The Vlasov continuum limit for the classical microcanonical ensemble
aa r X i v : . [ m a t h - ph ] J a n The Vlasov continuum limit for theclassical microcanonical ensemble
Michael K.-H. Kiessling
Department of Mathematics, Rutgers UniversityPiscataway NJ 08854, USA
Abstract
For classical Hamiltonian N -body systems with mildly regular pair in-teraction potential (in particular, L loc integrability is required) it isshown that when N → ∞ in a fixed bounded domain Λ ⊂ R , withenergy E scaling as E ∝ N , then Boltzmann’s ergodic ensemble en-tropy S Λ ( N, E ) has the asymptotic expansion S Λ ( N, N ε ) = − N ln N + s Λ ( ε ) N + o ( N ); here, the N ln N term is combinatorial in origin andindependent of the rescaled Hamiltonian, while s Λ ( ε ) is the system-specific Boltzmann entropy per particle, i.e. − s Λ ( ε ) is the minimumof Boltzmann’s H function for a perfect gas of energy ε subjected to acombination of externally and self-generated fields. It is also shown thatany limit point of the n -point marginal ensemble measures is a linearconvex superposition of n -fold products of the H -function-minimizingone-point functions. The proofs are direct, in the sense that (a) themap E S ( E ) is studied rather than its inverse S E ( S ); (b) no reg-ularization of the microcanonical measure δ ( E − H ) is invoked, and (c)no detour via the canonical ensemble. The proofs hold irrespective ofwhether microcanonical and canonical ensembles are equivalent or not.Typeset in L A TEX by the author. Version of August 11, 2009.Appeared in:
Reviews in Mathematical Physics , 1145–1195 (2009).c (cid:13) Introduction
The rigorous foundations of equilibrium statistical mechanics have largely beenlaid long ago [Rue69, Pen70, Len73, ML79], but the most basic problem inclassical statistical mechanics, namely the rigorous asymptotic evaluation ofGibbs’ microcanonical ensemble [Gib02] in the limit of a large number N of particles, has only been treated in an approximate way. The standardway of dealing with the microcanonical ensemble (a.k.a. Boltzmann’s ergodicensemble [Bol96]) in a rigorous manner [Rue69, Lan73, ML79] has been toreplace its singular ensemble measure by a regularized measure (usually alsoreferred to as microcanonical, although quasi-microcanonical would seem abetter name). In these approaches one cannot take the limit of vanishingregularization; yet, since one can approximate the singular measure as closelyas one pleases, “this is not completely unsatisfactory from a conceptual pointof view” ([Lan73], p.4). All the same, Lanford’s wording makes it plain that itis desirable to find a way to remove the regularization or to avoid it altogether.Recently [Kie09a] the author noticed that after only minor modifications,Ruelle’s method [Rue69] to establish the thermodynamic limit for Boltzmann’sergodic ensemble entropy, taken per volume (or per particle), works withoutthe need for any regularization of the ensemble measure; a follow-up work onthe thermodynamic limit of the correlation functions is planned. Taking “ the thermodynamic limit” [Rue69] means that the domain Λ grows “evenly” with N and such that N/ Vol (Λ) → ρ with ρ a fixed number density, and the energy E scales such that E / Vol (Λ) → ε (or E /N → ε , abusing notation), with ε ∈ R a fixed energy density (or energy per particle) — this limit covers systems ofinterest in condensed matter physics or chemical physics, such as those withhard core or Lennard-Jones interactions.In the present paper we will be concerned with another limit N → ∞ ,where Λ is fixed and E scales such that E /N → ε . This limit covers sys-tems of interest in plasma and astrophysics, such as those with Coulomb or(mollified) Newton interactions. It is variably known as a “thermodynamicmean-field limit,” a “self-averaging limit,” or “Vlasov limit.” We will studythe Boltzmann ensemble entropy and the correlation functions.The remainder of this paper is structured as follows. In section 2 we collectthe defining formulas of the ergodic / microcanonical ensemble for finite N andexplain which probabilistic quantities are of physical interest. In section 3 wegive a heuristic motivation for the Vlasov limit. In section 4 we state our maintheorems, ordered by increasing depth. Their proofs are given in sections 5.1to 5.3. Section 6 lists some spin-offs of our results, and section 7 closes ourpaper with an outlook on some open problems. The first two names refer to a Weiss-type “mean-field approximation” becoming exactin the limit, but we will not invoke any such approximation and speak of the
Vlasov limit . A brief review of the ergodic ensemble
For a Newtonian N -body system in a domain Λ ⊂ R with Hamiltonian H ( N )Λ ( p , . . . , q N ) = X ≤ i ≤ N | p i | + X X ≤ i 3f solids and liquids (etc.) are not revealed by “picturing” such systems asindividual points in R N , one associates each microstate X ( N ) with a uniquefamily of empirical n -point “densities” on ( R × Λ) n , n = 1 , , ..., N . The normalized one-point “density” with N atoms (empirical measure) is given by∆ (1) X ( N ) ( p , q ) = N X ≤ i ≤ N δ ( p − p i ) δ ( q − q i ) (6)and the normalized two-point density with N atoms ( U -statistic of order 2) by∆ (2) X ( N ) ( p , q ; p ′ , q ′ ) = N ( N − X X ≤ i = j ≤ N δ ( p − p i ) δ ( q − q i ) δ ( p ′ − p j ) δ ( q ′ − q j ); (7)similarly the empirical n -point densities with n = 3 , ..., N are defined. Themap X ( N ) → { ∆ ( n ) X ( N ) } Nn =1 is bijective if we insist that the particular labelinggiven to us algebraically with r.h.s.(6) or r.h.s.(7) etc. has an intrinsic mean-ing; however, considered purely measure theoretically as “density” on R n each∆ ( n ) X ( N ) is invariant under the permutation group applied to the particular label-ing, and since there are N ! distinct X ( N ) s obtained by permuting the particlelabels, the map X ( N ) → { ∆ ( n ) X ( N ) } Nn =1 is many-to-one in this sense. Understoodin this measure theoretic way the empirical n -point densities do not depend onthe unphysical (though mathematically convenient) labeling of the particles, and so are physically more natural than points in R N ; when n is small, say n = 1 or 2, then ∆ ( n ) X ( N ) is also “physically more manifest” than a point in R N .Hence, the probabilities of interest to physicists will be of the formProb (cid:16) ∆ ( n ) X ( N ) ∈ e B (cid:17) (8)for physically significant measurable sets e B in P s (( R × Λ) n ), the permutation-symmetric probability measures on ( R × Λ) n . Among the physically significantsets are balls (w.r.t. a suitable topology, still to be chosen) centered at a rep-resentative n -point density function for a solid, liquid, ... , or complements ofsuch balls. As for the topology, the fine ( T V ) topology for P s (( R × Λ) n ) is not suitable as it is equivalent to discriminating between different ∆ ( n ) X ( N ) w.r.t.the Borel sigma algebra of R N = /S N (see footnote 6). Practically accessible are only some considerably less finely resolved events, such as the empirical n -point densities ∆ ( n ) X ( N ) distinguished w.r.t. the weak topology, quantified by So, physically we can identify these N ! distinct X ( N ) s with a single N -point configurationin R × Λ, which is a point e X ( N ) ∈ R N × Λ N = /S N . The subscript = means that coincidencepoints are removed, and S N is the symmetric group of order N . We should also write ∆ ( n ) e X ( N ) ,with the understanding that as measure ∆ ( n ) e X ( N ) is given by ∆ ( n ) X ( N ) for any of the N ! points X ( N ) in the pre-image in R N of e X ( N ) . The map e X ( N ) → { ∆ ( n ) e X ( N ) } Nn =1 is bijective. Even if the balls in T V topology were practically accessible, for N ≫ d KR on P s (( R × Λ) n ). Two verydifferent points in R N = /S N , say e X ( N ) and e Y ( N ) , can map into two densities∆ ( n ) X ( N ) and ∆ ( n ) Y ( N ) which in weak topology on P s (( R × Λ) n ) are virtually indis-tinguishable; here X ( N ) and Y ( N ) are any two representative points out of the N ! points each, which constitute the pre-image in R N of e X ( N ) , respectively e Y ( N ) . So the probabilities of interest to physicists are typically of the formProb (cid:16) d KR (cid:16) ∆ ( n ) X ( N ) , f ( n )eq (cid:17) > δ (cid:17) , (9)where f ( n )eq ( p (1) , q (1) ; ... ; p ( n ) , q ( n ) ) ∈ ( P ∩ C b )(( R × Λ) n ) is an equilibrium densityfunction, defined — in the simplest of all cases — implicitly as the unique function for which (after rescaling of variables and parameters, if necessary)Prob (cid:16) d KR (cid:16) ∆ ( n ) X ( N ) , f ( n )eq (cid:17) > δ (cid:17) N →∞ −→ ∀ δ > . (10)In these simplest of all cases, (10) also explains what is meant by a “representa-tive n -point density function;” and whether f ( n )eq represents a solid, liquid, gas,etc., depends on the specific configurational correlations exhibited by f ( n )eq .In more complicated (and more interesting) situations, several “competing”equilibrium functions f ( n )eq may exist, and (10) has to be modified accordingly.The “simplest case” scenario just described was discovered by Boltzmann(p.442 of [Bol96]), based on his explicit evaluation of (2) for the perfect gas.He realized that when H ( N )Λ is the perfect gas Hamiltonian and N ≫ 1, thenbasically every point of { H ( N )Λ = E } (identified with an n -pt. density throughthe map X ( N ) → ∆ ( n ) X ( N ) ) lies in the vicinity (w.r.t. weak topology) of oneand the same equilibrium density function f ( n )eq at that energy E , and given n .When H ( N )Λ sports non-trivial pair interactions, Boltzmann’s description needsto be modified slightly to account for the phenomenon of phase transitions.While there can hardly be a doubt that Boltzmann’s insight into (2) iscorrect, the rigorous results which support his assessment have been obtainednot for (2) but for some regularized approximation of this singular measure[Rue69, Len73, ML79]. In this paper we will finally vindicate Boltzmann’sideas in the Vlasov regime of the relevant class of Hamiltonians (1). For the ergodic ensemble to exhibit a Vlasov regime the Hamiltonian (1)needs to satisfy additional conditions. In particular, a necessary conditionon the symmetric and irreducible pair potential W Λ is local integrability, i.e. W Λ ( q , · ) ∈ L ( B r ( q ) ∩ Λ) ∀ q ∈ Λ. We remark that for the existence of a dynamical Vlasov regime the local integrability of the forces derived from W Λ 5s mandatory, viz. ∇ q W Λ ( q , · ) ∈ L ( B r ( q ) ∩ Λ) ∀ q ∈ Λ. Coulomb’s electri-cal and Newton’s gravitational interactions belong in either class. Physicallymeaningful external potentials V ( N )Λ are continuous for q ∈ Λ; it has minortechnical advantages to assume that V ( N )Λ is actually continuous also at theboundary, i.e. lim q ′ → q V ( N )Λ ( q ′ ) = V ( N )Λ ( q ) for all q ∈ ∂ Λ and q ′ ∈ Λ. Forconvenience we assume that inf H ( N )Λ ( p , . . . , q N ) = min H ( N )Λ ( p , . . . , q N ) = E g ( N ) > −∞ , and call E g ( N ) the N -body ground state energy; Newton’sgravitational interactions need to be regularized to achieve E g ( N ) > −∞ .In the introduction we have already mentioned that the Vlasov limit scalingfor such interactions is E ≍ N ε for N ≫ 1. We now explain why. Integrating(6) over p -space R gives a normalized one-point “density” (empirical measure)on Λ with N atoms, which by abuse of notation we denote as follows,∆ (1) X ( N ) ( q ) ≡ Z R ∆ (1) X ( N ) ( p , q )d p = N X ≤ i ≤ N δ ( q − q i ) . (11)Whenever Boltzmann’s simplest scenario holds, then there is an equilibriumdensity ρ E ,N ∈ ( P ∩ C b )(Λ), depending on N ( ≫ 1) and E , such that ∆ (1) X ( N ) ( q ) ≈ ρ E ,N ( q ) for overwhelmingly most X ( N ) distributed by (2), where “ ≈ ” meansthe two “densities” do not differ by much in a conventional Kantorovich-Rubinstein metric d KR . This suggests that when Λ ⊂ R is fixed and N → ∞ together with E → ∞ such that E /N α → ε for a yet-to-be determined α ,then ρ E ,N N →∞ −→ ρ ε ∈ ( P ∩ C b )(Λ) and ∆ (1) X ( N ) ( q )d p N →∞ −→ ρ ε ( q ), weakly. Im-plementing this law-of-large-numbers type scenario inevitably leads to α = 2,as is most easily seen if we assume for a moment that W Λ ∈ C b (Λ × Λ). Then q W Λ ( q , q ) is a bounded continuous function in Λ and we can write H ( N ) ( X ( N ) ) = N ZZ | p | ∆ (1) X ( N ) ( p , q )d p d q + N ZZ (cid:16) V ( N )Λ ( q ) − W Λ ( q , q ) (cid:17) ∆ (1) X ( N ) ( p , q )d p d q (12)+ N ZZZZ W Λ ( q , ˜ q )∆ (1) X ( N ) ( p , q )d p d q ∆ (1) X ( N ) (˜ p , ˜ q )d ˜ p d ˜ q, Presumably boundedness below is not technically necessary. We expect that pair inter-actions which diverge logarithmically to −∞ can be accommodated but require additionalweak compactness estimates, e.g. in some L p space; cf. [KiLe97]. R R ∆ (1) X ( N ) ( p , q )d p ≈ ρ ε ( q ), we find H ( N ) ( X ( N ) ) ≈ N ZZ | p | ∆ (1) X ( N ) ( p , q )d p d q + N Z (cid:16) V ( N )Λ ( q ) − W Λ ( q , q ) (cid:17) ρ ε ( q )d q (13)+ N Z Z W Λ ( q , ˜ q ) ρ ε ( q ) ρ ε (˜ q )d q d ˜ q. The last term clearly scales ∝ N because W Λ and ρ ε are independent of N .In a sense this already establishes the E ∝ N scaling. However, we have yetto consider the terms on the first two lines on the r.h.s. of (13). It wouldseem that these scale ∝ N and so, for large N , would become insignificantas compared to the one in the last line, but only the N R W Λ ( q , q ) ρ ε ( q )d q contribution will surely become insignificant for large N , for the same reasonsfor why the last one scales ∝ N ( W Λ and ρ ε do not depend on N ). As for theexternal potential V ( N )Λ ( q ), the superscript ( N ) indicates that we may want toadjust it to the number of particles in the system on which it acts in order toretain a noticeable effect when N becomes large. So in particular we can set V ( N )Λ ( q ) = N V Λ ( q ) [or = ( N − V Λ ( q )], with V Λ ( q ) independent of N , andfind N R V ( N )Λ ( q ) ρ ε ( q )d q = N R V Λ ( q ) ρ ε ( q )d q [+ O ( N )], scaling ∝ N [inleading order], hence remaining significant in (13) as N becomes large. And asto the kinetic energy term, it is important to realize that R R ∆ (1) X ( N ) ( p , q )d p ≈ ρ ε ( q ) ∈ ( P ∩ C b )( R ) does not imply that ∆ (1) X ( N ) ( p , q ) ≈ f ε ( p , q ) ∈ ( P ∩ C b )( R × Λ). For instance, we can have that N / ∆ (1) X ( N ) ( N / p , q ) ≈ f ε ( p , q ) ∈ ( P ∩ C b )( R × Λ) so that a significant fraction of the energy will be distributedover the kinetic degrees of freedom, and then, up to terms of O ( N ), we find H ( N ) ( X ( N ) ) ≈ N (cid:18)ZZ (cid:0) | p | + V Λ ( q ) (cid:1) f ε ( p , q )d p d q (14)+ ZZZZ W Λ ( q , ˜ q ) f ε ( p , q ) f ε (˜ p , ˜ q )d p d q d ˜ p d ˜ q (cid:19) . This scaling scenario can be verified explicitly for the perfect gas ( W Λ ≡ 0) byinspecting Boltzmann’s calculations, and it is reasonable to expect that it willcontinue to hold for a physically interesting class of W Λ V ( N ) = N V means that N / ∆ (1) X ( N ) ( N / p , q ) N →∞ −→ f ε ( p , q ) weakly in P ( R × Λ), with Incidentally, this indicates that the Vlasov limit does not require the continuity of W Λ ,the only purpose of which was to furnish identity (12) which involves W Λ ( q , q ). Unless E is the ground state energy for which all particle momenta vanish, indeed. ε ( p , q ) ∈ ( P ∩ C b )( R × Λ), and N − H ( N ) ( X ( N ) ) N →∞ −→ E ( f ε ) = ε > ε g , where E ( f ) = ZZ (cid:0) | p | + V Λ ( q ) (cid:1) f ( p , q )d p d q + ZZZZ W Λ ( q , ˜ q ) f ( p , q ) f (˜ p , ˜ q )d p d q d ˜ p d ˜ q (15)is the “energy of f ,” and where ε g = inf f ∈ P ( R × Λ) E ( f ) is given by ε g = inf ρ ∈ P (Λ) (cid:16) Z V Λ ( q ) ρ ( q )d q + ZZ W Λ ( q , ˜ q ) ρ ( q ) ρ (˜ q )d q d ˜ q (cid:17) . (16) We now state our main results about the Vlasov scaling limit for Boltzmann’sergodic ensemble of N -body systems in a format which will be recognized asthe familiar folklore by anyone with a joint expertise in Vlasov theory andstatistical mechanics. We will also utilize some less familiar notions.In the following, Λ ⊂ R is a bounded, connected domain (open) whichdoes not depend on N . The upshot of the previous section is that if we wantthe external potential to remain significant when N gets large, then our N -body dynamics in Λ will be governed by Hamiltonians (1) of the special type H ( N )Λ ( p , . . . , q N ) = X ≤ i ≤ N (cid:0) | p i | + ( N − V Λ ( q i ) (cid:1) + X X ≤ i 1) is a consequence for W Λ of Newton’s “actio equals re-actio,”plus the symmetrized added contribution of V Λ , both of which need no furthercommentary. Hypothesis ( H 2) is satisfied by many important pair interac-tions invoked in physics, though not by all. For instance, the Coulomb pairpotential U Coul Λ (ˇ q , ˆ q ) = 1 / | ˇ q − ˆ q | for ˇ q = ˆ q satisfies ( H 2) after also setting U Coul Λ ( q , q ) ≡ u for any particular u ∈ R . On the other hand, the New-ton pair potential U Newt Λ (ˇ q , ˆ q ) = − U Coul Λ (ˇ q , ˆ q ) does not satisfy ( H 2) for anychoice of u ; however, the regularized Newton pair potential U Newt Λ ,reg (ˇ q , ˆ q ) = − ( χ Br ∗ U Coul Λ ∗ χ Br )(ˇ q , ˆ q ) (where f ∗ g denotes the conventional convolutionproduct of f and g ) does satisfy ( H H N -dependentground state energy E g ( N ), i.e. H ( N )Λ ≥ E g ( N ) > −∞ , but the ground stateconfiguration can have some unwanted features. Hypothesis ( H 3) eliminatesthe possibility of energetically isolated ground states, thus guaranteeing theexistence of a fat set of minimizing sequences of configurations. Hypothesis( H 4) is a little stronger than necessary, but it allows us to make convenientuse of Chebychev’s inequality to prove a law of large numbers for the pair-specific interaction energy; the important Coulomb potential satisfies ( H H 4) implies local L integrability of U Λ , which is needed in var-ious integrals featuring in the Vlasov limit. Note also that by ( H H N -independent ε g ∈ R defined by (16). In Appendix A weshow that ( H H 2) guarantee that the pair-specific ground state energy E g ( N ) / [ N ( N − ≡ ε g ( N ) is monotonic increasing with N , and using also( H 3) and ( H 4) we show that ε g ( N ) ր ε g as N → ∞ . In Appendix A wealso show that if U Λ ≥ 0, then also E g ( N ) /N ≡ ˜ ε g ( N ) ր ε g as N → ∞ . For instance, in our example of the amended Coulomb pair potential one can choose u = 0, but then Thomson’s problem on S ⊂ R [Tho04] yields as ground state configurationalways the spurious one (up to SO (3) action) for which all particle positions coincide. Toavoid these spurious ground state configurations it is advisable to choose u > H 5) is really inherited from the dynamical theory of N particlesin Λ ⊂ R , where one sets V ( N )Λ = + ∞ for q Λ to dynamically model con-finement in a container; ( H 5) has a minor notational advantage by allowingus to treat physical space integrals like momentum space integrals as over all R , the spatial cutoff to Λ automatically being provided by the potential V Λ through U Λ . Usually, ( H 5) is not listed explicitly as a hypothesis on the in-teractions even when spatial integrations are explicitly restricted to Λ. Thisconcludes our commentary on the list of hypotheses ( H − ( H U Λ ≥ 0, so that ε g ≥ 0. Since U Λ has a minimum in Λ , by ( H U Λ , we may assume that U Λ ≥ H thermodynamic functions . In the1960s and hence, techniques based on monotonicity, convexity and super-additivity estimates have been developed to prove their existence and regular-ity in the limit N → ∞ which avoids having to control the more sophisticatedobjects of interest, which are the correlation functions. For the traditionalthermodynamic limit scaling, see Ruelle’s book [Rue69] and [Kie09a] for a re-cent extension of Ruelle’s arguments to Boltzmann’s Ergode proper. For theVlasov scaling of the canonical ensemble, see [Kie93]. To extend these argu-ments to Boltzmann’s Ergode proper with Vlasov scaling, our first goal is toshow that the logarithm of the structure function (3) for the Hamiltonian (19),which yields Boltzmann’s ergodic ensemble entropy (cf. eq.(305) in [Gib02]), S H ( N )Λ ( E ) = ln Ω ′ H ( N )Λ ( E ) , (20)admits the correct type of asymptotic expansion for N → ∞ with E = N ε ,and has the correct qualitative ε dependence. The usual strategy can be putto work if we assume just a little more than ( H H Theorem 1. Let H ( N )Λ be given in (19), with U Λ satisfying conditions ( H and ( H , but with ( H , ( H , ( H replaced by the single stronger condition: ( H Continuity: U Λ (ˇ q , ˆ q ) is continuous on Λ × Λ . (21) Let ε > ε g , with ε g ≥ defined as before. Then the ergodic ensemble entropy(20) has the following asymptotic expansion for N ≫ , S H ( N )Λ ( N ε ) = − N ln N + N s Λ ( ε ) + o ( N ) , (22) where s Λ ( ε ) is the system-specific Boltzmann entropy per particle. The function ε s Λ ( ε ) is continuous and strictly increasing for ε > ε g . Entropy is measured in units of k B , where k B is Boltzmann’s constant. 10e remark that the leading term of r.h.s.(22) is purely combinatorial inorigin and independent of the Hamiltonian H ( N )Λ — it is solely due to the N !in (3). System-specific information begins to show in the next to leading term,which is O ( N ). The o ( N ) term in (22) is presumably O (ln N ).We will also prove two upgrades of Theorem 1 (Theorems 1 + and 1 ++ )which involve the decomposition of the system-specific Boltzmann entropyper particle s Λ ( ε ) into a “kinetic” and an “interaction” contribution. Thediscussion of this more technical material is postponed until section 5.1.While they do yield valuable qualitative information about the thermo-dynamic functions for the systems under study, in this case s Λ ( ε ), existencetheorems such as Theorem 1 and their “proofs by sub-additivity” have thedisadvantage that they do not characterize the limit objects in a way whichwould allow their systematic evaluation for physically interesting irreduciblepair potentials W Λ and external one-body potentials V Λ . It is this type ofcharacterization that we are after, and in section 5.2 we prove that s Λ ( ε ) sat-isfies the familiar maximum entropy variational principle for the entropy perparticle of a perfect gas in a combination of self- and externally generatedfields. More precisely, we prove the following strengthening of Theorem 1. Theorem 2. Let H ( N )Λ be given in (19), with U Λ ≥ satisfying ( H – ( H .Let ε > ε g . Then the Boltzmann entropy (20) has the asymptotic expansion S H ( N )Λ ( N ε ) = − N ln N + N s Λ ( ε ) + o ( N ) (23) for N ≫ , and the system-specific Boltzmann entropy per particle is given by s Λ ( ε ) = − H B ( f ε ) , (24) where H B ( f ) is “Boltzmann’s H function” of f , which reads H B ( f ) = ZZ f ( p , q ) ln( f ( p , q ) /e )d p d q, (25) and where f ε is any minimizer of this H functional over the set of trial densities A ε = { f ∈ ( P ∩ L ∩ L ln L )( R × Λ) : E ( f ) = ε } , where E ( f ) now reads E ( f ) = ZZ | p | f ( p , q )d p d q + ZZZZ U Λ ( q , ˜ q ) f ( p , q ) f (˜ p , ˜ q )d p d q d ˜ p d ˜ q. (26) Any minimizer f ε of H B ( f ) over the set A ε is of the form f ε ( p , q ) = σ ε ( p ) ρ ε ( q ) , (27) We remark that Euler’s number e in (25) is inherited from the N ! term in (20). here ρ ε ( q ) solves the following fixed point equation on q space, ρ ε ( q ) = exp (cid:0) − ϑ ε ( ρ ε ) − R Λ U Λ ( q , ˜ q ) ρ ε (˜ q )d ˜ q (cid:1)R Λ exp (cid:0) − ϑ ε ( ρ ε ) − R Λ U Λ (ˆ q , ˜ q ) ρ ε (˜ q )d ˜ q (cid:1) dˆ q (28) with ϑ ε ( ρ ) given by ϑ ε ( ρ ) = ε − ZZ U Λ ( q , ˜ q ) ρ ( q ) ρ (˜ q )d q d ˜ q, (29) and where σ ε ( p ) = σ ( ρ ε )( p ) , with σ ( ρ )( p ) defined whenever ϑ ε ( ρ ) > , by σ ( ρ )( p ) = (2 πϑ ε ( ρ )) − exp (cid:0) − | p | /ϑ ε ( ρ ) (cid:1) . (30)Evidently, every minimizer of H B ( f ) over A ε factors into a product of a Maxwellian on p space and a purely space-dependent “self-consistent Boltz-mann factor .” However, the Maxwellian in (27) is not autonomous from theBoltzmann factor in (27), as is manifest by the functional dependence of the(rescaled) temperature ϑ = ϑ ε ( ρ ε ) on ρ ε , see (30). For a subset of ε values theminimizer of H B ( f ) over A ε may not be unique, but all minimizers produce thesame asymptotic formula (23). In such a case of non-uniqueness of minimizers,they always seem to constitute either a finite set (typically a first order phasetransition) or a continuous group orbit of a compact group (e.g., when Λ is in-variant under SO (2) or SO (3) and a minimizer breaks that symmetry), to thebest of our knowledge; this seems to cover all physically relevant possibilities.In addition to the minimizers of H B ( f ) there may be non-minimizing criticalpoints of H B ( f ) satisfying (27)–(30), but these are irrelevant for (23).Our Theorem 3, proved in section 5.3 with input from section 5.2, charac-terizes the Vlasov limit N → ∞ of the marginal measures n µ ( N ) E (cid:0) d n X (cid:1) = µ ( N ) E (cid:0) d n X × ( R × Λ) N − n (cid:1) , n = 1 , , ... ( n fixed) (31)in terms of the f ε . We note that the object of interest in (mathematical)physics is not (2) itself but only the collection of its first few marginal measures(31). To state our theorem, we introduce P s (( R × Λ) N ), the permutation-symmetric probability measures on the set of infinite sequences in R × Λ. Atheorem of de Finetti [deF37], Dynkin [Dyn53], and Hewitt–Savage [HeSa55](see also [Ell85], App.A.9.) states that P s (( R × Λ) N ) is uniquely presentableas an average of infinite product measures; i.e., for each µ ∈ P s (( R × Λ) N )there exists a unique probability measure ν ( dτ | µ ) on P ( R × Λ), such that n µ (d n p d n q ) = Z P ( R × Λ) τ ⊗ n (d p d q · · · d p n d q n ) ν (d τ | µ ) ∀ n ∈ N , (32) The expression conventionally known as “Boltzmann factor” results when W Λ ≡ U Λ ( q , ˜ q ) = V Λ ( q ) + V Λ (˜ q ), i.e. for the perfect gas acted on by an external potential V Λ . n µ is the n -th marginal measure of µ , and τ ⊗ n (d p d q · · · d p n d q n ) ≡ τ (d p d q ) ⊗ · · · ⊗ τ (d p n d q n ). Equation (32) is also the extremal decompo-sition for the convex set P s (( R × Λ) N ), see [HeSa55]. Theorem 3. Under the same assumptions as in Theorem 2, consider (2) withHamiltonian (19) as extended to a probability on ( R × Λ) N . Then the sequence { µ ( N ) N ε } N ∈ N is tight, so one can extract a subsequence { µ ( ˙ N [ N ]) N ε } N ∈ N such that lim N →∞ n µ ( ˙ N [ N ])˙ N ε (d n p d n q ) = n ˙ µ ε (d n p d n q ) ∈ P s (( R × Λ) n ) ∀ n ∈ N . (33) The decomposition measure ν ( dτ | ˙ µ ε ) of each such limit point ˙ µ ε is supported bythe subset of P ( R × Λ) which consists of the probability measures τ ε (d p d q ) = f ε ( p , q )d p d q which minimize the H functional H B ( f ) over A ε . We have stated our Theorems 1,2,3 entirely in terms of the familiar quanti-ties of kinetic theory. These are the one-body density function f ε ( p , q ) whichminimizes Boltzmann’s H -function H ( f ) under the familiar energy functionalconstraint E ( f ) = ε , and the system-specific Boltzmann entropy per particle s Λ ( ε ) which is given as the negative of Boltzmann’s H -function evaluated with f ε . However, in this format our theorems give essentially symmetric weight tothe p and q variables, which ignores the fact that the p -space integrations in-volved in (31) and (20) can be carried out explicitly in the same fashion as forthe perfect gas. As a consequence the problem reduces to studying the large N asymptotics of the expressions which result from these p -space integrations. In fact, all the hard analytical work goes into controlling the q -space integra-tions. This is certainly the case as far as the entropy per particle goes, yet alsoeach minimizer f ε of H B ( f ) over the set A ε is uniquely determined by ρ ε , whichsignals that all of our Theorems 1 to 3 will be essentially straightforward corol-laries of theorems about certain q -space expressions. Those theorems take aless familiar form, presumably, which is why their statements have been rele-gated into this section where we prove Theorems 1 to 3. To prove Theorem 1 we first formulate and then prove an upgraded version(Theorem 1 + ), whose proof also proves Theorem 1. All Boltzmann needed for this was that (1 + x/n ) n ≍ e x ; cf. [Bol96], part II, ch. 3.Of course, things are not quite as straightforward with an irreducible W Λ 0, or elseBoltzmann would not have had to have W Λ .1.1 Theorem 1 + and its proof Carrying out the p integrations in Ω ′ H ( N )Λ ( E ) given by (3), with H ( N )Λ given in(19), Boltzmann’s ergodic ensemble entropy (20) becomes S H ( N )Λ ( E ) = ln (cid:16) (2 /N ) N/ N (cid:12)(cid:12) S N − (cid:12)(cid:12) Ψ ′ I ( N )Λ ( E ) (cid:17) (34)with (cid:12)(cid:12) S N − (cid:12)(cid:12) the standard measure of the unit 3 N − S N − , and withΨ ′ I ( N )Λ ( E ) = (3 / N − Z (cid:16) E − I ( N )Λ ( q , ..., q N ) (cid:17) N − χ n I ( N )Λ < E o d N q, (35)where we introduced the interaction Hamiltonian I ( N )Λ ( q , . . . , q N ) = X X ≤ i 0, recalling that (cid:12)(cid:12) S N − (cid:12)(cid:12) = π N/ / Γ(3 N/ S H ( N )Λ ( N ε ) = − N ln N + N ln (cid:16) | Λ | (cid:0) πe ε (cid:1) / (cid:17) + O (ln N )+ ln Z (cid:16) − εN I ( N )Λ ( q , ..., q N ) (cid:17) N − λ (d N q ) . (37)where ( · · · ) + means the positive part of ( · · · ); moreover, λ (d N q ) is the N -foldproduct of the normalized Lebesgue measure λ (d q ) = | Λ | − d q on Λ. Forbrevity we wrote | Λ | for the volume Vol (Λ) of Λ.When I ( N )Λ ≡ N , then H ( N )Λ becomes the Hamiltonian of the perfectgas without external fields, abbreviated as K ( N )Λ (for kinetic Hamiltonian).In this case the second line in (37) vanishes, and (37) becomes the asymptoticexpansion of the entropy of the spatially uniformly distributed perfect gas, viz. S K ( N )Λ ( N ε ) = − N ln N + N ln (cid:16) | Λ | (cid:0) πe ε (cid:1) / (cid:17) + O (ln N ) . (38)The coefficient of the O ( N ) term in (38) gives the system-specific Boltzmannentropy per particle of the spatially uniform perfect gas, which we denote by s Λ , K ( ε ) = ln (cid:16) | Λ | (cid:0) πe ε (cid:1) / (cid:17) . (39) It is understood that d N X etc. now involves the p variables used in (19). It is tacitly understood that the cutoff provided by I ( N )Λ remains effective, so that theconfigurational integrations in (37) are still over Λ N . I ( N )Λ O ( N ) and so contributesadditively to the system-specific Boltzmann entropy per particle, and providedit has the right monotonicity and regularity. This is expressed in Proposition 1. Under the assumptions stated in Theorem 1, there holds lim N →∞ N ln Z (cid:16) − εN I ( N )Λ ( q , ..., q N ) (cid:17) N − λ (d N q ) = s Λ , I ( ε ) . (40) The function ε s Λ , I ( ε ) is continuous and increasing for ε > ε g ≥ . This concludes the pretext for our first upgrade of Theorem 1, stated next. Theorem 1. + Theorem 1 holds, with s Λ ( ε ) = s Λ , K ( ε ) + s Λ , I ( ε ) , (41) where s Λ , K ( ε ) is given in (39), and s Λ , I ( ε ) in (40).Proof of Theorem 1 + : Clearly, Proposition 1 and formula (37) imply Theorem 1 and the splittingof the system-specific Boltzmann entropy per particle s Λ ( ε ) in (22) into a sumof a kinetic and an interaction component, (41). Proposition 1 also adds a pieceof information about s Λ , I ( ε ) which does not just re-express what is stated inTheorem 1. In fact, by the known strict increase of ε ln ε , the increaseof ε s Λ , I ( ε ) implies the strict increase of ε s Λ ( ε ), but the increase of ε s Λ , I ( ε ) does not follow from the properties of ε ln ε and the strictincrease of ε s Λ ( ε ). So Theorem 1 + holds and extends Theorem 1. Proof of Proposition 1: By hypothesis ( H U Λ is bounded continuous on Λ × Λ, so we can write N − I ( N )Λ ( q , ..., q N ) = ZZ U Λ (ˇ q , ˆ q )∆ (1) X ( N ) (ˇ q )∆ (1) X ( N ) (ˆ q )d ˇ q d ˆ q − N Z U Λ ( q , q )∆ (1) X ( N ) ( q )d q, (42)and we may abbreviate the first term of r.h.s.(42) in bilinear form notation, ZZ U Λ (ˇ q , ˆ q )∆ (1) X ( N ) (ˇ q )∆ (1) X ( N ) (ˆ q )d ˇ q d ˆ q ≡ (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11) . (43)The above integrals extend over R , and we set I ( N )Λ ( q , ..., q N ) = ∞ as wellas (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11) = ∞ if any q k Λ. Also by ( H O ( N − ) for all X ( N ) ∈ Λ N . Recalling our claim (which we15romised to prove) that the limit N → ∞ for the ensemble does not change ifthe Hamiltonian is changed by an additive term of order O ( N − ) relative tothe leading terms, we now introduce the configurational integralΥ ( N )Λ ( ε ) ≡ ln Z (cid:16) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:17) N − λ (d N q ) (44)for all N > N U ( ε ) (to be defined). Note that the integral (44) is gener-ally not well-defined for all N ∈ N because (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11) is bigger than N − I ( N )Λ ( q , ..., q N ) by the absolute value of the second line of r.h.s.(42), whichreads precisely N − P Nk =1 12 U Λ ( q k , q k ). And while this term = O (1 /N ), when N is not large enough then it is possible that (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11) > ε everywherein Λ N , in which case the integral in (44) vanishes, and its logarithm = −∞ ,then. Yet, when N > N U ( ε ) the integral (44) is well-defined, and we concludethat (modulo the proof of precisely the just re-uttered claim that O (1 /N ) con-tributions to the Hamiltonian drop out when N → ∞ ) our proposition 1 isproved if we can prove the following proposition. Proposition 2. Under the hypotheses on U Λ in Thm. 1, when N ≫ N U ( ε ) then Υ ( N )Λ ( ε ) = N γ Λ ( ε ) + o ( N ) . (45) The function ε γ Λ ( ε ) is continuous and increasing for ε > ε g ≥ .Proof of Proposition 2: We will establish uniform bounds and super-additivity estimates.For 0 < n < N , we set X ( N ) ≡ ( X ( n ) , Y ( N − n ) ), which also defines Y ( N − n ) .We note the convex linear decomposition∆ (1) X ( N ) ( q ) = nN ∆ (1) X ( n ) ( q ) + (1 − nN )∆ (1) Y ( N − n ) ( q ) . (46)Since U Λ ≥ (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11) ≤ nN (cid:10) ∆ (1) X ( n ) , ∆ (1) X ( n ) (cid:11) + (1 − nN ) (cid:10) ∆ (1) Y ( N − n ) , ∆ (1) Y ( N − n ) (cid:11) . (47)We of course also have 1 = nN + (1 − nN ), and so we conclude that (cid:16) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:17) + ≥ (cid:16) nN h − ε (cid:10) ∆ (1) X ( n ) , ∆ (1) X ( n ) (cid:11)i + (1 − nN ) h − ε (cid:10) ∆ (1) Y ( N − n ) , ∆ (1) Y ( N − n ) (cid:11)i(cid:17) + . (48)Next we recall that, if ϕ is some function on a domain D , and if Σ( ϕ + )denotes the support of its positive part, and χ Σ( ϕ +) is the characteristic function16f Σ( ϕ + ), then the inclusion Σ( ϕ + ) ∩ Σ( ϑ + ) ⊂ Σ(( ϕ + ϑ ) + ) for any two suchfunctions ϕ and ϑ yields the estimate( ϕ + ϑ ) + = ( ϕ + ϑ ) χ Σ(( ϕ + ϑ )+) ≥ ( ϕ + ϑ ) χ Σ( ϕ +) χ Σ( ϑ +) = ( ϕ + + ϑ + ) χ Σ( ϕ +) χ Σ( ϑ +) . (49)Set ϕ = nN h − ε (cid:10) ∆ (1) X ( n ) , ∆ (1) X ( n ) (cid:11)i and ϑ = (1 − nN ) h − ε (cid:10) ∆ (1) Y ( N − n ) , ∆ (1) Y ( N − n ) (cid:11)i . Then inequality (49) applies to r.h.s. (48). Applying next the classical in-equality between the arithmetic and the geometric means of any two positivenumbers A and B , viz. αA + (1 − α ) B ≥ A α B (1 − α ) for any α ∈ [0 , (cid:16) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:17) + ≥ h − ε (cid:10) ∆ (1) X ( n ) , ∆ (1) X ( n ) (cid:11)i nN + h − ε (cid:10) ∆ (1) Y ( N − n ) , ∆ (1) Y ( N − n ) (cid:11)i − nN + . (50)We now use (50) to estimate r.h.s.(44). For this, let N ≫ N U ( ε ) and let N U ( ε ) < n < N − N U ( ε ). Noting that the resulting integral over Λ N factorsinto two integrals, one over Λ n and another over Λ N − n , and working out thepowers, we findln Z (cid:16) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:17) N − λ (d N q ) ≥ (51)ln Z (cid:16) − ε (cid:10) ∆ (1) X ( n ) , ∆ (1) X ( n ) (cid:11)(cid:17) n − nN + λ (d n q ) +ln Z (cid:16) − ε (cid:10) ∆ (1) X ( N − n ) , ∆ (1) X ( N − n ) (cid:11)(cid:17) N − n )2 − nN + λ (d N − n ) q ) , where we also relabeled the integration variables under the second integral onr.h.s.(51) from Y ( N − n ) to X ( N − n ) . Noting next that 0 < nN < 1, we resort againto Jensen’s inequality, this time w.r.t. the λ measures in the two integrals onr.h.s.(51). Also using ln( · · · ) a = a ln( · · · ), we arrive atln Z (cid:16) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:17) N − λ (d N q ) ≥ (52) (cid:16) − n/N n − (cid:17) ln Z (cid:16) − ε (cid:10) ∆ (1) X ( n ) , ∆ (1) X ( n ) (cid:11)(cid:17) n − λ (d n q ) + (cid:16) n/N N − n ) − (cid:17) ln Z (cid:16) − ε (cid:10) ∆ (1) X ( N − n ) , ∆ (1) X ( N − n ) (cid:11)(cid:17) N − n )2 − λ (d N − n ) q ) . Formula (52) writes shorter thusly,Υ ( N )Λ ( ε ) ≥ (cid:16) − n/N n − (cid:17) Υ ( n )Λ ( ε ) + (cid:16) n/N N − n ) − (cid:17) Υ ( N − n )Λ ( ε ) . (53)17o N Υ ( N )Λ ( ε ) is almost super-additive.To be able to create a properly super-additive function we establish upperand lower bounds of ℓ Υ ( ℓ )Λ ( ε ) which are linear in ℓ , whenever ℓ > N U ( ε ); wewill need those bounds with ℓ ∈ { n, N − n } , with ℓ > 1. As a by-product, theupper bound with ℓ = N will also guarantee convergence of the constructedsuper-additive function.The upper bound is trivial. Recall that by hypothesis (cid:10) ∆ (1) X ( ℓ ) , ∆ (1) X ( ℓ ) (cid:11) ≥ ℓ ∈ N . So for ℓ > N U ( ε ) and ε > ε g ≥ ℓ − ln Z (cid:16) − ε (cid:10) ∆ (1) X ( ℓ ) , ∆ (1) X ( ℓ ) (cid:11)(cid:17) ℓ − λ (d ℓ q ) ≤ . (54)As for the lower bound, we distinguish two cases, (a): (cid:10) | Λ | − , | Λ | − (cid:11) < ε ,and (b): (cid:10) | Λ | − , | Λ | − (cid:11) ≥ ε . In case (a) we apply Jensen’s inequality w.r.t. λ to the convex map x (1 − x ) θ + (for θ ≥ ℓ − < ℓ , to getln (cid:20)Z (cid:16) − ε (cid:10) ∆ (1) X ( ℓ ) , ∆ (1) X ( ℓ ) (cid:11)(cid:17) ℓ − λ (d ℓ q ) (cid:21) ℓ − ≥ ln (cid:20) − ℓ ε Z U Λ ( q , q ) λ (d q ) − ε ZZ U Λ (ˇ q , ˆ q ) λ (d ˇ q ) λ (d ˆ q ) (cid:21) + , (55)and r.h.s.(55) ≥ − C > −∞ when ℓ > ℓ crit ( ε ) (given U Λ ), with C > ℓ . Since the interaction entropy exists when ℓ > N U ( ε ), clearly ℓ crit ≥ N U ( ε ), but after at most an adjustment of C , we can conclude thatl.h.s.(55) ≥ − C > −∞ when ℓ > N U ( ε ), with C > ℓ . Incase (b), inequality (55) is still true but now trivial, for r.h.s.(55)= −∞ for all ℓ > 1, then. So instead we now proceed as follows. By hypoth-esis ( H (cid:10) ∆ (1) X ( ℓ ) , ∆ (1) X ( ℓ ) (cid:11) takes its minimum ε ∗ g ( ℓ ) ≥ ε g .Clearly, ε ∗ g ( ℓ ) = ˜ ε g ( ℓ ) + O ( ℓ − ), where ˜ ε g ( ℓ ) := min ℓ − I ( ℓ )Λ ( q , ..., q ℓ ), and since˜ ε g ( ℓ ) ≤ ε g (as proved in Appendix A), we have that ε ∗ g ( ℓ ) ≤ ε g + O ( ℓ − ); ofcourse, we also assume that ℓ > N U ( ε ) so that ε ∗ g ( ℓ ) < ε . By permutation sym-metry there are many equivalent minimizers, but possibly also several distinctpermutation group orbits of minimizers. We pick any particular minimizer Q ( ℓ ) g and let q ( ℓ ) g,k ∈ Λ denote the k -th coordinate vector in Q ( ℓ ) g . By ( H 6) again, wecan vary all the q k in the minimizing configuration a little bit, say, each q k in B δ ( q ( ℓ ) g,k ) ∩ Λ, where B δ ( q ) is a ball centered at q , with radius δ > k and ℓ but chosen small enough (given ε ) so that (cid:10) ∆ (1) X ( ℓ ) , ∆ (1) X ( ℓ ) (cid:11) does not change by more than ( ε − ε g + O ( ℓ − )) / 2. For brevity we write B δ [ k ]for B δ ( q ( ℓ ) g,k ); let χ B δ [ k ] be the characteristic function of B δ [ k ]. We use that λ (d q k ) = χ B δ [ k ] λ (d q k ) + χ B cδ [ k ] λ (d q k ) where B cδ [ k ] = Λ \ B δ [ k ] is the com-plement in Λ of B δ [ k ], then use that both terms in this decomposition arenon-negative so that we get an upper estimate by dropping the contribution18rom χ B cδ [ k ] λ (d q k ) for each k . After this step the restriction to the positivepart of (1 − ε h . , . i ) is eventually tautological when ℓ is sufficiently large sothat the O ( ℓ − ) term has gotten sufficiently small. We next apply Jensen’s in-equality w.r.t. the probability measure Q ≤ k ≤ ℓ ( R B δ [ k ] ∩ Λ λ (d q )) − χ B δ [ k ] λ (d q k )to the convex map x + x θ + (for θ ≥ ≤ ε g < ε , and get (cid:20)Z (cid:16) − ε (cid:10) ∆ (1) X ( ℓ ) , ∆ (1) X ( ℓ ) (cid:11)(cid:17) ℓ − λ (d ℓ q ) (cid:21) ℓ − ≥ "Z (cid:16) − ε (cid:10) ∆ (1) X ( ℓ ) , ∆ (1) X ( ℓ ) (cid:11)(cid:17) ℓ − Y ≤ k ≤ ℓ χ B δ [ k ] λ (d q k ) ℓ − ≥| C δ | − /ℓ Z (cid:16) − ε (cid:10) ∆ (1) X ( ℓ ) , ∆ (1) X ( ℓ ) (cid:11)(cid:17) Y ≤ k ≤ ℓ χ Bδ [ k ] R Bδ [ k ] ∩ Λ λ (d q ) λ (d q k ) ≥| C δ | − /ℓ (cid:0) − (cid:0) ε g ε (cid:1) + O ( ℓ − ) (cid:1) ≥ C > ℓ large enough; here C δ = min q ′ Z B δ ( q ′ ) ∩ Λ λ (d q ) > . (57)In summary, our list of inequalities (54), (55) and (56), and the finitenessof the number of ℓ until “ ℓ is large enough,” establishes that when ℓ > N U ( ε ),then for some ℓ -independent constant C ∗ > − (cid:0) ℓ − (cid:1) C ∗ ≤ Υ ( ℓ )Λ ( ε ) ≤ 0; (58)incidentally, our (55) and (56) produce an upper estimate for N U ( ε ).Recall that in this proof we assume that N ≫ N U ( ε ), and that N U ( ε ) 0. And using (54)with N , we also see that N − (cid:0) Υ ( N )Λ ( ε ) + C (cid:1) is bounded above, and so, bystandard facts about super-additive functions, N − (cid:0) Υ ( N )Λ ( ε ) + C (cid:1) converges as N → ∞ ,lim N →∞ N − (cid:16) Υ ( N )Λ ( ε ) + C (cid:17) = sup N ∈ N N − (cid:16) Υ ( N )Λ ( ε ) + C (cid:17) , (60)and since N − C N →∞ −→ 0, we conclude that N − Υ ( N )Λ ( ε ) converges as well, i.e.lim N →∞ N Υ ( N )Λ ( ε ) = γ Λ ( ε ) . (61)19his proves (45).To prove continuity of γ Λ ( ε ), we establish upper and lower bounds on thederivative of the functions ε N − Υ ( N )Λ ( ε ) which are uniform in N > N U ( ε ).Differentiating the functions ε N − Υ ( N )Λ ( ε ) + (cid:0) − N (cid:1) ln ε , we obtain N Υ ( N )Λ ′ ( ε ) = (cid:0) − N (cid:1) ε R(cid:0) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:1) N − λ (d N q ) R(cid:0) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:1) N − λ (d N q ) − . (62)To get a lower bound, we split off a factor (cid:0) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:1) + in theintegrand of the denominator of r.h.s.(62), and using that ε > ε g ≥ 0, thepositivity of the bilinear form now gives (cid:0) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:1) + ≤ 1, and so N Υ ( N )Λ ′ ( ε ) ≥ (cid:0) − N (cid:1) ε [1 − 1] = 0; (63)incidentally, this shows once again monotonicity ↑ of ε γ Λ ( ε ). To get an N -independent upper bound to (62), note that N − (cid:0) N − (cid:1) (cid:0) − N − (cid:1) and that 0 < (cid:0) − N − (cid:1) < N > 1, then apply Jensen’s inequality w.r.t. λ to pull the power (cid:0) − N − (cid:1) out of the integral in the numerator, then notea cancellation versus the denominator. Since 0 < (cid:0) − N − (cid:1) < N > N Υ ( N )Λ ′ ( ε ) ≤ − N ε "(cid:20)Z (cid:16) − ε (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:17) N − λ (d N q ) (cid:21) − N − − (64)whenever N > N U ( ε ) (so that the integral is non-zero). By the first inequalityin (58) with ℓ = N , the r.h.s.(64) is bounded above independently of N . Thecontinuity of ε γ Λ ( ε ) follows.Proposition 2 is proved.To complete the proof of Proposition 1 we still need to show that theomission of N R U Λ ( q , q )∆ (1) X ( N ) ( q )d q from (42) was justified. This is nowstraightforward. By hypothesis ( H U Λ ( ≥ 0) is a bounded continuous func-tion on Λ × Λ. So there exists an N -independent constant B > ≤ Z U Λ ( q , q )∆ (1) X ( N ) ( q )d q ≤ B, (65)as long as X ( N ) ∈ Λ N . Thus, and abbreviating the expression in the secondline on r.h.s.(37) by S I ( N )Λ ( N ε ), we have the two-sided estimateΥ ( N )Λ ( ε ) ≤ S I ( N )Λ ( N ε ) ≤ Υ ( N )Λ (cid:0) ε + BN − (cid:1) . (66)But (cid:12)(cid:12)(cid:12) Υ ( N )Λ (cid:0) ε + BN − (cid:1) − Υ ( N )Λ ( ε ) (cid:12)(cid:12)(cid:12) ≤ Z ε + BN − ε (cid:12)(cid:12)(cid:12) Υ ( N )Λ ′ ( ς ) (cid:12)(cid:12)(cid:12) d ς ≤ BC, (67)20he last inequality by (64) and by the first inequality in (58), with ℓ = N , andby ε ≤ ς ≤ ε . So we conclude that for any B > N →∞ N Υ ( N )Λ (cid:0) ε + BN − (cid:1) = γ Λ ( ε ) . (68)Hence, and by (66),lim N →∞ N S I ( N )Λ ( ε ) = γ Λ ( ε ) , (69)and Proposition 1 is proved, with s Λ , I ( ε ) = γ Λ ( ε ).This also completes the proof of Theorem 1. ++ and its proof Ruelle’s proof [Rue69] of the traditional thermodynamic limit for (20) pervolume proceeded along somewhat different lines, and when adapted to theVlasov scaling it yields an interesting alternate proof of Theorem 1 whichcharacterizes s Λ ( ε ) in terms of a variational principle (VP) involving s Λ , K ( ε )and yet another (auxiliary) “interaction entropy,” which we denote by s Λ , I ( ε ).For technical reasons we now need to assume that ε g > ε g ≥ Ξ I ( N )Λ ( E ) = Z χ n I ( N )Λ < E o λ (d N q ) . (70)Up to a purely numerical factor, (70) is quasi the “3 N/ E of Ψ I ( N )Λ ( E ), the first derivative of which is given in (35). For conveniencewe rewrite (70), with E = N ε , asΞ I ( N )Λ ( N ε ) = Z (cid:16) ε − N − I ( N )Λ ( q , ..., q N ) (cid:17) λ (d N q ) . (71) Proposition 3. Assume the hypotheses of Theorem 1, but now let ε g > .Then the following limit exists, lim N →∞ N ln Ξ I ( N )Λ ( N ε ) = s Λ , I ( ε ) , (72) and s Λ , I ( ε ) ≤ is an increasing, right-continuous, function of ε > ε g . Actually, Ruelle discussed the entropy of a regularized microcanonical ensemble measure[Rue69]. In [Kie09a] the author showed that a minor modification of Ruelle’s approachestablishes the thermodynamic limit for (20) per volume without regularization. Instead of the normalized Lebesgue measure λ (d N q ), Ruelle [Rue69] uses N ! − d N q which gives equivalent results in the thermodynamic limit; not so in the Vlasov limit. roof of Proposition 3: Simplest things first, we note that Ξ I ( N )Λ ( E ) ≤ I ( N )Λ ( N ε ) ≤ N , and so s Λ , I ( ε ) ≤ ε > ε g > N − I ( N )Λ ( q , ..., q N )by (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11) in (71) and study its logarithm. For this we need once againto assume that N ≫ N U ( ε ). Inspection of our proof of Proposition 2 revealsthat we can recycle inequality (50), take its vanishing power, integrate andtake logarithms, and for N U ( ε ) < n < N − N U ( ε ), in place of (51) we now findln Z (cid:16) ε − (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:17) λ (d N q ) ≥ (73)ln Z (cid:16) ε − (cid:10) ∆ (1) X ( n ) , ∆ (1) X ( n ) (cid:11)(cid:17) λ (d n q ) +ln Z (cid:16) ε − (cid:10) ∆ (1) X ( N − n ) , ∆ (1) X ( N − n ) (cid:11)(cid:17) λ (d N − n ) q ) , which proves super-additivity of N ln R(cid:0) ε − (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:1) λ (d N q ) with-out further ado. Furthermore, since ( · · · ) is either 1 or 0, we conclude thatln R(cid:0) ε − (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:1) λ (d N q ) ≤ 0. This upper bound and super-additivitynow yield that the following limit exists,lim N →∞ N ln Z (cid:16) ε − (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11)(cid:17) λ (d N q ) = ˜ s Λ , I ( ε ); (74)moreover, ˜ s Λ , I ( ε ) ≤ s Λ , I ( ε ) as function of ε and thenconclude the proof as at the end of the proof of Theorem 1, but so far a proofof continuity of ˜ s Λ , I ( ε ) has eluded us. Fortunately we can bypass this obstaclebecause ˜ s Λ , I ( ε ) is a monotonic increasing function of ε . We define˜ s Λ , I ( ε + ) = inf x> ˜ s Λ , I ( xε ) (75)and show thatlim N →∞ N ln Z (cid:16) − εN I ( N )Λ ( q , ..., q N ) (cid:17) λ (d N q ) = ˜ s Λ , I ( ε + ) , (76)which proves Proposition 3, with s Λ , I ( ε ) = ˜ s Λ , I ( ε + ).To accomplish this, we recall (42) and (43) and rewrite (71) asΞ I ( N )Λ ( N ε ) = Z (cid:16) ε − (cid:10) ∆ (1) X ( N ) , ∆ (1) X ( N ) (cid:11) + N (cid:10) ∆ (1) X ( N ) (cid:11)(cid:17) λ (d N q ) , (77)22here we also introduced the abbreviation (cid:10) ∆ (1) X ( N ) (cid:11) = Z U Λ ( q , q )∆ (1) X ( N ) ( q )d q. (78)Since now ε g > 0, there exist constants B, B satisfying 0 < B < B < ∞ so that B ≤ (cid:10) ∆ (1) X ( N ) (cid:11) ≤ B. (79)But then, for all N > N U ( ε ) big enough, we have N ln Ξ I ( N )Λ ( N ε ) ≥ ˜ s Λ , I (cid:0) ε + N − B (cid:1) + o (1) ≥ ˜ s Λ , I (cid:0) ε + (cid:1) + o (1) (80)where o (1) → N → ∞ . Solim inf N →∞ N ln Ξ I ( N )Λ ( N ε ) ≥ ˜ s Λ , I (cid:0) ε + (cid:1) . (81)On the other hand, for all N > N U ( ε ) we also have that N ln Ξ I ( N )Λ ( N ε ) ≤ ˜ s Λ , I (cid:0) ε + N − B (cid:1) + o (1) , (82)and solim sup N →∞ N ln Ξ I ( N )Λ ( N ε ) ≤ ˜ s Λ , I (cid:0) ε + (cid:1) . (83)The estimates (81) and (83) prove (76).So s Λ , I ( ε ) = ˜ s Λ , I ( ε + ). Of course, s Λ , I ( ε ) = ˜ s Λ , I ( ε ) at all ε which are points ofcontinuity of ˜ s Λ , I ( ε ), and the two functions share their points of discontinuity.At such points s Λ , I ( ε ) is right-continuous and may or may not agree with˜ s Λ , I ( ε ).Proposition 3 is proved.We are now ready to state our second upgrade of our Theorem 1. Theorem 1. ++ Under the hypotheses of Proposition 3, Theorem 1 holds andthe system-specific Boltzmann entropy per particle s Λ ( ε ) given in (22) satisfiesthe variational principle s Λ ( ε ) = sup ≤ x ≤ (cid:16) s Λ , K ( xε ) + s Λ , I ([1 − x ] ε ) (cid:17) . (84) Proof of Theorem 1 ++ : Integration by parts yields, for any ℓ > ε > ε g , Z (cid:16) − εN I ( N )Λ (cid:17) ℓ + λ (d N q ) = Z Z (cid:16) − − x ] εN I ( N )Λ (cid:17) λ (d N q )d x ℓ , (85)23here we suppressed the arguments ( q , ..., q N ) from I ( N )Λ ( q , ..., q N ). Setting ℓ = N − 1, recalling (71), and using that N − ln (cid:0) N − (cid:1) → 0, we find s Λ , I ( ε ) = lim N →∞ N ln Z Ξ I ( N )Λ ( N [1 − x ] ε ) x N − d x. (86)Proposition 3 and Laplace’s method (cf. sect. II.7 in [Ell85]) now yield s Λ , I ( ε ) = sup ≤ x ≤ (cid:16) ln x + s Λ , I ([1 − x ] ε ) (cid:17) ; (87)note that (87) implies that ε s Λ , I ( ε ) is continuous even when s Λ , I ( ε ) is not.Recalling next the definition (39) of s Λ , K ( ε ) as well as (41) of Theorem 1 + , wesee that Theorem 1 ++ is proved.We end this subsection by pointing out that our method of proving The-orem 1 ++ not only avoids the regularization of Dirac’s δ measure, we alsotackled the map E S ( E ) directly rather than its inverse S E ( S ) [Rue69].The strategy to tackle S E ( S ) is due to Griffiths [Gri65]. Since formula (37) holds also under the assumptions ( H H 5) on the inter-actions, and since it is well-known that the system-specific Boltzmann entropyper particle of the perfect gas (39) minimizes Boltzmann’s H functional underthe constraint of prescribing the value of the kinetic Hamiltonian, it sufficesto study the interaction entropy of Boltzmann’s ergodic ensemble , S I ( N )Λ ( E ) = ln Z (cid:0) − E I ( N )Λ ( q , ..., q N ) (cid:1) + N − λ (d N q ) . (88)Note that (88) is non-positive, and under hypotheses ( H H 5) we also have (cid:20)Z (cid:16) − ε N I ( N )Λ ( q , ..., q N ) (cid:17) + N − λ (d N q ) (cid:21) N − ≥ "Z (cid:16) − ε N I ( N )Λ ( q , ..., q N ) (cid:17) N − Y ≤ k ≤ N χ B δ [ k ] λ (d q k ) N − ≥| C δ | − /N Z (cid:16) − ε N I ( N )Λ ( q , ..., q N ) (cid:17) Y ≤ k ≤ N χ Bδ [ k ] R Bδ [ k ] ∩ Λ λ (d q ) λ (d q k ) ≥| C δ | (cid:0) − (cid:0) ε g ε (cid:1)(cid:1) > , (89)where again C δ is given in (57), but now with δ ( ε ) independent of k and N chosen so that N − I ( N )Λ ( q , ..., q N ) ≤ ˜ ε g ( N ) + ( ε − ε g ) / q k B δ ( q g,k ) ∩ Λ, where ( q g, , ..., q g,N ) is a ground state configuration for I ( N )Λ ( q , ..., q N ) with a fat neighborhood, which exists by ( H H ε g ( N ) = min N − I ( N )Λ ( q , ..., q N ) ≤ ε g (see Appendix A). So N − S I ( N )Λ ( E ) ≥ ln (cid:16) | C δ | (cid:0) − (cid:0) ε g ε (cid:1)(cid:1)(cid:17) > −∞ (90)for all N > 1. The estimate (90) guarantees the existence of limit points ofthe (negative) interaction entropy per particle as N → ∞ . We want to showthat the interaction entropy per particle actually has a limit and characterizethe limit by the variational principle stated in Theorem 2.We begin by characterizing (88) by its own maximum entropy principle.We introduce the quasi-interaction energy of ̺ ( N ) ∈ P s (Λ N ), defined by Q ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) = N − Z ln (cid:16) − εN I ( N )Λ ( q , ..., q N ) (cid:17) + ̺ ( N ) (d N q ) (91)whenever supp ̺ ( N ) ⊂ supp (cid:0) ε − N − I ( N )Λ (cid:1) + ; else we set Q ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) = −∞ .The entropy of ̺ ( N ) relative to ̺ ( N ) ap ∈ P s (Λ N ) is defined as usual by R ( N ) (cid:0) ̺ ( N ) | ̺ ( N ) ap (cid:1) = − Z ln d̺ ( N ) d̺ ( N ) ap ! ̺ ( N ) (d N q ) (92)if ̺ ( N ) is absolutely continuous w.r.t. the a-priori measure ̺ ( N ) ap , and providedthe integral in (92) exists. In all other cases, R ( N ) (cid:0) ̺ ( N ) | ̺ ( N ) ap (cid:1) = −∞ . Finally,we define what we call the interaction entropy of ̺ ( N ) by S ( N ) I/ε ( ̺ ( N ) ) ≡ R ( N ) (cid:0) ̺ ( N ) | λ (cid:1) + Q ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) . (93)We are now ready to state our variational principle. Proposition 4. For ε > ε g ≥ , the interaction entropy functional (93)achieves its supremum. The maximizer is the unique probability measure ̺ ( N ) N ε (d N q ) = (cid:16) − εN I ( N )Λ ( q , ..., q N ) (cid:17) + N − d N q Z (cid:16) − εN I ( N )Λ (˜ q , ..., ˜ q N ) (cid:17) + N − d N ˜ q ∈ ( P s ∩ L ∞ )(Λ N ); (94) thus max ̺ ( N ) ∈ P s (Λ N ) S ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) = S ( N ) I/ε (cid:0) ̺ ( N ) N ε (cid:1) . (95) Moreover, S ( N ) I/ε (cid:0) ̺ ( N ) N ε (cid:1) = S I ( N )Λ ( N ε ) . (96) Our physicists’ sign convention of relative entropy is opposite to the probabilists’ one. roof of Proposition 4: Under our hypotheses on I ( N )Λ the measure ̺ ( N ) N ε is absolutely continuousw.r.t. λ and bounded whenever ε > ε g , so the standard convexity argu-ment due to Boltzmann [Bol96], cf. [Rue69, Ell85], applies and shows that S ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) − S ( N ) I/ε (cid:0) ̺ ( N ) N ε (cid:1) ≤ 0, with equality holding if and only if ̺ ( N ) = ̺ ( N ) N ε .Identity (96) is verified by explicit calculation.Since ultimately we are interested in the limit N → ∞ of our finite- N results, we recall the formalism of probabilities on infinite sequences Λ N , asencountered already in section 3 for (Λ × R ) N . Thus, by P s (Λ N ) we de-note the permutation-symmetric probability measures on the set of infiniteexchangeable sequences in Λ. Let { n ̺ } n ∈ N denote the sequence of marginals ofany ̺ ∈ P s (Λ N ). The de Finetti [deF37] – Dynkin [Dyn53] – Hewitt-Savage[HeSa55] decomposition theorem for P s (Λ N ) states that every ̺ ∈ P s (Λ N )is uniquely presentable as a linear convex superposition of infinite productmeasures, i.e., for each ̺ ∈ P s (Λ N ) there exists a unique probability measure ς ( dρ | ̺ ) on P (Λ), such that for each n ∈ N , n ̺ (d n q ) = Z P (Λ) ρ ⊗ n (d q · · · d q n ) ς (d ρ | ̺ ) , (97)where n ̺ is the n -th marginal measure of ̺ , and where ρ ⊗ n (d q · · · d q n ) ≡ ρ (d q ) × · · · × ρ (d q n ). Also, (97) expresses the extreme point decompositionof the convex set P s (Λ N ), see [HeSa55].Next we would like to formulate the N = ∞ analogue of (93), but thenaive manipulation of the formulas is not recommended. The functional Q ( N ) I/ε is well-defined by (91) and its accompanying text for all N ∈ N ; however, sinceour conditions on I ( N )Λ ( q , ..., q N ) allow it to be unbounded above when twopositions q k and q l approach each other (for example: Coulomb interactions),we find that Q ( N ) I/ε (cid:0) ρ ⊗ n (cid:1) = −∞ for all product measures ρ ⊗ n , but these are ex-actly the N -point marginals of the extreme points of our set of exchangeablemeasures on the infinite Cartesian product Λ N . This obstacle can be circum-vented by noting that the finite- N quasi-interaction energy defined in (91) andthe line ensuing (91) is the monotone limit of a family of concave functionalsin which the integrand function ln(1 − x ) + (with ln 0 = −∞ understood) isreplaced by ln(1 − x ) χ { x< − α } + [ln α + (1 − α − x ) /α ] χ { x ≥ − α } ; thus α Q ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) = N − Z (cid:16) ln (cid:16) − εN I ( N )Λ (cid:17) χ n I ( N )Λ <εN (1 − α ) o (98)+ h ln α + α (cid:16) − εN I ( N )Λ − α (cid:17)i χ n I ( N )Λ ≥ εN (1 − α ) o (cid:17) ̺ ( N ) (d N q ) , where we omitted the argument ( q , ..., q N ) from I ( N )Λ , for brevity, and Q ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) = lim α ↓ α Q ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) . (99)26e also define α S ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) precisely like S ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) except that Q ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) is replaced by α Q ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) . We have α S ( N ) I/ε (cid:0) ρ ⊗ n (cid:1) > −∞ for all ρ ∈ ( P ∩ L ln L )(Λ), and lim α ↓ α S ( N ) I/ε (cid:0) ρ ⊗ n (cid:1) = −∞ whenever I/ε 1. By α ̺ ( N ) N ε wedenote the unique maximizer of α S ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) , easily proven to exist as done for S ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) . Equally easily we find lim α ↓ α S ( N ) I/ε (cid:0) α ̺ ( N ) N ε (cid:1) = S ( N ) I/ε (cid:0) ̺ ( N ) N ε (cid:1) .We are now ready to formulate the N = ∞ analogue of (93).To define the mean quasi-interaction energy of ̺ ∈ P s (Λ N ), we introducethe subset P sU (Λ N ) ⊂ P s (Λ N ) for which the expected value of U is finite;i.e. R U ( q , q ′ ) ̺ (d q d q ′ ) < ∞ , where ̺ (d q d q ′ ) is the second marginalmeasure of ̺ ∈ P sU (Λ N ). Also, by P U (Λ) we denote the subset of P (Λ) whichconsists of Lebesgue-absolutely continuous probability measures ρ for which R U ( q , q ′ ) ρ ⊗ (d q d q ′ ) < ∞ , which implies h ρ , ρ i < ∞ ; here we recycled thebilinear form notation (43) for lower semi-continuous (rather than continuous) U Λ . If ̺ ∈ P sU (Λ N ), then the decomposition measure ς (d ρ | ̺ ) is concentratedon P U (Λ); this can be shown by adapting arguments from [HeSa55]; cf. also[MeSp82]. The mean quasi-interaction energy of ̺ ∈ P sU (Λ N ) is defined as Q I/ε ( ̺ ) ≡ lim α ↓ lim n →∞ n α Q ( n ) I/ε (cid:0) n ̺ (cid:1) . (100)We show that Q I/ε ( ̺ ) is well-defined. By the linearity of n ̺ α Q ( n ) I/ε (cid:0) n ̺ (cid:1) ,the presentation (97) yields α Q ( n ) I/ε (cid:0) n ̺ (cid:1) = Z α Q ( n ) I/ε ( ρ ⊗ n ) ς (d ρ | ̺ ) , (101)and on P sU (Λ N ) the conventional law of large numbers for U statistics applies(see [Hoe48]) and yieldslim n →∞ n α Q ( n ) I/ε ( ρ ⊗ n ) = h ln (cid:2) − ε (cid:10) ρ , ρ (cid:11)(cid:3) χ {h ρ , ρ i <ε (1 − α ) } (102)+ (cid:0) ln α + α (cid:2) − α − ε (cid:10) ρ , ρ (cid:11)(cid:3)(cid:1) χ {h ρ , ρ i≥ ε (1 − α ) } i . Clearly, when α ↓ −∞ is assigned to all ρ for which h ρ , ρ i ≥ ε ; the α ↓ h ρ , ρ i < ε . We conclude with: Lemma 1. The mean quasi-interaction energy (100) is well-defined and affinelinear. For ̺ ∈ P sU (Λ N ) having decomposition measure ς (d ρ | ̺ ) supportedentirely by ρ for which h ρ , ρ i < ε , we have (100) given by Q I/ε ( ̺ ) = Z Q I/ε ( ρ ) ς (d ρ | ̺ ) , (103)27 here Q I/ε ( ρ ) ≡ ln (cid:2) − ε (cid:10) ρ , ρ (cid:11)(cid:3) ; (104) otherwise, Q I/ε ( ̺ ) = −∞ . The N = ∞ analogue of (92) is the well-known mean (relative) entropy of ̺ ∈ P s (Λ N ), which is well-defined as limit R ( ̺ ) ≡ lim n →∞ n R ( n ) (cid:0) n ̺ | λ (cid:1) . (105)Here, R ( n ) (cid:0) n ̺ | λ (cid:1) , n ∈ { , , ... } , is the relative entropy of n ̺ , as defined in(92); we also set R ( − k ) (cid:0) − k ̺ | λ (cid:1) ≡ k ∈ N . The limit (105) exists or is −∞ . This is a consequence of the next lemma, which holds for ̺ ∈ P s (Λ N )or ̺ ∈ P s (Λ N ). If ̺ = ̺ ( N ) , it is understood that k ≤ N in ̺ ( N ) k . Lemma 2. Relative entropy n R ( n ) (cid:0) n ̺ | λ (cid:1) has the following properties:(A) Non-positivity : For all n , R ( n ) (cid:0) n ̺ | λ (cid:1) ≤ 0; (106) (B) Monotonic decrease : If n > m then R ( n ) (cid:0) n ̺ | λ (cid:1) ≤ R ( m ) (cid:0) m ̺ | λ (cid:1) ; (107) (C) Strong sub-additivity : For m, n ≤ ℓ , and k = ℓ − m − n , R ( ℓ ) (cid:0) ℓ ̺ | λ (cid:1) ≤ R ( m ) (cid:0) m ̺ | λ (cid:1) + R ( n ) (cid:0) n ̺ | λ (cid:1) + R ( k ) (cid:0) k ̺ | λ (cid:1) − R ( − k ) (cid:0) − k ̺ | λ (cid:1) . (108)The proof of Lemma 2 is a straightforward adaptation from a proof byRobinson and Ruelle [RoRu67] (section 2, proof of proposition 1) for thestandard-thermodynamic-limit problem to the Vlasov limit, studied here, cf.[Kie93].The next lemma also has an elementary proof which likewise is an adaptionfrom [RoRu67], proof of their proposition 3, cf. [Kie93]. Lemma 3. The mean entropy functional (105) is affine linear. Lemma 3 in conjunction with the de Finetti [deF37] – Dynkin [Dyn53]– Hewitt-Savage [HeSa55] decomposition theorem for P s (Λ N ) yields a keyformula for the mean entropy which does not hold for the finite- N entropy.Namely, as a consequence of Lemma 3, the extremal decomposition of ̺ yields R ( ̺ ) = Z R ( ρ | λ ) ς (d ρ | ̺ ) , (109)where we also set R ( ρ | λ ) ≡ R (1) ( ρ | λ ).Lemma 4, also proved by adaption of a corresponding proof in [RoRu67],proposition 4, ends the listing of properties of mean relative entropy (105).28 emma 4. The mean entropy functional is weakly upper semi-continuous. Finally we define the mean interaction entropy of ̺ ∈ P s (Λ N ), S I/ε ( ̺ ) ≡ R ( ̺ ) + Q I/ε ( ̺ ) . (110)By (109) and (103) we have S I/ε ( ̺ ) = Z P (Λ) S I/ε ( ρ ) ς (d ρ | ̺ ) , (111)where we introduced the functional S I/ε ( ρ ) ≡ R ( ρ | λ ) + Q I/ε ( ρ ) , (112)which is well-defined and finite whenever ρ ∈ ( P ∩ L ln L )(Λ) and (cid:10) ρ , ρ (cid:11) < ε ;else we have S I/ε ( ρ ) = −∞ . Note that S I/ε ( ρ ) ≤ 0, for R (cid:0) ̺ | λ (cid:1) ≤ Q I/ε ( ρ ) ≤ 0, the latter because U Λ ≥ S I/ε ( ̺ ) reduces to maximizing S I/ε ( ρ ) given in (112). Proposition 5. S I/ε ( ρ ) is weakly upper semi-continuous for ε > ε g ≥ andtakes its finite non-positive maximum at a solution of the fixed point equation ρ ( q ) = exp (cid:0) − ϑ − ε ( ρ ) R Λ U Λ ( q , ˜ q ) ρ (˜ q )d ˜ q (cid:1)R Λ exp (cid:0) − ϑ − ε ( ρ ) R Λ U Λ (ˆ q , ˜ q ) ρ (˜ q )d ˜ q (cid:1) dˆ q , (113) where ϑ ε ( ρ ) = (cid:0) − ε (cid:10) ρ , ρ (cid:11)(cid:1) ε > . (114) Proof of Proposition 5: Since relative entropy R ( ρ | λ ) is weakly upper semi-continuous ([ReSi80],Suppl. to IV.5; [Ell85], chpt.VIII), and since the functional Q I/ε ( ρ ) is weaklyupper semi-continuous as a consequence of hypothesis ( H 2) and the positivityof U Λ , so is S I/ε ( ρ ). Since Λ is compact, S I/ε ( ρ ) now takes its maximum, whichis non-positive because S I/ε ( ρ ) ≤ 0, and finite (i.e. > −∞ ) because of thefollowing. Let k ρ ( k ) in ( P ∩ C ∞ )(Λ) be a minimizing sequence for (cid:10) ρ , ρ (cid:11) .Since ε > ε g ≥ 0, by ( H 3) there is a K such that ε g < (cid:10) ρ ( k ) , ρ ( k ) (cid:11) < ε for all k ≥ K . Then max ρ S I/ε ( ρ ) ≥ S I/ε ( ρ ( K ) ) = R ( ρ ( K ) | λ ) + Q I/ε ( ρ ( K ) ) > −∞ .Let q ρ ε ( q ) denote any maximizer for S I/ε ( ρ ). Suppose (cid:10) ρ ε , ρ ε (cid:11) ≥ ε .Then Q I/ε ( ρ ε ) = −∞ , and because R ( ρ ε | λ ) ≤ S I/ε ( ρ ε ) = −∞ .Therefore (cid:10) ρ ε , ρ ε (cid:11) < ε strictly, and since ε > 0, this proves (114).The standard variational argument now shows that the maximizer satisfiesthe Euler-Lagrange equation for S I/ε ( ρ ), which is (113).29 orollary 1. The functional S I/ε ( ̺ ) given in (110) achieves its supremum.If ̺ ε is a maximizer of S I/ε ( ̺ ) , then the support of its decomposition measure ς (d ρ | ̺ ε ) is the set of maximizers { ρ ε } of the functional S I/ε ( ρ ) given in (112).Proof of Corollary 1: Abstractly, by Lemma 4 and the linearity of the mean quasi-interactionenergy functional, the mean interaction entropy functional S I/ε ( ̺ ) given in(110) is weakly upper semi-continuous, and so achieves its supremum over thecompact set of permutation symmetric probabilities P sU (Λ N ).Alternatively, by (111) and two obvious estimates, we have right away that S I/ε ( ρ ε ) = S I/ε ( ρ N ε ) ≤ sup ̺ S I/ε ( ̺ ) ≤ max ρ S I/ε ( ρ ) = S I/ε ( ρ ε ) , (115)so sup ̺ S I/ε ( ̺ ) = max ̺ S I/ε ( ̺ ) = S I/ε ( ρ N ε ). Now let ̺ ε maximize S I/ε ( ̺ ) andsuppose that supp ς (d ρ | ̺ ε ) is not a subset of the maximizers { ρ ε } of S I/ε ( ρ ).Then S I/ε ( ̺ ε ) = Z P (Λ) S I/ε ( ρ ) ς (d ρ | ̺ ε ) < max ρ S I/ε ( ρ ) = S I/ε ( ρ N ε ) , (116)so ̺ ε is not a maximizer — a contradiction to the supposition.We now relate the sequence of maximizers { ̺ ( N ) N ε } N ∈ N of { S ( N ) I/ε } to the set ofmaximizers { ρ ε } of S I/ε . We begin with the maxima of S ( N ) I/ε (cid:0) ̺ ( N ) (cid:1) and S I/ε ( ρ ). Proposition 6. We have lim N →∞ N S ( N ) I/ε (cid:0) ̺ ( N ) N ε (cid:1) = S I/ε ( ρ ε ) . (117) Proof of Proposition 6: For all α ∈ (0 , α S ( N ) I/ε (cid:0) α ̺ ( N ) N ε (cid:1) ≥ α S ( N ) I/ε (cid:0) ρ ⊗ nε (cid:1) . (118)We compute α S ( N ) I/ε (cid:0) ρ ⊗ nε (cid:1) = N R (1) (cid:0) ρ ε | λ (cid:1) + α Q ( N ) I/ε (cid:0) ρ ⊗ nε (cid:1) . (119)Since (cid:10) ρ ε , ρ ε (cid:11) < ε , when α ∈ (0 , 1) is sufficiently small we have by ( H 4) andProposition 5 thatlim N →∞ N α Q ( N ) I/ε (cid:0) ρ ⊗ nε (cid:1) = ln (cid:2) − ε (cid:10) ρ ε , ρ ε (cid:11)(cid:3) . (120)Hence, for all sufficiently small α ∈ (0 , N →∞ N α S ( N ) I/ε (cid:0) ρ ⊗ nε (cid:1) = S I/ε (cid:0) ρ ε (cid:1) . (121)30hus lim inf N →∞ N α S ( N ) I/ε (cid:0) α ̺ ( N ) N ε (cid:1) ≥ S I/ε ( ρ ε ) (122)for all sufficiently small α ∈ (0 , N →∞ N S ( N ) I/ε (cid:0) ̺ ( N ) N ε (cid:1) ≥ S I/ε ( ρ ε ) . (123)Now consider (94) as extended to a probability on Λ N . Since Λ is bounded,Λ is compact, and then the sequence { ̺ ( N ) N ε } N ∈ N is weakly compact, solim N →∞ n ̺ ( ˙ N [ N ])˙ N ε = n ˙ ̺ ε ∈ P s (Λ n ) ∀ n ∈ N , (124)after extraction of a subsequence { ̺ ( ˙ N [ N ]) } N ∈ N ; note that the { n ˙ ̺ ε } n ∈ N form acompatible sequence of marginals. Furthermore, we have R ∂ Λ 1 ˙ ̺ ε (d q ) = 0, orelse R ( ˙ ̺ ε ) = −∞ , a contradiction; so n ˙ ̺ ε ∈ P s (Λ n ).Following [MeSp82, Kie93] we now use sub-additivity of relative entropy(property ( C ) in Lemma 2) and then negativity of relative entropy (property( A ) in Lemma 2) (valid also with ̺ ( ˙ N )˙ N ε in place of ̺ ), and obtain R ( ˙ N ) (cid:0) ̺ ( ˙ N )˙ N ε | λ (cid:1) ≤ j ˙ Nn k R ( n ) (cid:0) n ̺ ( ˙ N )˙ N ε | λ (cid:1) + R ( m ) (cid:0) m ̺ ( ˙ N )˙ N ε | λ (cid:1) ≤ j ˙ Nn k R ( n ) (cid:0) n ̺ ( ˙ N )˙ N ε | λ (cid:1) (125)where ⌊ a/b ⌋ is the integer part of a/b , and where m < n . Upper semi-continuity for the relative entropy giveslim sup N →∞ R ( n ) (cid:0) n ̺ ( ˙ N [ N ])˙ N ε | λ (cid:1) ≤ R ( n ) ( n ˙ ̺ ε | λ ) , (126)while N j ˙ Nn k → n . Hence, dividing (125) by ˙ N [ N ] and letting N → ∞ giveslim sup N →∞ N R ( ˙ N ) (cid:0) ̺ ( ˙ N )˙ N ε | λ (cid:1) ≤ n R ( n ) ( n ˙ ̺ ε | λ ) ∀ n ∈ N , (127)and now taking the supremum over n (equivalently: the limit n → ∞ ) we getlim sup N →∞ N R ( ˙ N ) (cid:0) ̺ ( ˙ N )˙ N ε | λ (cid:1) ≤ R ( ˙ ̺ ε ) . (128)Lastly, using (109) in (128) yieldslim sup N →∞ N R ( ˙ N ) (cid:0) ̺ ( ˙ N ) N ε | λ (cid:1) ≤ Z R ( ρ | λ ) ς (d ρ | ˙ ̺ ε ) (129)31here ς (d ρ | ˙ ̺ ε ) be the Hewitt–Savage decomposition measure for ˙ ̺ ε . For each ρ ∈ supp ς (d ρ | ˙ ̺ ε ) we can choose a family of ̺ ( ˙ N ) [ ρ ] ∈ P s (Λ ˙ N ) satisfyinglim N →∞ n ̺ ( ˙ N ) [ ρ ] = ρ ⊗ n (130)for each n ∈ N , such that for each ˙ N [ N ], with N ∈ N , we have ̺ ( ˙ N )˙ N ε = Z ̺ ( ˙ N ) [ ρ ] ς (d ρ | ˙ ̺ ε ) . (131)In contrast to the de Finetti-Dynkin-Hewitt-Savage decomposition, this fi-nite N decomposition is not unique, but this is immaterial. We remarkthat in the physically (presumably) most important situations, namely whensupp ς (d ρ | ̺ ε ) is either a finite set or a continuous group orbit of a compactgroup, then a decomposition (131) satisfying (130) can easily be constructedexplicitly, as shown in Appendix B.By (131), the linearity of the map ̺ ( ˙ N ) α Q ( ˙ N ) I/ε (cid:0) ̺ ( ˙ N ) (cid:1) gives α Q ( ˙ N ) I/ε (cid:0) ̺ ( ˙ N ) N ε (cid:1) = Z α Q ( ˙ N ) I/ε ( ̺ ( ˙ N ) [ ρ ]) ς (d ρ | ˙ ̺ ε ) , (132)and by the concavity of the map I α Q ( ˙ N ) I/ε (cid:0) ̺ ( ˙ N ) (cid:1) , Jensen’s inequality gives α Q ( ˙ N ) I/ε ( ̺ ( ˙ N ) [ ρ ]) ≤ N − (cid:20) ln (cid:2) − ε U ( N ) ( ̺ ( ˙ N ) ) (cid:3) χ { U ( N )( ̺ ( ˙ N )) <ε (1 − α ) } (133)+ (cid:16) ln α + α (cid:2) − α − ε U ( N ) ( ̺ ( ˙ N ) ) (cid:3)(cid:17) χ { U ( ̺ ( ˙ N )) ≥ ε (1 − α ) } (cid:21) , where U ( N ) ( ̺ ( ˙ N ) ) = (cid:16) − ˙ N − (cid:17)Z U Λ (ˇ q , ˆ q ) ̺ ( ˙ N ) [ ρ ](d ˇ q d ˆ q ) . (134)The weak lower semi-continuity of U Λ now giveslim inf N →∞ Z U Λ 2 ρ ( ˙ N ) [ ρ ]d q ≥ (cid:10) ρ, ρ (cid:11) , (135)and since ˙ N − → 0, we find for each convergent subsequence of measures thatlim sup N →∞ N α Q ( ˙ N ) I/ε (cid:0) ̺ ( ˙ N ) [ ρ ] (cid:1) ≤ h ln (cid:2) − ε (cid:10) ρ , ρ (cid:11)(cid:3) χ {h ρ , ρ i <ε (1 − α ) } (136)+ (cid:0) ln α + α (cid:2) − α − ε (cid:10) ρ , ρ (cid:11)(cid:3)(cid:1) χ {h ρ , ρ i≥ ε (1 − α ) } i . for each α ∈ (0 , (cid:10) ρ , ρ (cid:11) ≥ ε ; then r.h.s.(136) ↓ −∞ as α ↓ 0, in which case by (136) and (132) also α Q ( ˙ N ) I/ε (cid:0) ̺ ( ˙ N ) N ε (cid:1) ↓ −∞ as α ↓ (cid:10) ρ , ρ (cid:11) < ε for every ρ ∈ supp ς (d ρ | ˙ ̺ ε ), and so, and recalling (104), we conclude thatlim sup N →∞ N Q ( ˙ N ) I/ε (cid:0) ̺ ( ˙ N ) N ε (cid:1) ≤ Z ln (cid:2) − ε (cid:10) ρ , ρ (cid:11)(cid:3) ς (d ρ | ˙ ̺ ε ) = Z Q I/ε ( ρ ) ς (d ρ | ˙ ̺ ε ) . (137)The estimates (137) and (129) and two obvious estimates now givelim sup N →∞ N S ( ˙ N ) I/ε (cid:0) ̺ ( ˙ N )˙ N ε (cid:1) ≤ lim sup N →∞ N R ( ˙ N ) (cid:0) ̺ ( ˙ N )˙ N ε | λ (cid:1) + lim sup N →∞ N Q ( ˙ N ) I/ε (cid:0) ̺ ( ˙ N )˙ N ε (cid:1) ≤ Z R ( ρ | λ ) ς (d ρ | ˙ ̺ ε ) + Z Q I/ε ( ρ ) ς (d ρ | ˙ ̺ ε )= Z S I/ε ( ρ ) ς (d ρ | ˙ ̺ ε ) ≤ max ρ S I/ε ( ρ ) , (138)and since this holds for each limit point ˙ ̺ ε we can drop the dot to getlim sup N →∞ N S ( N ) I/ε (cid:0) ̺ ( N ) N ε (cid:1) ≤ S I/ε ( ρ ε ) . (139)By (123) and (139), Proposition 6 is proved.By Propositions 4, 5, and 6, the interaction entropy per particle for Boltz-mann’s ergodic ensemble converges as follows, s Λ , I ( ε ) ≡ lim N →∞ N S I ( N )Λ ( N ε ) = S I/ε ( ρ ε ) , (140)and s Λ , I ( ε ) is characterized by its own variational principle expressed in Propo-sition 5. Moreover, by formula (37), which holds under assumptions ( H H s Λ ( ε ) = s Λ , K ( ε ) + s Λ , I ( ε ) , (141)where s Λ , K ( ε ) is given in (39), and s Λ , I ( ε ) in (140) and Proposition 5. ByProposition 5, any maximizer ρ ε of S I/ε ( ρ ) satisfies (28) and (29).Lastly, one readily verifies that the just described s Λ ( ε ) equals the negativeminimum of Boltzmann’s H functional over the set of trial densities A ε = { f ∈ ( P ∩ L ∩ L ln L )( R × Λ) : E ( f ) = ε } . This is done by explicitly carryingout the standard variational argument for H ( f ), taking the constraints intoaccount with the help of Lagrange multipliers which are then eliminated withthe help of the very functionals of ρ ε displayed in Theorem 2.This completes the proof of Theorem 2.33 .3 Proof of Theorem 3 We begin with the observation that (94) is clearly the N -th configurationalmarginal measure of (2), i.e. (94) is (2) with the Hamiltonian given by (19),integrated over all the p variables in (19). Put differently, (94) is the joint N -point distribution on configuration space Λ N of an N -body system withHamiltonian (19) chosen w.r.t. the a-priori measure (2) on ( R × Λ) N . Hence,our proof of Theorem 2 also proves the following weaker version of Theorem 3. Theorem 3. − Under the same assumptions as in Theorem 2, consider (2)for the Hamiltonian (19) as extended to a probability on ( R × Λ) N . Then thesequence { ̺ ( N ) N ε } N ∈ N of its configuration space marginals, obtained by integrat-ing over all the p variables in (19) and given in (94), is weakly compact in P s (Λ N ) , so one can extract a subsequence { ̺ ( ˙ N [ N ])˙ N ε } N ∈ N such that lim N →∞ ̺ ( ˙ N [ N ])˙ N ε = ˙ ̺ ε ∈ P s (Λ N ) , (142) in the sense that lim N →∞ n ̺ ( ˙ N [ N ])˙ N ε = Z P (Λ) Y ≤ k ≤ n ρ ( q k )d q k ς (d ρ | ˙ ̺ ε ) ∀ n ∈ N . (143) The decomposition measure ς (d ρ | ˙ ̺ ε ) of each such limit point ˙ ̺ ε is supported onthe subset of P (Λ) which consists of the probability measures ρ ε ( q )d q whichmaximize the functional S I/ε ( ρ ) . Since each limit point ˙ ̺ ε of (94) is a convex linear superposition of infi-nite product measures on Λ N consisting of “Boltzmann factors” ρ ε ( q ) on Λ,satisfying (28) with (29) and maximizing the interaction entropy functional S I/ε ( ρ ), and since each such Boltzmann factor is associated with a unique“Maxwellian” σ ε ( p ) on R through (30), each such Boltzmann factor therebydefines a unique Maxwell–Boltzmann distribution σ ε ( p ) ρ ε ( q ) on R × Λ givenby the product of this Boltzmann factor with its associated Maxwellian. Sothe very decomposition measure ς (d ρ | ˙ ̺ ε ) of each limit point ˙ ̺ ε on Λ N allowsus to define a unique probability measure ˙ µ ε on ( R × Λ) N , viz. n ˙ µ ε (d n p d n q ) = Z P (Λ) Y ≤ k ≤ n σ ( ρ )( p k ) ρ ( q k )d p k d q k ς (d ρ | ˙ ̺ ε ) ∀ n ∈ N , (144)and this measure ς (d ρ | ˙ ̺ ε ) on P (Λ) can be mapped into a unique measure ν (d τ | ˙ µ ε ) on P ( R × Λ) which is concentrated on those τ ∈ P ( R × Λ) whichare of the form τ (d p d q ) = σ ( ρ )( p ) ρ ( q )d p d q , with σ ( ρ ) given by (30) and ρ satisfying (28) with (29) and maximizing S I/ε ( ρ ), thus n ˙ µ ε (d n p d n q ) = Z P ( R × Λ) Y ≤ k ≤ n τ (d p k d q k ) ν (d τ | ˙ µ ε ) ∀ n ∈ N . (145)34he corresponding infinite product measures Q ≤ k ≤∞ τ (d p k d q k ) on ( R × Λ) N are extreme points of P s (( R × Λ) N ), and so (145) is the extremal representationof ˙ µ ε . So, having Theorem 3 − and its consequence (145), all we need to do tofinish the proof of Theorem 3 is to show that each such defined ˙ µ ε is indeed alimit point of (2) under the stated hypotheses.To see this, we use that by Theorem 3 − we already know that (143) holds,and we know the support of ς (d ρ | ˙ ̺ ε ). Writing n ̺ ( ˙ N [ N ])˙ N ε explicitly gives n ̺ ( ˙ N )˙ N ε (d n q ) = Z (cid:16) − ε ˙ N I ( ˙ N )Λ ( q , ..., q ˙ N ) (cid:17) + N − d 3( ˙ N − n ) q Z (cid:16) − ε ˙ N I ( ˙ N )Λ (˜ q , ..., ˜ q ˙ N ) (cid:17) + N − d N ˜ q d n q, (146)where the integral in the numerator runs over the variables q n +1 to q ˙ N . Forany ˙ N and 1 ≤ n < ˙ N we now write I ( ˙ N )Λ ( q , ..., q ˙ N ) = I ( n )Λ ( q , ..., q n )+ I ( n | ˙ N )Λ ( q , ..., q ˙ N )+ I ( ˙ N − n )Λ ( q n +1 , ..., q ˙ N ) (147)which defines I ( n | ˙ N )Λ ( q , ..., q ˙ N ). Henceforth we omit the arguments from the I s to keep the formulas within sight; by (147) the superscripts convey whichvariables are used. With the help of (147) we rewrite the integrands thusly, (cid:16) − ε ˙ N I ( ˙ N )Λ (cid:17) + = − I ( n )Λ + I ( n | ˙ N )Λ ˙ N (cid:0) ε − ˙ N − I ( ˙ N − n )Λ (cid:1) ! + (cid:16) − ε ˙ N I ( ˙ N − n )Λ (cid:17) + . (148)Now (cid:0) − ( ε ˙ N ) − I ( ˙ N )Λ (cid:1) + vanishes in an open neighborhood of configurations( q , ..., q n ) ∞ for which I ( ˙ N )Λ = ∞ , so that I ( n )Λ < ∞ on the support of (148).And for any configuration ( q , ..., q n ) for which I ( n )Λ < ∞ , we have ˙ N − I ( n )Λ → N [ N ] → ∞ . Moreover, by our Theorem 3 − and its explication (143),we have that (cid:16) − ε ˙ N I ( ˙ N − n )Λ (cid:17) + N − d 3( ˙ N − n ) q , interpreted as a measure on theconvex set of probability measures P (Λ) with support in the set of empiricalone-point “densities” with ˙ N − n atoms, converges (up to normalization) to We are using that ( f g ) + = f + g + + f − g − for two arbitrary functions f and g , and thatin our case f cannot be strictly negative if g is, giving ( f g ) + = f + g + ; in our case, f and g are the respective expressions between the two pairs of big parentheses at r.h.s.(148). If U Λ is bounded continuous on Λ , then we already know that we can rewrite I ( N )Λ asa sum of a bilinear and a linear form on P (Λ) evaluated at a normalized empirical one-point “density” with N atoms; see (42). If U Λ is only lower semi-continuous this particularidentification ceases to make sense, but happily we can always interpret I ( N )Λ as a linearform on the convex set of probability measures P (Λ ) (cf. (133) and (135)), evaluated at anormalized empirical two-point “density” with N atoms (7), and we note that any empiricaltwo-point “density” with N atoms (7) is uniquely determined by its associated empiricalone-point “density” with N atoms (6). (d ρ | ˙ ̺ ε ), and for any ρ in the support of ς (d ρ | ˙ ̺ ε ) we have that ˙ N − I ( ˙ N − n )Λ →h ρ, ρ i while ˙ N − I ( n | ˙ N )Λ → P ≤ k ≤ n R Λ U Λ ( q k , ˜ q ) ρ (˜ q )d q when ˙ N [ N ] → ∞ . Sofor any ρ in the support of the decomposition measure ς (d ρ | ˙ ̺ ε ) we have − I ( ˙ n )Λ + I ( n | ˙ N )Λ ˙ N (cid:0) ε − ˙ N − I ( ˙ N − n )Λ (cid:1) ! N − → Y ≤ k ≤ n exp (cid:18) − ϑ ε ( ρ ) Z Λ U Λ ( q k , ˜ q ) ρ (˜ q )d ˜ q (cid:19) (149)with ϑ ε ( ρ ) = ε − h ρ, ρ i . After this preparation, we now explicitly computethe marginal n µ ( ˙ N )˙ N ε and find n µ ( ˙ N )˙ N ε (d n p d n q ) = Z (cid:16) − ε ˙ N (cid:16) K ( n | N ) + I ( ˙ N )Λ (cid:17)(cid:17) + 3( ˙ N − n )2 − d 3( ˙ N − n ) q Z (cid:16) − ε ˙ N (cid:16) K ( n | N ) + I ( ˙ N )Λ (cid:17)(cid:17) + 3( ˙ N − n )2 − d N ˜ q d n ˜ p d n p d n q, (150)where K ( n | N ) ( p , ..., p n ) = N P ≤ k ≤ n | p k | . Using (147) we factor the inte-grands as in (148), though now we get − K ( n | N ) + I ( ˙ N )Λ ε ˙ N ! + = − K ( n | N ) + I ( n )Λ + I ( n | ˙ N )Λ ˙ N (cid:0) ε − ˙ N − I ( ˙ N − n )Λ (cid:1) ! + (cid:16) − ε ˙ N I ( ˙ N − n )Λ (cid:17) + (151)and by following essentially verbatim the arguments which lead from (148) to(149), we now find that for any ρ ∈ supp ς (d ρ | ˙ ̺ ε ), − K ( n | N ) + I ( ˙ n )Λ + I ( n | ˙ N )Λ ˙ N (cid:0) ε − ˙ N − I ( ˙ N − n )Λ (cid:1) ! 3( ˙ N − n )2 − → Y ≤ k ≤ n exp (cid:18) − | p k | + R Λ U Λ ( q k , ˜ q ) ρ (˜ q )d ˜ qϑ ε ( ρ ) (cid:19) (152)Our Theorem 3 is proved. In this section we list a number of corollaries of our results. Whenever H B ( f ) has a unique minimizer f ε over A ε , then necessarily all limitpoints in (33) coincide, i.e. any ˙ µ ε = µ ε . By the weak compactness of P s (Λ N )(in product topology) we then in fact do have weak convergence,lim N →∞ n µ ( N ) N ε (d n p d n q ) = n µ ε (d n p d n q ) ∈ P s (( R × Λ) n ) ∀ n ∈ N . (153)36ince in this case the decomposition measure ν ( dτ | µ ε ) is a singleton, the limit µ ε = { n µ ε } n ∈ N is of the form n µ ε (d n p d n q ) = Y ≤ k ≤ n f ε ( p k , q k )d p k d q k (154)with f ε ( p , q ) = σ ε ( p ) ρ ε ( q ) as defined in Theorem 2. As discussed in [Spo91],the factorization property (154) is equivalent to a weak law of large numbers— or to an ergodic theorem, depending on ones point of view. Since the singleparticle momentum P and position Q of an individual N -body system pickedfrom Boltzmann’s Ergode (2), with Hamiltonian (19), are random variables,any bounded continuous single-particle test function θ on R × Λ defines a newrandom variable Θ = θ ( P , Q ), and so does its sample mean over a single N -body system, (cid:10) Θ (cid:11) N ≡ N N X j =1 θ ( P j , Q j ) . (155)Theorem 3 in the special case (154) implies that, for all such θ ,lim N →∞ (cid:10) Θ (cid:11) N = Z R × Λ θ ( p , q ) f ε ( p , q )d p d q , (156)in probability. The generalization to n -body test functions holds as well. A second corollary, or actually a whole family of corollaries, is the existenceof the Vlasov limit for the thermodynamic potentials of the canonical andgrandcanonical ensembles under the same hypotheses. We only discuss theVlasov limit for the thermodynamic potential of the canonical ensemble.Thus, taking the Laplace transform of (3), i.e. multiplying by e − β E andintegrating over E , yields what is known as the canonical partition function , Z H ( N )Λ ( β ) = N ! Z exp (cid:16) − βH ( N )Λ ( X ( N ) ) (cid:17) d N X. (157)The Hamiltonian H ( N )Λ ( X ( N ) ) is given in (19). Clearly (157) factors as follows, Z H ( N )Λ ( β ) = Z K ( N ) ( β ) Z I ( N )Λ ( β ) (158)where Z I ( N )Λ ( β ) = Z exp (cid:16) − βI ( N )Λ ( q , ..., q N ) (cid:17) λ (d N q ) (159)37s the canonical configurational integral , with λ (d q ) = | Λ | − d q the normalizedLebesgue measure introduced in section 5, and λ (d N q ) its N -fold product, and Z K ( N ) ( β ) = | Λ | N N ! Z exp (cid:0) − βK ( N ) ( p , ..., p N ) (cid:1) d N p (160)is the canonical partition function of a spatially uniform perfect gas in Λ, aGaussian on the Cartesian product of the p spaces, which evaluates to Z K ( N ) ( β ) = | Λ | N N ! (cid:0) πϑ (cid:1) N/ ; (161)here, we introduced N ϑ = β − , with ϑ independent of N , not to be confusedwith ϑ ε which is a functional of ρ . Since β − receives the meaning of a tem-perature of a heat bath (up to the absorbed factor k B ), it needs to grow ∝ N to compensate for the growth of the system’s energy E ∝ N . Taking thelogarithm of (157) gives what we call the canonical thermodynamic potential (canonical T -potential, for short) Φ H ( N )Λ ( β ). Using (158) and (161) as wellas β = Nϑ yields the asymptotic expansionΦ H ( N )Λ ( Nϑ ) = − N ln N + N ln (cid:0) e | Λ | (2 πϑ ) / (cid:1) + O (ln N )+ ln Z I ( N )Λ ( Nϑ ) . (162)Again, the N ln N term is due to Gibbs’ N ! and purely combinatorial in origin.In the absence of interactions (save the confinement to Λ) (162) reduces toΦ K ( N ) ( Nϑ ) = − N ln N + N ln (cid:0) e | Λ | (2 πϑ ) / (cid:1) + O (ln N ) , (163)the asymptotic expansion of the canonical T -potential of the spatially uniformperfect gas. The coefficient of the O ( N ) term in (163) is the system-specificHelmholtz T -potential per particle of the uniform perfect gas in Λ, denoted by φ Λ , K ( ϑ ) = ln (cid:0) e | Λ | (2 πϑ ) / (cid:1) . (164)The system-specific interaction Helmholtz T -potential per particle is definedby φ Λ , I ( ϑ ) = lim N →∞ N ln Z I ( N )Λ ( Nϑ ) . (165)The limit (165) exists for Hamiltonians satisfying ( H H H 2) is replaced by bounded continuity of theinteraction, as explained earlier, then we can also infer the existence of the Multiplying the canonical T -potential by the temperature of the heat bath yields thenegative of what is usually called the canonical free energy, which in the thermodynamiclimit yields the Helmholtz free energy of the physical systems. N ln Z H ( N )Λ ( β ) = N ln Z e − β E + S ( E ) d E (166)where S ( E ) is shorthand for S H ( N )Λ ( E ). Setting E = N ε and β = Nϑ and ex-panding S ( E ) using (23) (if U Λ is bounded continuous on Λ we can alternatelyuse (22)), we find N ln Z H ( N )Λ ( Nϑ ) = − ln N + ln (cid:20)Z e N ( − ϑ − ε + s Λ ( ε ) ) + o ( N ) d ε (cid:21) N + O (cid:0) ln NN (cid:1) . (167)Clearly, k g k N → k g k ∞ as N → ∞ , and so the following asymptotic expansionfor the canonical T -potential results,Φ H ( N )Λ ( Nϑ ) = − N ln N + N φ Λ ( ϑ ) + o ( N ) (168)with φ Λ ( ϑ ) ≡ sup ε>ε g (cid:0) − ϑ − ε + s Λ ( ε ) (cid:1) . (169)By (162) and (168), we also have N − ln Z I ( N )Λ ( Nϑ ) N →∞ −→ φ Λ , I ( ϑ ), with φ Λ , I ( ϑ ) = φ Λ ( ϑ ) − φ Λ , K ( ϑ ) . (170)This concludes our demonstration that the Vlasov limit for the system-specificHelmholtz T -potential per particle follows from our theorems about the Vlasovlimit of the system-specific Boltzmann entropy per particle.Next we notice that also the familiar “minimum free energy principle” for − φ Λ ( ϑ ) follows from combining the Legendre–Fenchel transform (169) withour “maximum entropy principle” in Theorem 2. Thus, for the system-specificHelmholtz T -potential per particle we find the variational principle − ϑφ Λ ( ϑ ) = inf f ∈ A F ϑ ( f ) , (171)with A = { f ∈ ( P U Λ ∩ L ∩ L ln L )( R × Λ) } the admissible trial densities,and F ϑ ( f ) = E ( f ) + ϑ H B ( f ) , (172)the Helmholtz free energy functional of f , where H B ( f ) is Boltzmann’s H function of f , given in (25), and E ( f ) is the energy functional given in (26).39t also follows directly from our results that F ϑ ( f ) takes its infimum over theset A , and that any minimizer f ϑ of F ϑ ( f ) over A is of the form f ϑ ( p , q ) = σ ϑ ( p ) ρ ϑ ( q ) , (173)where σ ϑ ( p ) = (2 πϑ ) − exp (cid:0) − ϑ − | p | (cid:1) , (174)while ρ ϑ ( q ) now solves the following fixed point equation on q space, ρ ϑ ( q ) = exp (cid:0) − ϑ R Λ U Λ ( q , ˜ q ) ρ ϑ (˜ q )d ˜ q (cid:1)R Λ exp (cid:0) − ϑ R Λ U Λ (ˆ q , ˜ q ) ρ ϑ (˜ q )d ˜ q (cid:1) dˆ q (175)with ϑ > possible relationships between the set of maxi-mizers of the maximum entropy variational principle and the set of minimizersof the minimum free energy variational principle have been discussed in greatdetail in [EHT00, CETT05]. Note that this can be (and was) done withoutproving that the maximum entropy variational principle characterizes the limitpoints of Boltzmann’s Ergode (2) proper.We also remark that the existence of the system-specific Helmholtz T -potential per particle in the Vlasov limit for the canonical ensemble was shownpreviously by various techniques. Sub-additivity arguments, such as those usedto prove Theorem 1, are used in [Kie93]. The very strategy which we appliedto prove Theorems 2 and 3, which not only yields the variational principle forthe system-specific Boltzmann entropy but also identifies the limit points ofthe sequence of ergodic ensemble measures as convex linear superpositions ofinfinite products of the optimizers for this maximum entropy principle, wasoriginally applied in [MeSp82] to the canonical ensemble for Lipschitz continu-ous interactions I ( N )Λ ; subsequently in [Kie93] and in [CLMP92] this approachto the canonical ensemble was generalized to less regular interactions includingthe ones studied here; and in [KiSp99] the limit N → ∞ of N − ln Z I ( N )Λ (1 /ϑ )was obtained by adapting this strategy (note the different N scaling of β ).We emphasize that none of these canonical results implies the existence ofthe Vlasov limit for the system-specific Boltzmann entropy per particle, norcaptures the limit points of the ergodic ensemble measures, unless it is a prioriknown that the ensembles are (convexly) equivalent, i.e. unless it is knownthat ε s Λ ( ε ) is concave (more on that in section 7). Our results, by contrast,hold irrespective of whether ε s Λ ( ε ) is concave or not. Another spin-off, or in this case rather a variation on the theme of our mi-crocanonical results is the straightforward generalization of our Theorems to40ubensembles whose invariant measures are concentrated on sub-manifolds of { H = E } determined by further isolating integrals of the Hamiltonian (19),such as angular momentum if the domain Λ is rotationally symmetric, orthe Lynden-Bells’ invariant [LBLB99, LBLB04] which occurs in a generaliza-tion of the Calogero–Moser model to particles moving in R confined by aquadratic potential. Hypothesis ( H 4) does not hold for these interactions, butcan be replaced by a weaker one at the expense of some extra work. In thosecases the entropy maximizer factors into a product of a locally (at q ) shiftedMaxwellian on p space and a purely space-dependent Boltzmann factor . Theshifted Maxwellian which generalizes (27) to include angular momentum isknown as a “rotating Maxwellian;” in the case of the Lynden-Bells’ Hamilto-nian one finds a “rotating-dilating Maxwellian.” An announcement of theseresults was made in [Kie08]; details will appear in [KiLa09]. In this last section of our paper we point out some open problems related tothe ones treated here. To the best of the author’s knowledge, the maximum interaction entropy prin-ciple formulated in Proposition 5 is new. As made clear in Theorem 2 it offersa way to directly evaluate the usual variational principle of maximum entropywith energy constraint. By contrast, the standard approach to evaluate thisconstrained maximum entropy principle has been rather indirect. Namely,a Lagrange parameter (basically ϑ ) is introduced for the energy constraint,yielding the corresponding fix point equation (175) for the stationary pointsof the free energy functional. After finding all solution families (not just theminimizers of the free energy functional), a parameter representation of energyand entropy along the various solution families of (175) results, among whichthe one with highest entropy for given energy has then to be selected. Clearlyour new variational approach appears to be more economical than that.One of the simplest tasks would be to prove the existence of a unique solu-tion to (28) at sufficiently high energies ε . For Coulomb interactions a uniquesolution is expected for all energies, while for (regularized) Newton interac-tions multiplicity of solutions is expected for sufficiently low energies. This issuggested by the detailed numerical evaluations of the standard principle ofmaximum entropy with constraints for related equations, cf. [SKS95, Cha02].41 .2 Convergence of the ergodic ensemble measures We already pointed out in subsection 6.1 that the sequence of ergodic ensem-ble measures converges whenever a unique optimizer exists for the maximuminteraction entropy variational principle in Theorem 2 and Proposition 5. Wedon’t see any reason why the sequence of ergodic ensemble measures shouldnot converge when the entropy maximizer is not unique, and so we expectthat the mere existence of limit points concluded in this paper by using weakcompactness can actually be upgraded to the existence of a limit. As also noted in subsection 6.1, the decomposition measure ν ( dτ | µ ε ) is a single-ton whenever a unique optimizer exists for the maximum interaction entropyvariational principle in Theorem 2. In more general situations we have lit-tle information on the decomposition measure ν ( dτ | µ ε ), beyond knowing thatit reduces to ς (d ρ | ̺ ε ) and that ς (d ρ | ̺ ε ) is supported on the maximizers ofthe maximum interaction entropy principle formulated in Proposition 5. Ofcourse, we already mentioned earlier that experience with explicitly studiedphysical systems suggests that supp ς (d ρ | ̺ ε ) is either a finite set or a contin-uous group orbit of a compact group, but a general proof or disproof seemsnot available. More is known for the canonical ensemble [KuTa84], and theirapproach should apply to the microcanonical ensemble to determine ν ( dτ | µ ε ). Whenever H B ( f ) has a unique minimizer f ε over A ε , then Theorems 2 and 3imply thatProb (cid:16) d KR (cid:16) ∆ ( n ) X ( N ) , f ⊗ nε (cid:17) > δ (cid:17) N →∞ −→ ∀ δ > , (176)where “Prob” refers to the ensemble measure (2) with Hamiltonian (19). It isdesirable to improve (176) to a large deviation principle, a rigorous variationon the theme of Einstein’s fluctuation formula. Heuristically we expectProb (cid:16) d KR (cid:16) ∆ ( n ) X ( N ) , f ⊗ nε (cid:17) > δ (cid:17) ≍ sup f ∈ A δε e − N ( H B ( f ) − H B ( f ε )) ∀ δ > , (177)where A δε = { f ∈ ( P ∩ L ∩ L ln L )( R × Λ) : E ( f ) = ε }\ e B δ ( f ε ). In[EySp93, EHT00, CETT05] such a feat was accomplished for the regularizedmicrocanonical ensembles at the level of the 1-point functions. The recent ar-ticle [EiSch02] establishes some nice large deviation principles for the n -point42unctions in a strong topology which allows one to handle some singular in-teractions. We expect that the conjectured large deviation principle can beproved along their lines.We also refer to Lanford’s article [Lan73] and the books by Varadhan[Var84] and Ellis [Ell85] for mathematical background on large deviation prin-ciples and their applications to statistical mechanics, and to [Tou08] for a morerecent review. Using the very strategy used in this paper to prove our Theorems 2 and 3,the Vlasov limit for the canonical ensemble measures associated with (157)was established in [MeSp82, CLMP92, Kie93] under various hypotheses on theinteractions, covering our ( H H By hypothesis ( H 2) we allow the pair interactions to diverge when two parti-cles approach each other infinitely closely. However, W Λ ( q , ˜ q ) is only allowedto diverge to + ∞ , which happens with the repulsive Coulomb interactionswhen q → ˜ q . Divergence of W Λ ( q , ˜ q ) to −∞ is excluded from our analysis,because our postulates imply that I ( N )Λ is bounded below by E g ( N ) > −∞ .In particular, the −∞ singularity of the attractive Newton interactions in R will have to be regularized.The canonical ensemble and regularized microcanonical ensembles havebeen controlled under weaker hypotheses, allowing in particular the interac-tions to diverge logarithmically to −∞ , see [CLMP92, Kie93] for the canonicaland [CLMP95, KiLe97, Kie00] for the regularized microcanonical ensembles.It should be possible to adapt the technical arguments in these papers to estab-lish the Vlasov limit for (2) for negative logarithmically singular interactions.43 .7 Unbounded domains In [KiSp99] and [ChKi00], unbounded Λ where allowed for the canonical en-semble, and our microcanonical theorems should similarly be extendible tounbounded domains under a suitable confinement hypothesis which replaceshypothesis ( H H ′ ) Confinement : e − U Λ ( q , ˜ q ) ∈ L (Λ × Λ) . (178)Incidentally, ( H ′ ) not only imposes on behavior of U Λ as any of its two argu-ments is sent to infinity, it also restricts the manner in which U Λ can divergeto −∞ , e.g. when its two arguments approach each other infinitely closely,allowing logarithmic divergence. Our analysis does not cover ergodic ensembles of quasi-particle systems likepoint vortices moving in two dimensions whose Kirchhoff Hamiltonian is of thetype (1) without the sum of | p | terms. The ergodic point vortex ensemblemeasures are of the type µ ( N ) E (d N X ) = (cid:0) N !Ω ′ I ( N )Λ ( E ) (cid:1) − δ (cid:0) E − I ( N )Λ ( X ( N ) ) (cid:1) d N X , (179)where X ( N ) := ( q , ..., q N ) ∈ Λ N , where now Λ ⊂ R , and d N X is 2 N -dimensional Lebesgue measure, and the pair interactions now feature posi-tive logarithmic singularities (for a single specie of point vortices). Onsager[Ons49] observed that for such systems a critical E value exists such that themap E S ( E ) is decreasing when E > E crit , giving rise to negative ensembletemperatures. Regularized microcanonical measures for such vortex Hamilto-nians have been analyzed in [CLMP95] under an equivalence assumption tothe canonical ensemble, and in [KiLe97, Kie00] without such an equivalenceassumption. It is desirable to find a way to handle the proper ergodic en-semble for point vortex and other quasi-particle systems for which the sum ofsquares of kinematical momenta is absent from their Hamiltonian, but clearlythis will require the introduction of new technical ideas. Incidentally, thislast sentence applies verbatim also to other scalings than Vlasov scaling, inparticular to the conventional thermodynamic limit scaling explained in theintroduction. The authors of [CLMP95] use the primitive Ω I ( N )Λ ( E ) of Ω ′ I ( N )Λ ( E ) (i.e. (3) with H ≡ I )to define a quasi-microcanonical ensemble entropy when E < E crit , and for E > E crit theyuse Ω I ( N )Λ ( ∞ ) − Ω I ( N )Λ ( E ). In [KiLe97, Kie00] a Gaussian approximation to δ ( I − E ) is used.We also mention [EySp93] where the approximation Ω I ( N )Λ ( E ) − Ω I ( N )Λ ( E − △ E )is used; theseauthors also regularize the logarithmic singularity of the interactions. E crit of a point vortex system it is a priori known that all the n -pointmeasures have densities given by (1 / | Λ | ) ⊗ n . Taking advantage of this fact,O’Neil and collaborators [ONR91, CON91] found that for a neutral two-species system the vicinity of E crit ∝ N ln N can be analyzed directly using δ ( I − E );it turns out to be a small-entropy regime where S , not S/N , converges to alimit when N → ∞ , with E − CN ln N ∝ N . Interestingly enough, this scalingfalls in between the conventional thermodynamic limit and the Vlasov scaling.To the author’s knowledge, so far these are the only results for point vorticesobtained for δ ( I − E ) proper, i.e. without regularization of the Dirac measure. Acknowledgment : The author thanks Carlo Lancellotti for his careful read-ing of the manuscript and for his comments. This paper was written withsupport from the NSF under grant DMS-0807705. Any opinions expressed inthis paper are entirely those of the author and not necessarily those of theNSF. 45 Monotonicity of the ground state energy In this appendix we will prove two monotonic convergence results about theground state energy which are used in the setup of our construction of theVlasov limit N → ∞ . The results and their proofs are rather elementaryand presumably known, and quite likely to be found in the vast literatureon U statistics; however, my (certainly incomplete) perusal of the pertinentliterature has not yet met with success. Here is our first proposition. Proposition 7. Let Λ ⊂ R D be a bounded and connected domain. Assumethe following hypotheses regarding U Λ ( q , ˜ q ) : ( H Symmetry: U Λ (ˇ q , ˆ q ) = U Λ (ˆ q , ˇ q )( H Lower Semi-Continuity: U Λ (ˇ q , ˆ q ) is l . s . c . on Λ × Λ( H Sublevel Set Regularity: λ ⊗ (cid:16)n U Λ (ˇ q , ˆ q ) − min U Λ < ǫ o(cid:17) > H Local Square Integrability: U Λ ( q , · ) ∈ L ( B r ( q ) ∩ Λ) ∀ q ∈ Λ where λ is normalized Lebesgue measure for Λ . For N ≥ define the pair-specific ground state energy by ε g ( N ) ≡ min { q ,..., q N } N ( N − X X ≤ i We begin with the mandatory observation that under hypotheses ( H H 2) the pair-specific ground state energy ε g ( N ) defined in (180) is well-defined; i.e. ε g ( N ) ∈ R (note that ( H H 4) are immaterial here). In fact, I originally did not expect monotonicity results of the type proved here to holdat all. I was prompted to conjecture the results, and then to prove them, by analyzingthe numerical results of the computations of the (conjectured) ground state energies E g ( N )for Thomson’s problem [Tho04] reported in [Aetal97, Petal97], which – divided by either N or N ( N − 1) – arranged themselves monotonically increasing when plotted vs. N . Aninteresting spin-off of the monotonicity of the pair-specific Thomson energies is a necessarycriterion for minimality which can be used as a test for the empirical numerical experiments.After the present paper was submitted I successfully carried out such a test; see [Kie09b]. 46e next prove the monotonicity of N ε g ( N ), with N ≥ 2. Elementary(combinatorial) identities and the single inequality that the minimum of a sumis not less than the sum of the minima shows that ε g ( N + 1) ≥ ε g ( N ), viz. ε g ( N + 1) = min { q ,..., q N +1 } N +1) N X X ≤ i 2) and ( H ε g definedin (181) is well-defined; actually, for this issue we can even relax ( H 4) to theweaker L loc (Λ) condition which is implied by ( H N -point measures in the weakly compact set of all probabilitymeasures on Λ , and the existence (by ( H H ∈ C b (Λ) for h ρ, ρ i , to prove convergence ε g ( N ) ր ε g . We let ∆ (2) X ( N ) g denotethe 2-point measure in Λ for a ground state X ( N ) g = (0 , q ; ... ; 0 N , q N ) g of N points in Λ (which need not be unique), and let ∆ (2) X ( N ) be any other 2-pointmeasure on Λ with N support points.We define the linear functional ρ U ( ρ ) by U ( ρ ) = ZZ U Λ (ˇ q , ˆ q ) ρ (d D ˇ q d D ˆ q ) . (183)Note that for product measures ρ = ρ ⊗ we have U ( ρ ⊗ ) = h ρ, ρ i . (184)Note furthermore that the functional ρ U ( ρ ) is generally not continuous,because we have only (weak) lower semi-continuity of U Λ . In particular, whileany continuous change in the supporting points of the 2-point measure ∆ (2) X ( N ) 47n Λ results in a weakly continuous change of the 2-point measure, the func-tional U evaluated at these 2-point measures, i.e. U (∆ (2) X ( N ) ), generally changesdiscontinuously. However, we do have ε g ( N ) = U (∆ (2) X ( N ) g ) ≤ U (∆ (2) X ( N ) ) . (185)Now let { ρ n } n ∈ N be a minimizing sequence in ( P ∩ C b )(Λ) for h ρ, ρ i = U ( ρ ⊗ );note that it is not necessary to postulate also that ρ n → ρ for any actualminimizer ρ , as this will follow automatically from the proof. Then, by ( H ǫ > n ǫ such that U ( ρ ⊗ n ) ≤ ε g + ǫ whenever n ≥ n ǫ .So pick any ǫ > 0, let n = n ǫ , and let { q k } k ∈ N be i.i.d. with a-priori measure ρ n ǫ ∈ ( P ∩ C b )(Λ) for each q k . Then by ( H 4) the weak law of large numbersfor U statistics (of order 2) holds [Hoe48], and so, in probability, U (∆ (2) X ( N ) ) N →∞ −→ (cid:10) ρ n ǫ , ρ n ǫ (cid:11) ≤ ε g + ǫ (186)for each ǫ > 0. By (186) and (185) we havelim sup N →∞ ε g ( N ) = lim sup N →∞ U (∆ (2) X ( N ) g ) ≤ ε g . (187)On the other hand, by the compactness of Λ and the weak ∗ compactness of P (Λ) we can extract a ∗ -weakly convergent subsequence ∆ (2) X ( ˙ N ) g → ˙ ρ ∈ P (Λ ).Moreover, since any convergent sequence of n -point measures ∆ ( n ) X ( ˙ N ) g necessarilyconverges to an n -fold product measure, we have ˙ ρ = ˙ ρ ⊗ . Now the weak lowersemi-continuity of U giveslim inf ˙ N →∞ U (∆ (2) X ( ˙ N ) g ) ≥ (cid:10) ˙ ρ, ˙ ρ (cid:11) ≥ ε g . (188)Estimates (187) and (188) prove convergence ε g ( N ) → ε g .Convergence and the earlier proved monotonicity of N ε g ( N ) completesthe proof of Proposition 7.We notice that our proof of Proposition 7 yields as a “byproduct” that (cid:10) ˙ ρ, ˙ ρ (cid:11) = ε g . Thus we have the following noteworthy corollary: Corollary 2. Any limit point ˙ ρ ⊗ of the sequence of ground state 2-pointmeasures { ∆ (2) X ( N ) g } N ∈ N minimizes the bilinear form U ( ρ ⊗ ) = h ρ, ρ i . Here is our second proposition. Proposition 8. Assume the hypotheses on U Λ ( q , ˜ q ) stated in the previousproposition, and in addition assume that U Λ ≥ . Then the quasi pair-specificground state energy, defined by ˜ ε g ( N ) ≡ min q ,..., q N N X X ≤ i 1, the limit of ˜ ε g ( N ) coincides with that of ε g ( N ).This concludes the proof of Proposition 8. B Decomposition of the finite N measures Let ̺ ε ∈ P s (Λ N ) be the weak limit of { ̺ ( N ) N ε ∈ P s (Λ N ) } N ∈ N , and let ς (d ρ | ̺ ε )be its unique de Finetti-Dynkin-Hewitt-Savage decomposition measure. (If { ̺ ( N ) N ε } N ∈ N has several limit points, as accounted for in the main text, thefollowing considerations are valid for the associated converging subsequencesof finite N measures.) We now show that if supp ς (d ρ | ̺ ε ) is either a finite set ora continuous group orbit of a compact group, then for each ρ ∈ supp ς (d ρ | ̺ ε )we can explicitly construct a family of ̺ ( N ) [ ρ ] ∈ P s (Λ N ) satisfyinglim N →∞ n ̺ ( N ) [ ρ ] = ρ ⊗ n (192)for each n ∈ N , such that for each N ∈ N , ̺ ( N ) N ε = Z ̺ ( N ) [ ρ ] ς (d ρ | ̺ ε ) . (193) B.1 The support of ς (d ρ | ̺ ε ) is a finite set In the simplest case ς (d ρ | ̺ ε ) is a singleton, so that ̺ ε = ρ ⊗ N ε , i.e.lim N →∞ n ̺ ( N ) N ε = ρ ⊗ nε ∀ n ∈ N . (194)In this case R ̺ ( N ) [ ρ ] ς (d ρ | ̺ ε ) = ̺ ( N ) [ ρ ε ] = ̺ ( N ) N ε , and we are done.Next, assume that ς (d ρ | ̺ ε ) is an arithmetic mean of two singletons, viz. ς (d ρ | ̺ ε ) = ν δ ρ (d ρ ) + ν δ ρ (d ρ ) (195)49ith 0 < ν = 1 − ν < 1, and let d KR ( ρ , ρ ) = D > ρ and ρ . Let B D/ ( ρ k ) be the KR -open ball in P (Λ) which is centered at ρ k and has radius D/ 2. Now decomposeΛ N = Λ N ∪ Λ N , where Λ ∩ Λ = ∅ and ̺ ( N ) N ε (Λ Nk ) = ν k , such that Λ Nk containsall points for which ∆ ( N ) ∈ B D/ ( ρ k ); when N is too small there may be nosuch points, but by the weak density in P (Λ) of the empirical one-point mea-sures the set of such points ∈ Λ N has positive ̺ ( N ) N ε measure when N is largeenough. In fact, since by hypothesis the weak limit of { ̺ ( N ) N ε ∈ P s (Λ N ) } N ∈ N isgiven by ̺ ε = ν ρ ⊗ N + ν ρ ⊗ N ∈ P s (Λ N ), it follows that when N ր ∞ then theprobability w.r.t. ̺ ( N ) N ε that ∆ ( N ) ∈ B D/ ( ρ k ) approaches ν k . So if we define ̺ ( N ) [ ρ k ] = ν − k ̺ ( N ) N ε χ Λ Nk (196)and recall that ̺ ( N ) N ε (Λ Nk ) = ν k , it follows thatlim N →∞ n ̺ ( N ) [ ρ k ] = ρ ⊗ nk (197)for each n ∈ N and k = 1 or 2, and such that for each N ∈ N , ̺ ( N ) N ε = ν ̺ ( N ) [ ρ ] + ν ̺ ( N ) [ ρ ] , (198)which is (193) in the case that ς is the arithmetic mean of two singletons.The general case of supp ς being a finite set is treated similarly in an obviousmanner, with D now the minimum of the set of distances between any pair( ρ k , ρ l ) picked from the support of ς . B.2 The support of ς (d ρ | ̺ ε ) is a continuous group orbit For simplicity we assume that we are dealing with a one-parameter continuousgroup G acting on the base space, like SO (2) acting on Λ; the generalizationto more complicated situations (e.g. SO (3) acting on Λ) is straightforward. Inthis case we can pick any particular ρ ∈ supp ς and obtain every other (say) ρ θ ∈ supp ς by acting with a group element g θ ∈ G thusly, ρ θ = ρ ◦ g θ . 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