Λ Ξ ∅ Λ ( β, λ ) (cid:21) In conclusion, we getlim sup Λ ↑∞ | Λ | log Ξ ω Λ ( β, λ ) ≤ lim Λ ↑∞ | Λ | log (cid:20) e −| Λ | ( βG Λ | Λ | − λe βB ) + e βE Λ Ξ ∅ Λ ( β, λ ) (cid:21) 18 lim Λ ↑∞ | Λ | log h + e βE Λ Ξ ∅ Λ ( β, λ ) i + lim Λ ↑∞ | Λ | log e −| Λ | ( βG Λ | Λ | − λe βB ) e βE Λ Ξ ∅ Λ ( β, λ ) ≤ lim Λ ↑∞ | Λ | log h e βE Λ Ξ ∅ Λ ( β, λ ) i + lim Λ ↑∞ | Λ | log (cid:18) e −| Λ | ( βG Λ | Λ | − λe βB ) (cid:19) = lim Λ ↑∞ βE Λ | Λ | + lim Λ ↑∞ | Λ | log Ξ ∅ Λ ( β, λ )= lim Λ ↑∞ | Λ | log Ξ ∅ Λ ( β, λ )and thus inequality (23) is proved. This concludes the proof of Theorem 2.1. We start by proving the following preliminary lemma. Lemma 4.1 Let g be admissible and let ω ∈ Ω ∗ g . Then there exists a finite constant ¯ κ such that,for any x ∈ Λ such that d Λ x ≥ bE v + Λ ( x, ω ) ≤ ¯ κ (cid:2) W ( d Λ x ) + (1 + g ( L )) V ( d Λ x ) (cid:3) Proof . If ω ∈ Ω ∗ g , then there exists ρ ∈ [0 , ∞ ) such that ρ ωδ ( y ) ≤ ρg ( k y k ) for all y ∈ R d . Moreover,given x ∈ Λ such that d Λ x ≥ b , we have that v + ( x − y ) ≤ η ( k x − y k ) for any y ∈ Λ c . Therefore thuswe can bound E v + Λ ( x, ω ) ≤ X ∆ δ ⊂ Λ c sup y ∈ ∆ δ v + ( x − y ) | ω ∩ ∆ δ | ≤ δ d X ∆ δ ⊂ Λ c sup y ∈ ∆ δ η ( k x − y k ) ρ ωδ ( y )As we did previously (see (12)), we can find a constant C δ such that δ d sup y ∈ ∆ δ η ( k x − y k ) ρ ωδ ( y ) ≤ C δ Z ∆ d η ( k x − y k ) ρ ωδ ( y ) dy Therefore E v + Λ ( x, ω ) ≤ C δ X ∆ δ ⊂ Λ c Z ∆ d η ( k x − y k ) ρ ωδ ( y ) dy = C δ Z Λ c η ( k x − y k ) ρ ωδ ( y ) dy ≤ C δ ρ Z Λ c η ( k x − y k )(1 + g ( k y k )) dy ≤ C δ ρ Z Λ c η ( k x − y k )(1 + g ( k x − y k + k x k )) dy Now, using again (15), we get g ( k x − y k + k x k ) ≤ [ g ( k x − y k ) + g ( k x k )]19nd therefore E v + Λ ( x, ω ) ≤ C δ ρ Z Λ c h g ( k x − y k ) + g ( k x k ) i η ( k x − y k ) dy ≤ C δ ρ (cid:20)Z Λ c g ( k x − y k ) η ( k x − y k ) dy + (1 + g ( k x k )) Z Λ c η ( | x − y k ) dy (cid:21) ≤ C δ ρ "Z k x − y k≥ d Λ x g ( k x − y k ) η ( k x − y k ) dy + (1 + g ( k x k )) Z k x − y k≥ d Λ x η ( | x − y k ) dy ≤ C δ ρd (cid:2) W ( d Λ x ) + (1 + g ( L )) V ( d Λ x ) (cid:3) where in the last line we have again used definitions (13) and (20) and the fact that g ( k x k ) ≤ dg ( L )for any x ∈ Λ. (cid:3) Using Lemma 4.1 we can now conclude the proof of Theorem 2.2. By hypothesis there exists anincreasing continuous function h ( L ) such that lim L →∞ h ( L ) = ∞ , lim L →∞ h ( L ) /L = 0 andlim L →∞ g ( L ) V ( h ( L )) = 0 (46) glvl We take L sufficiently large in such a way that b < h ( L ) < L , and define Λ h = { x ∈ Λ : d Λ x > h ( L ) } so that Λ h is a cube centered at the origin with size 2( L − h ( L )) fully contained in Λ. Thereforewe have that Ξ ω Λ ( βλ ) ≥ Z Ω Λ h dµ λ ( ~x ) e − βv ( ~x ) − βE v Λ ( ~x,ω ) ≥ Z Ω Λ h dµ λ ( ~x ) e − βv ( ~x ) − βE v +Λ ( ~x,ω ) Now by definition, for all x ∈ Λ h we have that d Λ x ≥ h ( L ) > b and thus we can apply Lemma 4.1to bound, for any x ∈ Λ h E v + Λ ( x, ω ) ≤ ˜ κ h W ( h ( L )) + [1 + g ( L )] V ( h ( L )) i Moreover, since, by (14), (21) and (46), lim Λ ↑∞ [ W ( h ( L )) + [1 + g ( L )] V ( h ( L ))] = 0, for Λ largeenough and for any fixed ε > 0, we can bound E v + Λ ( x, ω ) ≤ ε so thatΞ ω Λ ( βλ ) ≥ Z Ω Λ h dµ λ ( ~x ) e − βv ( ~x ) − βε | ~x | = Ξ ∅ Λ h ( β, e − βε λ )Therefore, considering that lim Λ ↑∞ Λ h = + ∞ and that lim Λ ↑∞ | Λ h || Λ | = 1, we getlim inf Λ ↑∞ | Λ | log Ξ ω Λ ( β, λ ) ≥ lim Λ ↑∞ | Λ | log Ξ ∅ Λ h ( β, e − βε λ )= lim Λ ↑∞ | Λ h || Λ | | Λ h | log Ξ ∅ Λ h ( β, e − βε λ )= lim Λ ↑∞ | Λ h || Λ | lim Λ ↑∞ | Λ h | log Ξ ∅ Λ h ( β, e − βε λ )= lim Λ h ↑∞ | Λ h | Ξ ∅ Λ h ( β, e − βε λ )= βp ∅ ( β, e − βε λ )20ow, since the free-boundary condition infinite volume pressure p ∅ ( β, λ ) is continuous as a functionof β and λ , by the arbitrariness of ε we can conclude that,lim inf Λ ↑∞ | Λ | Ξ ω Λ ( β, λ ) ≥ βp ∅ ( β, λ ) = lim Λ ↑∞ | Λ | Ξ ∅ Λ ( β, λ ) (47) Okkk This ends the proof of Theorem 2.2. In this note we considered a d -dimensional system of classical particles confined in a cubic boxΛ interacting via a superstable pair potential in the Grand Canonical ensemble at fixed inversetemperature β > λ > 0. We proved that the thermodynamic limit of the finitevolume pressure of such system does not depend on boundary conditions generated by particles atfixed positions outside the volume Λ as long as these external particles are distributed according toa bounded density ρ ext (even larger as we please than the density ρ ( β, z ) of the system calculatedusing free boundary conditions). We also prove the independency of the thermodynamic limit ofthe pressure of the system in presence of boundary conditions whose density may increase with thedistance from the origin to a rate which depends on how fast the pair potential decays.A related open question (and possibly the subject of a project to come) is whether it is possibleto perform an absolutely convergent Mayer expansion of the pressure of the systems consideredin this note (i.e. interacting via a non-necessarily repulsive pair potential) for fugacities within aconvergence radius uniform in the boundary conditions when these are in the class described above. Acknowledgments A.P. has been partially supported by the Brazilian agencies Conselho Nacional de DesenvolvimentoCient´ıfico e Tecnol´ogico (CNPq - Bolsa de Produtividade em pesquisa, grant n. 306208/2014-8)and Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior (CAPES - Bolsa PRINT, grantn. 88887.474425/2020-00). 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