Large Deviations in the Superstable Weakly Imperfect Bose Gas
aa r X i v : . [ m a t h - ph ] S e p Large Deviations in the Superstable Weakly Imperfect Bose Gas Large Deviations in the Superstable Weakly Imperfect Bose Gas
J.-B. Bru a and V.A. Zagrebnov ba Fakult¨at f¨ur Physik, Universit¨at Wien, Boltzmanngasse 5, 1090 Vienna, Austria b Universit´e Aix-Marseille II and Centre de Physique Th´eorique - UMR 6207Luminy-Case 907, F-13288 Marseille, Cedex 09, France
Abstract
The superstable Weakly Imperfect Bose Gas (WIBG) was originally derived to solve the inconsistency ofthe Bogoliubov theory of superfluidity. Its grand-canonical thermodynamics was recently solved but not atpoint of the (first order) phase transition. This paper proposes to close this gap by using the large deviationsformalism and in particular the analysis of the Kac distribution function. It turns out that, as a functionof the chemical potential, the discontinuity of the Bose condensate density at the phase transition pointdisappears as a function of the particle density. Indeed, the Bose condensate continuously starts at the firstcritical particle density and progressively grows but the free-energy per particle stays constant until thesecond critical density is reached. At higher particle densities, the Bose condensate density as well as thefree-energy per particle both increase monotonously.
Keywords : WIBG, Bose-Einstein condensation, Kac distribution, large deviations, equivalence of ensembles.
1. Introduction
The proof of large deviations for the distribution of the particle density (the Kac distribution) in the Perfectand in the Mean-Field Bose gases goes back to [1]. In recent papers [2, 3], the authors addressed to the largedeviations in the particle density in a sub-domain both for the perfect and for rarified quantum gases (Fermi orBose). In the present paper we extend the study of Large Deviations (LD) principle to the superstable WeaklyImperfect Bose Gas (WIBG) [4], known also as the Superstable Bogoliubov model [5]. The study of this modelstarted in [4, 6] was recently completed in [7, 9, 10, 8].Actually, this model originates from a weaker truncation than that of the Bogoliubov one in the grand-canonical ensemble. This new system served to solve some inconsistencies between the grand-canonical Bogoli-ubov theory of superfluidity and the WIBG description. This non-diagonal boson model was rigorously solvedon the thermodynamic level for the grand-canonical ensemble in [8, 10]. It turns out that similar to the WIBGit manifests a phase transition with a non-conventional
Bose condensation at high densities ρ or high inversetemperatures β . Meantime, even for β ↑ + ∞ , i.e. at a zero-temperature, only a fraction of the full density isin the condensate: i.e. there is a coexistence of particles inside and outside the boson condensate. This lastphenomenon is known as depletion of the condensate. More interesting for our analysis is a discontinuity of theparticle density from ρ − > ρ + > ρ − related to a strictly positive jump of the condensate density at thephase transition defined by a fixed chemical potential µ c . This first-order phase transition as a function of thechemical potential µ sounds unusual and seems to be not quite clear as far as it concerns its physical relevance.In fact, the grand-canonical thermodynamics of the superstable WIBG is unknown at the point of coexistenceof the low- and high-density phases. This paper proposes to close this gap using large deviations techniquesdescription of the density distribution. For instance to answer the question, what is the value of the Bosecondensate density when ρ ∈ [ ρ − , ρ + ] ? In fact several scenarios are possible. Since this phase transition ischaracterized by the appearance of a non-conventional Bose condensation, which is due to particle interaction,a naive thought might be that there is no condensate at all in domain ρ ∈ ( ρ − , ρ + ), i.e. the condensatedensity jumps from zero to a strictly positive value for ρ > ρ + . In fact this scenario is wrong . Here we showthat this discontinuity is a subtle function of the total particle density ρ . Formally, at the point of the phasecoexistence, the corresponding quantum Gibbs state of the model is no more a pure state [11] but instead, aconvex combination of some of them. A similar observation was made for example in Section 4 of [1]. J.-B. Bru - V.A. Zagrebnov
Actually, we verify LD for the Bose condensate density for any given particle densities ρ in the grand-canonicalensemble, i.e. even at the point of the phase transition. A direct consequence of this study is a rigorous proofthat the discontinuity of the Bose condensate and its depletion, visible as a function of the chemical potential µ , does not appear in the same grand-canonical ensemble if it is considered as a function of the total particledensity ρ >
0. We show that the Bose condensate density continuously increases with ρ >
0. In others words,there is no jump and the phase transition in ρ > ρ (or theinverse temperature β ) exceeds the first critical value ρ − , the Bose condensate density continuously grows butthe free-energy per particle, i.e., the corresponding chemical potential µ ρ , stays constant: µ ρ = µ c in domain: ρ ∈ [ ρ − , ρ + ]. At higher particle densities (or inverse temperatures β ), the Bose condensate as well as thefree-energy per particle µ ρ > µ c both increase when ρ > ρ + .The structure of the paper is the following. In Section 2 we briefly review the grand-canonical thermodynamicsof the superstable WIBG for a given particle density ρ . Our main results are formulated in Section 3. Theproofs are collected in Section 4. For the reader convenience, we collect in Appendix (Section 5) some technicalresults as well as a short review on the LD principles.To conclude, we recall that throughout this paper β > µ and ρ > h−i H Λ for ( finite-volume ) grand-canonical Gibbs state corresponding to the Hamiltonian H Λ .
2. The Superstable Weakly Imperfect Bose Gas
Let an homogeneous gas of n spinless bosons with mass m be enclosed in a cubic box Λ ⊂ R of volume V := | Λ | . The one-particle energy spectrum is then ε k := ~ k / m and, using periodic boundary conditions,Λ ∗ := (2 π Z /V / ) ⊂ R is the set of wave vectors k . The considered system is with interactions defined via a(real) two-body soft potential ϕ ( x ) = ϕ ( || x || ) such that:(A) ϕ ( x ) ∈ L (cid:0) R (cid:1) (absolute integrability).(B) Its (real) Fourier transformation λ k = λ || k || satisfies: λ > ≤ λ k ≤ λ for k ∈ R .The Superstable WIBG (also known as the AVZ Hamiltonian [8] or the Superstable Bogoliubov Hamiltonian[7]), was proposed for the first time in [4]. It is defined by H SB Λ ,λ > := H B Λ , + U MF Λ . (2.1)Here the weakly imperfect Bose gas H B Λ , := X k ∈ Λ ∗ \{ } (cid:26) ε k a ∗ k a k + λ k (cid:18) a ∗ a V (cid:0) a ∗ k a k + a ∗− k a − k (cid:1) + a ∗ k a ∗− k a V + a ∗ V a k a − k (cid:19)(cid:27) (2.2)contains the kinetic-energy term ∗ plus diagonal and non-diagonal interactions. It is solved in the canonicalensemble in [9, 10]. The repulsive interaction ensuring the superstability of H SB Λ ,λ by assumptions (A)-(B) isthe “forward scattering” interaction U MF Λ := λ V X k ,k ∈ Λ ∗ a ∗ k a ∗ k a k a k = λ V (cid:0) N − N Λ (cid:1) , with N Λ := X k ∈ Λ ∗ a ∗ k a k (2.3)defined as the particle-number operator within the grand-canonical framework. Indeed, a ∗ k and a k are the usualboson creation / annihilation operators in the one-particle state † χ Λ ( x ) e ikx / √ V , acting on the boson Fock space F B Λ := + ∞ M n =0 H ( n ) B , with H ( n ) B := (cid:0) L (Λ n ) (cid:1) symm , H (0) B := C , (2.4) ∗ Recall that ε = 0 . † Here χ Λ ( x ) is the characteristic function of the box Λ . arge Deviations in the Superstable Weakly Imperfect Bose Gas n -particle Hilbert spaces, see [11, 12]. Remark 2.1
Let H ⊂ L (Λ) be the one-dimensional subspace generated by ψ k =0 ( x ) = 1 / √ V . Then F B Λ ≈F ⊗ F ′ Λ where F and F ′ Λ are the boson Fock spaces constructed out of H and of its orthogonal complement H ⊥ respectively. We consider here the grand-canonical ensemble ( β, µ ) defined by a given particle density ρ , or more preciselyby the chemical potential µ Λ ,ρ , which is a unique solution of the equation (2.5) below. In any finite volume,the corresponding particle density is strictly increasing by strict convexity of the pressure. Therefore, for any ρ > , there exists a unique µ Λ ,ρ such that (cid:28) N Λ V (cid:29) H SB Λ ,λ = ρ, (2.5)where h−i H SB Λ ,λ always represents the (finite volume) grand-canonical Gibbs states for H SB Λ ,λ taken at inversetemperature β and chemical potential µ Λ ,ρ . In the thermodynamic limit, µ Λ ,ρ converges to µ ρ ∈ R for any ρ >
0. In fact, µ ρ is strictly increasing except for ρ ∈ [ ρ − , ρ + ] where it equals µ c = µ c . Here ρ + > ρ − > β > . Additionally, µ ρ := α ρ + λ ρ with α ρ < ∂ λ α ρ = 0 for ρ ∈ [ ρ − , ρ + ] . Moreover, there is a non-conventional Bose condensation induced by the non-diagonal interaction U ND Λ forhigh particle densities: x ρ :=lim Λ (cid:28) a ∗ a V (cid:29) H SB Λ ,λ = (cid:26) = 0 for ρ < ρ − ,> ρ > ρ + , (2.6)with ∂ λ x ρ = 0 for ρ / ∈ [ ρ − , ρ + ] . When ρ ↓ ρ + , note that the Bose condensate density x ρ converges to x ρ + > . In particular, since µ ρ = µ c for ρ ∈ [ ρ − , ρ + ] , the Bose condensate density x µ as a function of the chemicalpotential µ jumps from 0 to x ρ + at µ = µ c . An illustration of the behavior of x ρ for a fixed density ρ (or x µ ata fixed chemical potential µ ) is performed in Figure 1. ρ x + ρ x + x To zero−temperature µµ c x ρ ρ ρ − + To zero−temperature Bose condensation Bose condensation ρ µ
Figure 1:
Illustration of the Bose condensate density, x ρ at fixed particle density ρ > or x µ at fixed chemicalpotential µ ∈ R . The dashed line closing continuously the gap between ρ − and ρ + in the illustration of x ρ is a consequence of results of the present paper. Here each of the asymptotic straight lines are : x ρ = ρ , or x µ = µ/λ . They correspond to the limits : x ρ →∞ , or x µ →∞ , with 100% of the Bose condensate. We would like to stress that coexistence of different types of condensations is a subtle matter. For example aslight modification of interaction, see [8, 9, 10], excludes any coexistence of non-conventional and conventionalBose condensation, as it appears for high densities in the Bogoliubov WIBG [13, 14].
J.-B. Bru - V.A. Zagrebnov
Below we consider coexistence a high and low density phases. For intermediate total density ρ / ∈ [ ρ − , ρ + ] wefind for the particle density outside the zero-modelim Λ V X k ∈ Λ ∗ \{ } h a ∗ k a k i H SB Λ ,λ = 1(2 π ) Z R ( f k E k [ e βE k −
1] + x ρ λ k E k [ f k + E k ] ) d k, (2.7)where f k := ε k − α ρ + x ρ λ k and E k := ( f k − x ρ λ k ) . Observe that the grand-canonical thermodynamic behavior of the superstable WIBG is unknown for ρ ∈ [ ρ − , ρ + ] at fixed β >
0. When β → + ∞ , i.e. at zero temperature, the critical densities ρ − and ρ + couldboth converge to zero depending on the interaction potential [8, 10], whereas the critical chemical potential µ c converges to a negative value. Moreover, we have a non-zero particle density outside the zero-mode for any fixed ρ > β → + ∞ lim Λ V X k ∈ Λ ∗ \{ } h a ∗ k a k i H SB Λ ,λ > β → + ∞ lim Λ (cid:28) a ∗ a V (cid:29) H SB Λ ,λ < ρ. (2.8)In other words, there is a depletion of the Bose condensate even at zero temperature.To conclude, the grand-canonical pressure associated with H SB Λ ,λ in the thermodynamic limit equals p SB ( β, µ ρ ) = sup x ≥ (cid:26) inf α ≤ (cid:26) p B ( β, α, x ) + ( µ ρ − α ) λ (cid:27)(cid:27) (2.9)= inf α ≤ (cid:26) p B ( β, α, x ρ ) + ( µ ρ − α ) λ (cid:27) (2.10)= p B ( β, α ρ , x ρ ) + λ ρ , (2.11)for any ρ > . In particular, x ρ and α ρ are solutions of the variational problems (2.9) and (2.10) respectively.For any x ≥ α ≤ p B ( β, α, x ) :=lim Λ βV ln Tr F ′ Λ n e − β ( H B Λ , ( x,α ) − αx ) o (2.12)is here the (infinite volume) pressure of the so-called Bogoliubov approximation ‡ H B Λ , ( x, α ) := X k ∈ Λ ∗ \{ } (cid:26) ( ε k − α ) a ∗ k a k + xλ k (cid:0) a ∗ k a k + a ∗− k a − k + a ∗ k a ∗− k + a k a − k (cid:1)(cid:27) (2.13)of { H B Λ , − α ( N Λ − a ∗ a ) } . Observe that H B Λ , ( x, α ) is defined on the boson Fock space F ′ Λ for non-zero momentumbosons, cf. Remark 2.1 and Section 3 for a rigorous definition of the so-called Bogoliubov approximation.Finally, note that the Hamiltonian H B Λ , ( x, α ) represents, via a unitary transformation, a perfect Bose gas ofquasi-particles with one-particle spectrum E k for k ∈ Λ ∗ \ { } , see for instance [8].
3. Large deviations for the Bose condensate at a fixed total particle density
To define the (finite volume) distribution D Λ ,ρ of the condensate, we first recall the rigorous definition ofthe Bogoliubov approximation due to Ginibre [15] and based on coherent vectors. For any complex c ∈ C , acoherent vector | c i is an element of the boson Fock space F for zero momentum bosons (cf. Remark 2.1),satisfying a | c i = c √ V | c i . In fact, if Ω is the vacuum of F B Λ , then | c i := exp {− V | c | / c √ V a ∗ } Ω for any c ∈ C . The Bogoliubov approximation of a self-adjoint operator A acting on F B Λ is the operator A( c ) definedon the boson Fock space F ′ Λ without the zero mode by its quadratic form h ψ ′ | A ( c ) | ψ ′ i := h c ⊗ ψ ′ | A | c ⊗ ψ ′ i , (3.1) ‡ Combined with a gauge transformation a k → e iϕ a k arge Deviations in the Superstable Weakly Imperfect Bose Gas | c ⊗ ψ ′ , i in the form-domain of A.Now, for any chemical potential µ ∈ R the (finite volume) grand-canonical pressure associated with H SB Λ ,λ equals p SB Λ ( β, µ ) := 1 βV ln Tr F B Λ { W Λ } , with W Λ := e − β ( H SB Λ ,λ − µN Λ ) . (3.2)By using the generating family of coherent vectors | c i for c ∈ C , we can rewrite the trace Tr above to observethat p SB Λ ( β, µ ) = 1 βV ln 12 π Z C Tr F ′ Λ { W Λ ( c ) } d c = 1 βV ln 12 π Z C e βV p SB Λ ( β,µ,c ) d c, (3.3)where d c := V π − d c d c with c := c + ic , W Λ ( c ) results from the Bogoliubov approximation (3.1) of thestatistical operator W Λ , and p SB Λ ( β, µ, c ) := 1 βV ln Tr F ′ Λ { W Λ ( c ) } (3.4)is the pressure defined by the partial trace. For any ρ >
0, the corresponding distribution D Λ ,µ related to theBose condensate number density, is now defined on the Borel subsets A of C by D Λ ,µ [ A ] := e − βV p SB Λ ( β,µ ) π Z A e βV p SB Λ ( β,µ,c ) d c. (3.5)Then, at fixed particle density ρ >
0, we express a large deviations principle (Section 5.2) for the condensatedistribution D Λ ,ρ := D Λ ,µ Λ ,ρ . Theorem 3.1 (LD principle for the condensate distribution at a fixed density ρ ) The sequence { D Λ ,ρ } satisfies a large deviation principle with speed βV and rate function D ρ ( x ) := inf α ≤ (cid:26) p B ( β, α, x ρ ) + ( µ ρ − α ) λ (cid:27) − inf α ≤ (cid:26) p B ( β, α, x ) + ( µ ρ − α ) λ (cid:27) , for x = | c | ≥ , cf. (2.9)-(2.10). This theorem shows in particular that the probability to observe a density n /V ∈ A of condensed bosonsenclosed in Λ for a fixed chemical potential µ ∈ R decreases exponentially with the volume V = | Λ | if thedistance between the Bose condensate density x ρ (2.6) and the set A ⊂ R is strictly positive. Now, the nextstep is to evaluate the limiting probability measure, in particular at the phase transition defined for a chemicalpotential ρ ∈ [ ρ − , ρ + ]. Recall that the Bose condensate density x ρ (2.6) converges to 0 when ρ ↑ ρ − but to astrictly positive value x ρ + > ρ ↓ ρ + . Theorem 3.2 (The condensate distribution outside the point of the phase transition)
The (finite volume) distribution D Λ ,ρ of the condensate converges weakly in the set of probability measures M ( C ) as Λ ↑ R towards the singular measures D ρ :=lim Λ D Λ ,ρ = 12 π π Z δ (cid:16) c − x / ρ e iθ (cid:17) d θ, for any ρ ∈ (0 , ρ − ) ∪ ( ρ + , + ∞ ) . For β → + ∞ , i.e. at zero-temperature, observe that ρ − and ρ + could both converge to zero, depending onthe interaction potential. But, at finite temperature, i.e. at β > , one always has ρ + > ρ − and the convergenceof D Λ ,ρ is not solved for ρ ∈ [ ρ − , ρ + ] . The corresponding result is therefore expressed in the next theorem.
J.-B. Bru - V.A. Zagrebnov
Theorem 3.3 (The condensate distribution at the point of the phase transition)
Let ρ + > ρ − . As Λ ↑ R , the (finite volume) distribution D Λ ,ρ of the condensate converges weakly in M ( C ) towards a convex combination of the singular measures D ρ :=lim Λ D Λ ,ρ = (1 − κ ρ ) δ ( c ) + κ ρ π π Z δ (cid:16) c − x / ρ + e iθ (cid:17) d θ, for any ρ ∈ [ ρ − , ρ + ] and with κ ρ := ( ρ − ρ − ) / ( ρ + − ρ − ) . Note that κ ρ is a strictly increasing and continuous function from [ ρ − , ρ + ] to [0 , . This result gives a strongevidence that, at the phase transition, the corresponding Gibbs state is not a pure state anymore [11] but aconvex combination of pure states, see for example Section 4 in [1].Integrating D Λ ,ρ with the function ϕ ( c ) = | c | , we finally obtain the Bose condensate density (2.6) inside thephase transition, i.e. for ρ ∈ [ ρ − , ρ + ]. Corollary 3.4 (Derivation of the Bose condensate density for any total density)
The Bose condensate density equals lim Λ (cid:28) a ∗ a V (cid:29) H SB Λ j,λ = ρ ≤ ρ − .ρ − ρ − ρ + − ρ − x ρ + for ρ ∈ [ ρ − , ρ + ] .x ρ > ρ ≥ ρ + . In particular, it is continuous as a function of ρ > and linearly increasing for ρ ∈ [ ρ − , ρ + ] , cf. figure 1. As a function of the density ρ > ρ + > ρ − whereas it is of order one as a function of the chemical potential. In particular, take ρ < ρ − , then thesystem behaves as the so-called Mean-Field Bose Gas, i.e. the model defined by the Hamiltonian H MF Λ := X k ∈ Λ ∗ ε k a ∗ k a k + λ V (cid:0) N − N Λ (cid:1) , (3.6)with no Bose condensations. Increase now the particle density. The free-energy per particle, i.e., the chemicalpotential µ β,ρ ≤ µ c , normally grows until we reach ρ = ρ − . By further increasing of the density, a Bosecondensation continuously appears to reach the value x ρ + for ρ = ρ + . Meanwhile, the corresponding chemicalpotential µ ρ stays constant at the phase transition: µ ρ = µ c for ρ ∈ [ ρ − , ρ + ] . Finally, at higher particle densities,i.e., for ρ > ρ + , the Bose condensate as well as the free-energy per particle µ ρ > µ c both increase.
4. Proofs: Large Deviations for a generalized Kac distribution
We are going to study the grand-canonical ensemble at a fixed total particle density ρ >
0. But before doingthis, we start our analysis at a fixed chemical potential µ . Then we prove the LD principle for the condensateplus “out of condensate” particle densities. The corresponding distribution K Λ ,µ is a combination of the so-called Kac distribution [1] for particles outside the condensate with the condensate distribution D Λ ,µ . This isexpressed by Theorem 4.1, which is therefore, a generalization of Theorem 3.1. To study the phase transition,we use the generalized quasi-average procedure [1] by taking a ”perturbed” chemical potential˜ µ c := µ c + γβV + o (cid:18) βV (cid:19) for γ ∈ R , (4.1)we analyze the thermodynamic limit of the generalized Kac distribution at this chemical potential, see Theorem4.2.As a consequence, the generalized quasi-average procedure (4.1) gives the finite volume behavior of thechemical potential µ Λ ,ρ solution of (2.5) at the phase transition, i.e. when ρ ∈ [ ρ − , ρ + ] if ρ + > ρ − . Indeed, by arge Deviations in the Superstable Weakly Imperfect Bose Gas K Λ ,µ to an appropriate function, we obtain the mean particle density at a chemicalpotential ˜ µ c for any γ ∈ R . This procedure will then imply that for ρ ∈ [ ρ − , ρ + ] there is a unique and explicit γ ρ such that µ Λ ,ρ = ˜ µ c with | γ ρ | = o ( V ), see Section 4.2.Meanwhile, the large deviation principle for K Λ ,µ given by Theorem 4.1 directly implies Theorem 3.1 for any ρ > . Applying the result of Theorem 4.2 to the chemical potential µ Λ ,ρ = ˜ µ c for γ = γ ρ , we also get Theorem3.3 for ρ ∈ [ ρ − , ρ + ]. If ρ / ∈ [ ρ − , ρ + ] , the generalized quasi-average procedure is not necessary and Theorem 3.2is a simple consequence of Theorem 3.1. We give now the promised proofs. The particle number density as a R -valued random variable, is well-defined via a well-known probabilitymeasure, the so-called Kac distribution [1]. We give here a generalized version of the Kac distribution associatedwith the condensate and its depletion. This distribution is defined, on the Borel subsets A ⊂ C and B ⊂ R + byintegration over the zero-mode coherent state: K Λ ,µ [ A ] := e − βV p SB Λ ( β,µ ) π Z A d c Z B ν Λ (d y ) e βV ( µ ( y + | c | ) − f SB Λ ( β,y,c ) ) , (4.2)with ν Λ (d y ) := + ∞ X n =1 δ ([ yV ] − n ) d y. (4.3)Here [ . ] is the integer part and f SB Λ ( β, y, c ) := − βV ln Tr H [ yV ] B,k =0 (cid:16) { W Λ ( c ) } ([ yV ] ,k =0) (cid:17) , (4.4)where W Λ , ( c ) results from the Bogoliubov approximation (3.1) of the statistical operator W Λ , (3.2) andA ( n,k =0) is the restriction of any operator A acting on the boson Fock space F ′ B to the space {H ⊥ } ( n ) of n non-zero momentum bosons. Now we express our first result concerning large deviations for the generalizedKac distribution K Λ ,µ . Theorem 4.1 (LD principle for the generalized Kac distribution)
The sequence { K Λ ,µ } satisfies a large deviation principle with speed βV and rate function K µ ( x, y ) := p SB ( β, µ ) + f B ( β, y, x ) + λ y + x ) − µ ( y + x ) . Here x = | c | ≥ , y ≥ and f B ( β, y, x ) :=sup α ≤ (cid:8) α ( y + x ) − p B ( β, α, x ) (cid:9) is the Legendre-Fenchel transform of p B ( β, α, x ) (2.12).Proof . Let us start by some observations. The pressure p B ( β, α, x ) defined in (2.12) and used in (2.9) can beexplicitly computed. Indeed, p B ( β, α, x ) = αx − β (2 π ) Z R ln (cid:16) − e − β √ ( ε k − α )( ε k − α +2 xλ k ) (cid:17) d k ++ 12 (2 π ) Z R n ε k − α + xλ k − p ( ε k − α ) ( ε k − α + 2 xλ k ) o d k, (4.5)for any α ≤
0. Since p B ( β, α, x ) is a convex function of α ≤
0, it is also the Legendre-Fenchel transform of f B ( β, y, x ) , i.e. p B ( β, α, x ) =sup y ≥ (cid:8) α ( y + x ) − f B ( β, y, x ) (cid:9) for any α ≤ . (4.6) J.-B. Bru - V.A. Zagrebnov
Combined with (2.9) this last inequality implies that p SB ( β, µ ) =sup x ≥ (cid:26) inf α ≤ (cid:26) sup y ≥ (cid:26) α ( y + x ) − f B ( β, y, x ) + ( µ − α ) λ (cid:27)(cid:27)(cid:27) . (4.7)We would like to bring the infimum over α ≤ x ≥ y, α ) := α ( y + x ) − f B ( β, y, x ) + ( µ − α ) λ (4.8)is a strictly concave function of y ≥ α ≤
0. Then, we obtain the uniquenessof the stationary point (˜ y, ˜ α ) solution of ∂ y Ψ ( y, α ) = 0 and ∂ α Ψ ( y, α ) = y + x + α − µλ = 0 . (4.9)In particular, we can commute the infimum over α ≤ y ≥ p SB ( β, µ ) = sup ( x,y ) ∈ R (cid:26) µ ( y + x ) − f B ( β, y, x ) − λ y + x ) (cid:27) . (4.10)This result is coherent with the rate function K µ ( x, y ). By explicit computations, observe also that there are M, B > x µ , y µ ) of the variational problem (4.10) verifies x µ < M and y µ < M whereasfor any x ≥ M and y ≥ M we have µ ( y + x ) − f B ( β, y, x ) − λ y + x ) ≤ − B ( y + x ) . (4.11)Now we are in position to analyze the LD principle for distribution K Λ ,µ (Section 5.2).From (4.5) the rate function K µ ( x, y ) is not identical ∞ and has compact level sets, i.e. for each m < ∞ , thesubset { ( x, y ) : K µ ( x, y ) ≤ m } is compact.Let a closed set C := C × C ⊂ C × R + . Remark that M can be taken arbitrary large (and B being thesame). Then, without lost of generality, we can assume that any c ∈ C and y ∈ C satisfy | c | < M and y < M respectively. By (4.11), we also obtain K Λ ,µ [ C ] ≤ π e βV sup C { µ ( y + | c | ) − f SB Λ ( β,y,c ) } − p SB Λ ( β,µ ) ff Z C d c Z C ν Λ (d y ) ++ 12 π e − βV { p SB Λ ( β,µ )+2 BM } Z C d c Z R + ν Λ (d y ) e − βV B ( | c | + y ) . (4.12)For large enough M, one has2 BM + sup C (cid:26) µ ( y + x ) − f B ( β, y, x ) − λ y + x ) (cid:27) > . (4.13)Consequently, the inequality (4.12) combined Lemma 5.2 implies thatlim sup Λ βV ln K Λ ,µ [ C ] ≤ − inf C K µ (cid:0) | c | , y (cid:1) . (4.14)In other words, the large deviations upper bound (5.22) for K Λ ,µ with speed βV and rate function K µ is verified.It remains to analyze the corresponding large deviations lower bound (5.23).Let G be an arbitrary open subset of C × R + . Note that K Λ ,µ [ G ] ≥ K Λ ,µ [ { ( c, y ) } ] = e − βV p SB Λ ( β,µ ) e βV { µ ([ yV ]+ | c | ) − f SB Λ ( β,y,c ) } , (4.15) arge Deviations in the Superstable Weakly Imperfect Bose Gas c, y ) ∈ G . From Lemma 5.2, it yields thatlim inf Λ βV ln K Λ ,µ [ G ] ≥ − K µ (cid:0) | c | , y (cid:1) . (4.16)Since the last inequality holds for each point of G , it means thatlim inf Λ βV ln K Λ ,µ [ G ] ≥ − inf G K µ (cid:0) | c | , y (cid:1) , (4.17)i.e. the corresponding large deviation lower bound (5.23) for K Λ ,µ holds with speed βV and rate function K µ . (cid:3) For µ = µ c we already know [8] that the variational problem (4.10) has a unique solution ( x µ , y µ ). Therefore,as a direct consequence of the fact that the sequence { K Λ ,µ } satisfies a large deviations principle with ratefunction K µ having a unique minimum in R at ( x µ , y µ ) for any µ = µ c , the distribution K Λ ,µ converges weaklyon the set of probability measures M ( C × R + ) as Λ ↑ R towards the singular measure K µ :=lim Λ K Λ ,µ = 12 π π Z δ (cid:16) c − x / µ e iθ (cid:17) δ ( y − y µ ) d θ, for µ = µ c . (4.18)Now, the next step is to evaluate the limiting probability measure at the phase transition defined for a chemicalpotential µ = µ c . Indeed, if ρ + > ρ − , the solution ( x µ , y µ ) jumps when µ cross the critical chemical potential µ c from (0 , ρ − ) to ( x ρ + , y ρ + ) with x ρ + > y ρ + := ρ + − x ρ + > ρ − . Theorem 4.2 (The generalized Kac distribution at the phase transition) If ρ + > ρ − , then the distribution K Λ , ˜ µ c converges weakly in M ( C × R + ) as Λ ↑ R towards lim Λ K Λ , ˜ µ c = ξ γ δ ( c ) δ ( y − ρ − ) + (1 − ξ γ )2 π π Z δ (cid:16) c − x / ρ + e iθ (cid:17) δ (cid:0) y − y ρ + (cid:1) d θ, (4.19) with ξ γ := (1 + e γ ( ρ + − ρ − ) ) − ∈ (0 , and ˜ µ c defined by (4.1) for any γ ∈ R .Proof. We have already mentioned that the rate function K µ c has two distinct minima in R at (0 , ρ − ) and( x ρ + , y ρ + ). To get around this complication, take ε ∈ (0 , x ρ + ) ∩ (0 , y ρ + − ρ − ) and define A − := (cid:8) c ∈ C : | c | ∈ (cid:0) , x ρ + − ε (cid:3)(cid:9) × (cid:0) ρ − , y ρ + − ε (cid:3) (4.20)and A + := (cid:8) c ∈ C : | c | ∈ (cid:0) x ρ + − ε, + ∞ (cid:1)(cid:9) × (cid:0) y ρ + − ε, + ∞ (cid:1) . (4.21)Now, let K µ − c and K µ + c be defined as the two restrictions of K µ c to A − and A + respectively and remark that K µ − c and K µ + c have both a unique minimizer in R , respectively (0 , ρ − ) and ( x ρ + , y ρ + ). Define the correspondingprobability measures L − Λ [ A ] := K Λ ,µ c [ A ∩ A − ] K Λ ,µ c [ A − ] and L +Λ [ A ] := K Λ ,µ c [ A ∩ A + ] K Λ ,µ c [ A + ] , (4.22)which satisfy a large deviations principle respectively with rate functions K µ − c and K µ + c (where x = | c | ). Takeany positive and continuous function ϕ ( c, y ) of ( c, y ) ∈ C × R + and observe that Z R ϕ ( c, y ) K Λ ,µ c, Λ (cid:0) d c (cid:1) = R C d c R R + ν Λ (d y ) ϕ ( c, y ) e βV ( ˜ µ c ( y + | c | ) − f SB Λ ( β,y,c ) ) R C d c R R + ν Λ (d y ) e βV ( ˜ µ c ( y + | c | ) − f SB Λ ( β,y,c ) ) = Φ − Λ + Φ +Λ , (4.23)0 J.-B. Bru - V.A. Zagrebnov with Φ − Λ : = R A − ϕ ( c, y ) e { γ + o (1) } ( y + | c | ) L − Λ (cid:0) d c d y (cid:1)R A − e { γ + o (1) } ( y + | c | ) L − Λ (d c d y ) + Θ Λ R A + e { γ + o (1) } ( y + | c | ) L +Λ (d c d y ) , (4.24)Φ +Λ : = R A + ϕ ( c, y ) e { γ + o (1) } ( y + | c | ) L +Λ (cid:0) d c d y (cid:1) Θ − R A − e { γ + o (1) } ( y + | c | ) L − Λ (d c d y ) + R A + e { γ + o (1) } ( y + | c | ) L +Λ (d c d y ) , (4.25)and Θ Λ := R A + e βV { µ c ( y + | c | ) − f SB Λ ( β,y,c ) } ν Λ (d y ) d c R A − e βV { µ c ( y + | c | ) − f SB Λ ( β,y,c ) } ν Λ (d y ) d c . (4.26)By Lemma 5.2 the function µ c ( y + | c | ) − f SB Λ ( β, y, c ) (4.27)converges in the thermodynamic limit to µ c ( y + | c | ) − f B (cid:0) β, y, | c | (cid:1) − λ (cid:0) y + | c | (cid:1) , (4.28)which has suprema at (0 , ρ − ) and ( e iθ x ρ + , y ρ + ) for any θ ∈ [0 , π ] . Consequently, the coefficient Θ Λ (4.26)converges to 1 in the thermodynamic limit. Since ρ + = y ρ + + x ρ + , it is then straightforward to see thatlim Λ Φ − Λ = ξ γ ϕ (0 , ρ − ) and lim Λ Φ +Λ = (1 − ξ γ )2 π π Z ϕ ( x / ρ + e iθ , y ρ + )d θ. (4.29)Let us apply these limits to the function ϕ ( c, y ) = e − t ( c + y ) with t > . Then, by bijectivity of the Laplacetransform, it follows that K Λ , ˜ µ c converges weakly on M ( C × R + ) as Λ ↑ R to (4.19). (cid:3) Notice that the function ξ γ : R → (0 ,
1) defined in Theorem 4.2 is strictly decreasing and in fact bijective.Therefore, by applying K Λ , ˜ µ c to ϕ ( c, y ) = | c | + y, we have shown that the particle density can converge to anyfixed density in the open set ( ρ − , ρ + ) :lim Λ (cid:28) N Λ V (cid:29) H SB Λ ,λ = ξ γ ρ − + (1 − ξ γ ) ρ + . (4.30)Note that all these results are coherent since we have ρ − = lim γ →−∞ { ξ γ ρ − + (1 − ξ γ ρ + ) } and ρ + = lim γ → + ∞ { ξ γ ρ − + (1 − ξ γ ρ + ) } . (4.31)In particular, if γ = γ Λ = o ( ± V ) in (4.1) diverges to ±∞ , then we would obtain one of the previous limit,depending if γ Λ ↓ −∞ or γ Λ ↑ + ∞ . Let us consider now the total particle density as a parameter that defines the grand-canonical ensemble.Theorems 3.1 and 3.3 are direct consequences respectively of Theorem 4.1 and (4.18) for the chemical potential µ ρ defined as the thermodynamic limit of µ Λ ,ρ (2.5). The only remaining question is to study the case of fixedparticle densities at the the point of phase transition, i.e. in domain: ρ ∈ ( ρ − , ρ + ) for ρ + > ρ − . From (4.30),we obtain that lim Λ (cid:28) N Λ V (cid:29) H SB Λ ,λ = ρ ∈ ( ρ − , ρ + ) , (4.32) arge Deviations in the Superstable Weakly Imperfect Bose Gas µ = µ c + γ ρ βV + o (cid:18) βV (cid:19) with γ ρ := 1 ρ + − ρ − ln (cid:18) ρ − ρ − ρ + − ρ (cid:19) , (4.33)cf. (4.1). Therefore, µ Λ ,ρ = µ c + γ ρ βV + o (cid:18) βV (cid:19) . (4.34)In particular, from (4.2) with γ = γ ρ we get Theorem 3.3 for ρ ∈ ( ρ − , ρ + ). Recall also (4.31). In other words, if ρ = ρ − then γ ρ < | γ ρ | = o ( V )) would diverges to −∞ , whereas if ρ = ρ + then γ ρ = o ( V ) → + ∞ . It followsthat Theorem 3.3 is proven for any ρ ∈ [ ρ − , ρ + ] .
5. Appendix
In this appendix, we first give supplementary results needed in the previous section. Next, for the convenienceof our reader, we shortly repeat the notion of large deviations principles.
The thermodynamic limit of p SB Λ ( β, µ, c ) (3.4) is first analyzed in order to obtain next the one of the free-energy density f SB Λ ( β, y, c ) (4.4), which is given in Lemma 5.2. Lemma 5.1 (The pressure p SB Λ ( β, µ ; c ) in the thermodynamic limit) For any c ∈ C , µ ∈ R and β > , the pressure p SB Λ ( β, µ, c ) converges towards p SB ( β, µ, c ) :=lim Λ p SB Λ ( β, µ, c ) = inf α ≤ (cid:26) p B ( β, α, x ) + ( µ − α ) λ (cid:27) . Here x = | c | ≥ and recall that p B ( β, α, x ) is defined in (2.12), cf. also (4.5).Proof. The proof is obtained by a comparison between suitable lower and upper bounds for p SB Λ ( β, µ, c ) . Westart by the lower bound. By taking any othonormal basis {h ψ ′ n |} ∞ n =1 of F ′ Λ ,Tr F ′ Λ { W Λ ( c ) } = ∞ X n =1 h c ⊗ ψ ′ n | e − β ( H SB Λ ,λ − µN Λ ) | c ⊗ ψ ′ n i , (5.1)and so, by the Peierls-Bogoliubov inequality we getTr F ′ Λ { W Λ ( c ) } ≥ sup { ψ ′ n } ∞ n =1 ( ∞ X n =1 e − β h c ⊗ ψ ′ n | H SB Λ ,λ − µN Λ | c ⊗ ψ ′ n i ) = Tr F ′ Λ n e − βH SB Λ ,λ ( c,µ ) o , (5.2)see e.g. [16, 17], where H SB Λ ,λ ( c, µ ) results from the Bogoliubov approximation (3.1) of { H SB Λ ,λ − µN Λ } . From[8] we already know thatlim Λ βV ln Tr F ′ Λ n e − βH SB Λ ,λ ( c,µ ) o = inf α ≤ (cid:26) p B ( β, α, | c | ) + ( µ − α ) λ (cid:27) . (5.3)Consequently, the inequality (5.2) implies in the thermodynamic limit the lower bound p SB ( β, µ, c ) ≥ inf α ≤ (cid:26) p B ( β, α, | c | ) + ( µ − α ) λ (cid:27) , (5.4)for any c ∈ C , µ ∈ R and β > . J.-B. Bru - V.A. Zagrebnov
To obtain an upper bound on p SB ( β, µ, c ), we follow the idea of [18], and use the coherent state representationof { H SB Λ ,λ − µN Λ } given by H SB Λ ,λ − µN Λ = Z C d c n ˆ H SB Λ ,λ ( c, µ ) | c i h c | o , (5.5)where the Hamiltonian ˆ H SB Λ ,λ ( c, µ ) is defined on F ′ Λ byˆ H SB Λ ,λ ( c, µ ) := H SB Λ ,λ ( c, µ ) + ∆ , (5.6)with ∆ := µ − λ | c | + λ V − V X k ∈ Λ ∗ \{ } ( λ + λ k ) a ∗ k a k . (5.7)Actually, ˆ H SB Λ ,λ ( c, µ ) is derived by replacing the operators a ∗ a , a a , a ∗ a ∗ , and a ∗ a ∗ a a in { H SB Λ ,λ − µN Λ } respectively by | V c | − , V c , V ¯ c and V | c | − V | c | + 2 . Let {h ψ ′ n ( c ) |} ∞ n =1 be an othonormal basis ofeigenvectors of ˆ H SB Λ ,λ ( c, µ ) . Since for any z, c ∈ C h z | c i = e − { (¯ z − ¯ c )( z − c )+¯ cz − ¯ zc } , (5.8)it follows thatTr F ′ Λ { W Λ ( c ) } = ∞ X n =1 h c ⊗ ψ ′ n ( c ) | e − β R C d z ˆ H SB Λ ,λ ( z,µ ) | z ih z | | c ⊗ ψ ′ n ( c ) i = ∞ X n =1 ∞ X m =1 ( − β ) m m ! Z C m d z · · · d z m e − V { R m ( z , ··· ,z m )+ i I m ( z , ··· ,z m ) } × m Y j =1 h ψ ′ n ( c ) | ˆ H SB Λ ,λ ( z j , µ ) | ψ ′ n ( c ) i , (5.9)with the two real-valued functions R m and I m of ( z , · · · , z m ) ∈ C m defined byR m ( z , · · · , z m ) := | z − c | + m P j =1 | z j − − z j | + | z m − c | , I m ( z , · · · , z m ) := i (¯ z c − ¯ cz ) + i m P j =1 (¯ z j z j − − ¯ z j − z j ) + i (¯ cz m − ¯ z m c ) . (5.10)Since I m ( c, · · · , c ) = 0 and inf ( z , ··· ,z m ) ∈ C m R m ( z , · · · , z m ) = R m ( c, · · · , c ) = 0 , (5.11)by virtue of (5.9) combined with large deviations arguments, one can obtain in the thermodynamic limit that p SB ( β, µ, c ) =lim Λ βV ln Tr F ′ Λ n e − β ˆ H SB Λ ,λ ( c,µ ) o . (5.12)Justification of the LD technique in sums (5.9) is based on the uniform domination theorem and it follows theline of reasoning developed in [18]. Meanwhile, by using the Bogoliubov convexity inequality [14] it follows thatTr F ′ Λ n e − β ˆ H SB Λ ,λ ( c,µ ) o ≤ βV ln Tr F ′ Λ n e − βH SB Λ ,λ ( c,µ ) o − V h ∆ i ˆ H SB Λ ,λ ( c,µ ) , (5.13)where h−i ˆ H SB Λ ,λ ( c,µ ) := Tr F ′ Λ n − e − β ˆ H SB Λ ,λ ( c,µ ) o Tr F ′ Λ n e − β ˆ H SB Λ ,λ ( c,µ ) o . (5.14) arge Deviations in the Superstable Weakly Imperfect Bose Gas k ∈ R , ≤ λ k ≤ λ by our assumption (B) on the interaction potential, we obtain fromthe inequality (5.13) together with (5.7) that1 βV ln Tr F ′ Λ n e − β ˆ H SB Λ ,λ ( c,µ ) o ≤ βV ln Tr F ′ Λ n e − βH SB Λ ,λ ( c,µ ) o + 2 | c | λ − µV − λ V + 2 λ V X k ∈ Λ ∗ \{ } h a ∗ k a k i ˆ H SB Λ ,λ ( c,µ ) . (5.15)The last term can be explicitly computed. We omit the details. In fact, for any µ ∈ R one can check that1 V X k ∈ Λ ∗ \{ } h a ∗ k a k i ˆ H SB Λ ,λ ( c,µ ) = O (1) as Λ ↑ R . (5.16)Therefore, from (5.15) together with (5.3) and (5.12) one deduces that p SB ( β, µ, c ) ≤ inf α ≤ (cid:26) p B ( β, α, | c | ) + ( µ − α ) λ (cid:27) . (5.17)Together with the lower bound (5.4), this inequality proves the lemma. (cid:3) Lemma 5.2 (The free-energy density f SB Λ ( β, y, c ) in the thermodynamic limit) For any c ∈ C , y ≥ and β > , the thermodynamic limit f SB ( β, y, c ) of the free-energy density f SB Λ ( β, y, c ) (4.4) equals f SB ( β, y, c ) :=lim Λ f SB Λ ( β, y, c ) = f B ( β, y, x ) + λ y + x ) , with x = | c | ≥ , and f B ( β, y, x ) defined as the Legendre-Fenchel transform of p B ( β, α, x ) (2.12), cf. Theorem4.1.Proof. The pressure p SB Λ ( β, µ, c ) (3.4) can be rewritten as p SB Λ ( β, µ, c ) = 1 βV ln Z R + e βV ( µy − f SB Λ ( β,y,c ) ) ν Λ (d y ) + µ | c | , (5.18)with ν Λ (d y ) defined in (4.3). It is then straightforward to check that the thermodynamic limit p SB ( β, µ, c ) of p SB Λ ( β, µ, c ) (3.4) equals p SB ( β, µ, c ) =sup y ≥ (cid:8) µy − f SB ( β, y, c ) (cid:9) + µ | c | , (5.19)with f SB ( β, y, c ) < ∞ for y ≥
0. The derivative of the pressure p SB ( β, µ, c ) is continuous as a function of µ ,cf. Lemma 5.1 and (4.5). Thus, by using the Tauberien theorem proven in [20], the existence of p SB ( β, µ, c )already implies the convexity of f SB ( β, y, c ) as a function of y ≥
0. In particular, it yields that f SB ( β, y, c ) =sup µ ∈ R (cid:8) µ ( y + | c | ) − p SB ( β, µ, c ) (cid:9) for y ≥ . (5.20)By using the explicit form of p SB ( β, µ, c ) given by Lemma 5.1, a straightforward computation then gives f SB ( β, y, c ) =sup α ≤ (cid:8) α ( y + | c | ) − p B ( β, α, | c | ) (cid:9) + λ y + x ) . (5.21) (cid:3) Let X denote a topological vector space. A lower semi-continuous function I : X → [0 , ∞ ] is called a ratefunction if I is not identical ∞ and has compact level sets, i.e., if I − ([0 , m ]) = { x ∈ X : I( x ) ≤ m } is compact4 J.-B. Bru - V.A. Zagrebnov for any m ≥
0. A sequence { X l } + ∞ l =1 of X -valued random variables X l or the corresponding sequence { P l } + ∞ l =1 of probability measures on the Borel subsets of X satisfy the large deviations upper bound with speed a l andrate function I if, for any closed subset C of X ,lim sup l → + ∞ a l ln P l ( X l ∈ C ) = lim sup l →∞ a l ln P l ( C ) ≤ − inf C I ( x ) , (5.22)and they satisfy the large deviations lower bound if, for any open subset G of X ,lim inf l → + ∞ a l ln P l ( X l ∈ G ) = lim sup l →∞ a l ln P l ( G ) ≤ − inf G I ( x ) . (5.23)If both, upper and lower bound, are satisfied, one says that { X l } + ∞ l =1 or { P l } + ∞ l =1 satisfy a large deviationsprinciple . The principle is called weak if the upper bound in (5.22) holds only for compact sets C . This notioneasily extends to the situation where the distribution of X l is not normalized, but a sub-probability distributiononly. Observe also that one of the most important conclusions from a large deviations principle is Varadhan’sLemma, which says that, for any bounded and continuous function ϕ : X → R ,lim l → + ∞ a l ln Z exp ( a l ϕ ( X l )) d P = − inf x ∈X { I ( x ) − ϕ ( x ) } . For a comprehensive treatment of the theory of large deviations, see [19].
Acknowledgements
We are most grateful to Joel Lebowitz for interest to this project, for his help and inspiring discussions. Wealso wish to thank him for hospitality extended to us in the Rutgers University (NJ) that allowed to discussdifferent aspects of the paper. The present version of the manuscript was finalized during the J.-B.Bru visit ofthe Centre de Physique Th´eorique - Luminy UMR-6207. He is thankful to Pierre Duclos for invitation to CPT.
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