Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs
aa r X i v : . [ m a t h . P R ] F e b GIBBS MEASURES AS UNIQUE KMS EQUILIBRIUM STATES OFNONLINEAR HAMILTONIAN PDEs
ZIED AMMARI AND VEDRAN SOHINGER
Abstract.
The classical Kubo-Martin-Schwinger (KMS) condition is a fundamental property ofstatistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It wasintroduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin-Lanford-Ruelle (DLR) equation. In this article, we consider this concept in the framework ofnonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs mea-sures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavincalculus and Gross-Sobolev spaces. The main feature of our work is the applicability of our resultsto the general context of white noise, abstract Wiener spaces and Gaussian probability spaces,as well as to fundamental examples of PDEs like the nonlinear Schr¨odinger, Hartree, and wave(Klein-Gordon) equations.
Contents
1. Introduction 12. KMS states and their main properties 52.1. General framework 52.2. KMS equilibrium states 72.3. Stationary and equilibrium hierarchies 103. Gaussian measures and KMS states 143.1. Gaussian measures on countably Hilbert nuclear spaces 143.2. Wiener and Gaussian probability spaces 164. The Gibbs-KMS equivalence 184.1. Finite dimensional dynamical systems 184.2. Linear infinite dimensional dynamical systems 224.3. Nonlinear infinite dimensional dynamical systems 275. Nonlinear PDEs 315.1. Nonlinear Schr¨odinger equations 325.2. Nonlinear wave (Klein-Gordon) equations 395.3. The focusing NLS and the local KMS condition 41Appendix A. Malliavin calculus 43Appendix B. Proofs of auxiliary facts from Section 5 45References 491.
Introduction
The
Kubo-Martin-Schwinger (KMS) condition emerged in quantum statistical mechanics as acriterion characterizing the equilibrium states for infinite quantum systems [13, 34]. Ever since,
Date : February 23, 2021.2020
Mathematics Subject Classification.
Primary 35L05, 35Q55, 37D35; Secondary 60H07, 28C20.
Key words and phrases.
KMS states, Gaussian measures, Malliavin calculus, Nonlinear PDEs. the concept has been considered as a cornerstone in the study of quantum dynamical systemsand more generally in C ∗ and W ∗ -algebras topic, see e.g. [48]. In the seventies, G. Gallavotti andE. Verboven, suggested a classical analogue to the quantum KMS condition suitable for classicalmechanical systems and they analysed its relationship with the Dobrushin-Lanford-Ruelle (DLR)equation, see [28] and also [1]. The work of Gallavotti and Verboven generated interest in thestudy of KMS states for infinite classical systems (see for instance [22, 46] and the referencestherein). To the best of our knowledge, only few results are concerned with nonlinear PDEs,namely [7, 20, 45].On the other hand, Gibbs measures for nonlinear Hamiltonian systems have attracted a lotof interest in the PDE community since [9, 10, 11, 37, 54], following on the analysis of thisproblem in the constructive quantum field theory literature [31, 49]. Indeed, these measuresturn to be an effective tool in the study of almost sure existence of global solutions with roughinitial data since they provide conservation laws beyond the classical energy spaces, see e.g.[16, 41, 43, 52, 53] and the references therein. In this approach, the main ingredient is theinvariance of the measure, rather than its statistical properties. In principle any invariant measurewould produce conceptually similar results. It is therefore desirable to bridge the statistical andPDE points of view with the aim of obtaining a better understanding of stability and ergodictheory for PDE dynamical systems. Moreover, in light of recent progress made in quantumstatistical mechanics it is quite tempting to investigate thoroughly the structure of classical KMSstates for such dynamical systems. In particular, the KMS states are a convenient tool for thestudy of thermodynamic limits, multi-phase behavior and ergodic properties.The purpose of this article is to introduce the concept of
KMS equilibrium states for Hamil-tonian PDEs and to study their main properties and general structure. In this respect, one ofthe primary problems that we shall consider is the relationship between KMS states and Gibbsmeasures. To answer such a question, we consider an abstract framework for Hamiltonian PDEswithin which it is possible to rigorously define a Gibbs measure. First, we show that such aGibbs measure is a KMS equilibrium state (Theorem 4.11). Second, we show that, under addi-tional assumptions, the Gibbs measure is the unique KMS equilibrium state of the HamiltonianPDE (Theorem 4.14). The general framework we consider encloses several fundamental examplesthat include white noise, abstract Wiener spaces, and Gaussian probability spaces, see Section3. Our analysis applies to nonlinear PDEs like the nonlinear Schr¨odinger, Hartree, and wave(Klein-Gordon) equations as illustrated in Section 5.Let us formally explain our setup. A dynamical system is described by a vector field X : S → S ,defined as a mapping over a phase-space S , and a field equation,˙ u ( t ) = X ( u ( t )) , where u : R → S is a solution satisfying a prescribed initial condition u (0) = u ∈ S . Thereare two general approaches for the study of the dynamics. A deterministic point of view aimsto establish local or global well-posedness results in different functional spaces (i.e.: existence,uniqueness and stability in Hadamard’s sense). The main related questions in this approachconcern periodic and soliton solutions, blow-up solutions and scattering. A second probabilistic point of view aims to study the dynamics of ensembles of initial data rather that of a single pointin the phase-space. This leads to the consideration of the Liouville equation, ddt Z S F ( u ) µ t ( du ) = Z S h∇ F ( u ) , X ( u ) i µ t ( du ) , (1.1)where h· , ·i denotes a given Euclidean structure on the phase-space S , ∇ F is a gradient of thesmooth function F and t ∈ R µ t is a probability measure solution with a prescribed initial IBBS MEASURES AS KMS EQUILIBRIUM STATES 3 condition µ . The main questions within this approach are about existence, uniqueness, asymp-totic statistical stability of solutions and chaotic or ergodic behavior of the dynamical system.In this context, probability measures on the phase-space S are regarded as classical states of thedynamical system and the classical KMS condition is a widely accepted criterion that singles outthe equilibrium states among all possible stationary states of the Liouville equation (1.1). In fact,we will say that µ is a KMS state at inverse temperature β > Z S { F, G } ( u ) µ ( du ) = β Z S h∇ F ( u ) , X ( u ) i G ( u ) µ ( du ) , (1.2)where F, G : S → R are two smooth functions and {· , ·} denotes a Poisson structure over thephase-space S . By taking the function G ≡ • We prove that the nonlinear vector fields of these equations make sense as the Malliavinderivative of energy functionals in the Gross-Sobolev spaces, thus highlighting the factthat global stochastic analysis is a well fitted tool for the study of such deterministicdispersive PDEs. • We prove that Gibbs measures are stationary solutions of the Liouville equation whichindicates that the techniques of kinetic theory and gradient flow on probability measurespaces would be fruitful in this problem, see e.g. [2]. • We show that the Gibbs measures are KMS equilibrium states of the dynamical systemand therefore it is very tempting to study the system near equilibrium and to investigateits statistical stability and asymptotic properties.Beyond the formal program sketched above, there are more precise motivations for our presentwork. First, characterizing Gibbs measures through the KMS condition would provide an alter-native method for the derivation of Gibbs measures from many-body quantum field theories, see[6] and [38, 25, 26, 27, 24, 39, 50]. Second, the Gibbs measure as a KMS state and a stationarysolution of the Liouville equation should generally yield the existence of a global flow definedalmost surely on the phase-space S , see [3, 4, 5]. These questions will be addressed elsewhere andhere we focus on the more fundamental properties of the KMS condition.For an illustration of our main results, consider the NLS equation on the 2-dimensional torus T defined through its classical Hamiltonian, H ( u ) = 12 Z T |∇ u | + | u | dx + 12 m Z T : | u | m : dx , where : : denotes Wick ordering (see Section 5). Furthermore, we note that the nonlinear func-tional h I : H − s ( T ) → R defined over the negative Sobolev space with s > h I ( u ) = 12 m Z T : | u | m : dx ZIED AMMARI AND VEDRAN SOHINGER belongs to the Gross-Sobolev space D , ( µ β, ) where µ β, is a centered Gaussian measure withcovariance operator β − ( − ∆ + ) − (see Definition 4.10 below). Moreover, we prove that theGibbs measure µ β = z − β e − βh I µ β, , for z β = µ β, ( e − βh I ) a normalization constant, is the unique equilibrium KMS state of the NLSdynamical system. In particular, µ β is a stationary solution of the corresponding Liouville equa-tion (1.1) with the vector field X ( u ) = − i ( − ∆ u + u + ∇ h I ( u )) , where ∇ h I is the Malliavin derivative of the nonlinear functional h I (see Lemma 4.9 for theprecise definition). The above statements are obtained as a consequence of a more general result(Theorem 4.14). Indeed, consider a complex Hamiltonian system, h ( u ) = h ( u ) + h I ( u ) , such that h ( u ) = h u, Au i with A a positive self-adjoint operator admitting a compact resolventsuch that for some α ≥
1, Tr[ A − α ] < ∞ . Moreover, assume that the nonlinear functional h I ∈ D , ( µ β, ) and e − βh I ∈ L ( µ β, ), where µ β, is the centered Gaussian measure with covariance operator β − A − . So, we prove that if µ is aKMS state for this dynamical system which is absolutely continuous with respect to µ β, with adensity ̺ = dµdµ β, ∈ D , ( µ β, ), then µ is the Gibbs measure µ β , i.e.: µ = µ β ≡ k e − βh I k L ( µ β, ) e − βh I µ β, . The proof of the above result is based on the derivation of a differential equation on the density ̺ given by ∇ ̺ + β̺ ∇ h I = 0 . To solve such an equation, one uses the Malliavin calculus in order to prove that ce − βh I are theonly solutions of the above equation. Uniqueness is then obtained using the normalization of thedensity ̺ . The details of these arguments are given in Section 4. Overview of the article:
We define, in Section 2, the notion of KMS states and study their mainproperties in a general framework. In particular, we establish a relationship between Kirkwood-Salzburg type hierarchy equations and KMS states and prove stationarity, convexity and char-acteristic identities. In Section 3, fundamental examples of KMS states are given in terms ofGaussian measures over countably Hilbert nuclear spaces and canonical Gaussian measures onWiener spaces and Gaussian probability spaces. Finally, we prove the equivalence between KMSstates and Gibbs measures in Section 4 for: • Finite dimensional dynamical systems; • Linear complex Hamiltonian systems; • Nonlinear complex Hamiltonian systems.To emphasize our main results and show their wide applicability, we consider in Section 5 severalexamples of nonlinear PDEs such as the nonlinear Schr¨odinger (NLS), Hartree and wave equa-tions. When studying the NLS in 1 D , we also address the problem of the focusing nonlinearity.In this case, we prove that suitably localized invariant measures of Gibbs type satisfy a local KMScondition , see Section 5.3 for the precise definition. In Appendix A, we provide a short review ofMalliavin calculus. In Appendix B, we prove some auxiliary facts about Sobolev embedding anddiscrete convolutions that we use in Section 5. IBBS MEASURES AS KMS EQUILIBRIUM STATES 5
Acknowledgements:
The authors would like to thank Andrew Rout for helpful discussions. V.S.acknowledges support of the EPSRC New Investigator Award grant EP/T027975/1.2.
KMS states and their main properties
In this section, we define classical KMS states in a general abstract framework and study theirmain properties.2.1.
General framework.
There are several possible settings for the notion of KMS states. Wefirst give the definition in the most general setting. In the sequel, we adapt this to the probabilityand PDE context. So, we start with a rigged Hilbert space setting together with a compatiblesymplectic structure. The latter allows us to define a Poisson structure on an appropriate algebraof smooth cylindrical functions.
Rigged Hilbert space:
Consider a rigged Hilbert space Φ ⊆ H ⊆ Φ ′ where ( H, h· , ·i ) is a realseparable Hilbert space, Φ is a dense subset of H equipped with the structure of a topologicalvector space such that the natural embedding i : Φ → H is continuous and Φ ′ is the dual of Φwith respect to the inner product of H . Two standard examples are S ( R d ) ⊆ L ( R d ) ⊆ S ′ ( R d )and H s ( R d ) ⊆ L ( R d ) ⊆ H − s ( R d ) where S ( R d ) is the Schwartz space and H s ( R d ) is the Sobolevspace with a non-negative exponent s ≥ B (Φ ′ ) denotes the Borel σ -algebra over Φ ′ where the latter space is equippedwith the weak- ∗ topology. Moreover, P (Φ ′ ) denotes the set of all Borel probability measures onΦ ′ . Symplectic structure:
Assume further that the Hilbert space H is endowed with a non-degenerate continuous symplectic structure σ : H × H → R , i.e.: σ is a continuous bilinearform satisfying σ ( u, v ) = − σ ( v, u ) for all u, v ∈ H and if σ ( u, v ) = 0 for all v ∈ H then u = 0.Therefore, there exists a unique bounded linear operator J : H → H such that σ ( u, v ) = h u, J v i , ∀ u, v ∈ H .
In particular, the transpose operator of J is J T = − J . Suppose further that J maps continuouslyΦ to itself and consequently J extends uniquely and continuously to J : Φ ′ → Φ ′ . For now thereis no need to introduce a compatible complex structure. Such an assumption will be required inSection 4. Smooth cylindrical test functions:
Let { e n } n ∈ N be a countable linearly independent subset of Φsuch that span R { e j , j ∈ N } is dense in H . Then one defines the spaces of smooth cylindrical testfunctions denoted respectively by C ∞ c,cyl (Φ ′ ), S cyl (Φ ′ ) and C ∞ b,cyl (Φ ′ ), as the sets of all functions F : Φ ′ → R such that there exist n ∈ N and a function ϕ : R n → R satisfying for all u ∈ Φ ′ , F ( u ) = ϕ ( h u, e i , . . . , h u, e n i ) , (2.1)with ϕ ∈ C ∞ c ( R n ), ϕ ∈ S ( R n ) or ϕ ∈ C ∞ b ( R n ) respectively. Here, we recall that the latterspace consists of smooth functions all of whose derivatives are bounded. Obviously, one has thefollowing inclusions C ∞ c,cyl (Φ ′ ) ⊂ S cyl (Φ ′ ) ⊂ C ∞ b,cyl (Φ ′ ) . Remark that C ∞ c,cyl (Φ ′ ) is stable under multiplication but not stable under addition of its elements,while C ∞ b,cyl (Φ ′ ) is a unital algebra over the field R . Although the representation formula (2.1) maynot be unique, these classes of smooth functions are quite convenient for the analysis. Indeed,they are endowed with a nice differential calculus. In fact, all the functions in C ∞ b,cyl (Φ ′ ) are ZIED AMMARI AND VEDRAN SOHINGER differentiable over Φ ′ in the direction of H . More precisely, taking F : Φ ′ → R as in (2.1) thenfor all v ∈ H , DF ( u )[ v ] = lim v → ,v ∈ H F ( u + v ) − F ( u ) || v || = n X j =1 ∂ j ϕ ( h u, e i , . . . , h u, e n i ) h e j , v i . (2.2)Furthermore, the differential DF ( u ) is regarded as a continuous R -linear form in L ( H, R ) ≃ H .In particular, the gradient of F is defined as ∇ F ( u ) := n X j =1 ∂ j ϕ ( h u, e i , . . . , h u, e n i ) e j ∈ Φ , (2.3)and the following product rule is true for all F, G ∈ C ∞ b,cyl (Φ ′ ) and u ∈ Φ ′ , ∇ ( F G )( u ) = ∇ F ( u ) G ( u ) + F ( u ) ∇ G ( u ) . (2.4)It is useful to write the gradient of a smooth cylindrical function using the Fourier transform. Lemma 2.1.
For any F ∈ S cyl (Φ ′ ) satisfying (2.1) , for some n ∈ N and ϕ ∈ S ( R n ) , ∇ F ( u ) = − (2 π ) − n n X j =1 Z R n t j e j (cid:18) ℜ e( ˆ ϕ )( t , . . . , t n ) sin h n X j =1 t j e j , u i + ℑ m( ˆ ϕ )( t , . . . , t n ) cos h n X j =1 t j e j , u i (cid:19) dt , where ˆ ϕ is the Fourier transform of ϕ .Proof. Using (2.3), one writes h∇ F ( u ) , v i = n X j =1 ∂ j ϕ ( h u, e i , . . . , h u, e n i ) h e j , v i = (2 π ) − n n X j =1 Z R n d ∂ j ϕ ( t , . . . , t n ) e i P nj =1 t j h u,e j i h e j , v i dt . Since the left hand side is real, then one shows h∇ F ( u ) , v i = i (2 π ) − n Z R n b ϕ ( t , . . . , t n ) e i P nj =1 t j h u,e j i h n X j =1 t j e j , v i dt = − (2 π ) − n Z R n (cid:18) ℜ e( b ϕ )( t , . . . , t n ) sin h n X j =1 t j e j , u i + ℑ m( b ϕ )( t , . . . , t n ) cos h n X j =1 t j e j , u i (cid:19) h n X j =1 t j e j , v i dt . The last equality proves the claimed identity. (cid:3)
Poisson structure:
The above differential calculus enables us to define a Poisson structureover the algebra C ∞ b,cyl (Φ ′ ). In fact, in this framework we define the Poisson bracket for all F, G ∈ C ∞ b,cyl (Φ ′ ) and all u ∈ Φ ′ as { F, G } ( u ) = σ ( ∇ F ( u ) , ∇ G ( u )) = h∇ F ( u ) , J ∇ G ( u ) i . (2.5) IBBS MEASURES AS KMS EQUILIBRIUM STATES 7
Using (2.3) one shows that { F, G } belongs to the algebra C ∞ b,cyl (Φ ′ ) and obviously the bracket isbilinear and skew symmetric. Moreover, one checks that the Leibniz rule and the Jacobi identityare satisfied for all F, G, R ∈ C ∞ b,cyl (Φ ′ ), • { R, F G } = { R, F } G + { R, G } F ; • { F, { G, R }} + { G, { R, F }} + { R, { F, G }} = 0 . The Poisson structure is one of the main ingredients that enters in the definition of KMS states.Similarly as in Lemma 2.1, one can express the Poisson bracket using the Fourier transform.
Lemma 2.2.
For all
F, G ∈ S cyl (Φ ′ ) such that (2.1) is satisfied for both F and G with n, m ∈ N and ϕ ∈ S ( R n ) , ψ ∈ S ( R m ) respectively. Then { F, G } ( u ) = − (2 π ) − ( n + m )2 Z R n Z R m ˆ ϕ ( t , . . . , t n ) ˆ ψ ( s , . . . , s m ) (cid:10) n X j =1 t j e j , J m X j =1 s j e j (cid:11) e i P nj =1 t j h u,e j i + i P mj =1 s j h u,e j i dtds . Proof.
This follows from Lemma 2.1 and (2.5). Since ϕ and ψ are real-valued functions, one has i Z R n t j b ϕ ( t , . . . , t n ) e i P nj =1 t j h u,e j i dt = Z R n ℜ e( ˆ ϕ )( t , . . . , t n ) sin h n X j =1 t j e j , u i + ℑ m( ˆ ϕ )( t , . . . , t n ) cos h n X j =1 t j e j , u i dt , and similarly i Z R m s j b ψ ( s , . . . , s n ) e i P mj =1 s j h u,e j i ds = Z R m ℜ e( ˆ ψ )( s , . . . , s m ) sin h m X j =1 s j e j , u i + ℑ m( ˆ ψ )( s , . . . , s m ) cos h m X j =1 s j e j , u i ds. Hence, using Lemma 2.1 and (2.5), one checks that the claimed identity holds true. (cid:3)
KMS equilibrium states.
Consider a Borel vector field X : Φ ′ → Φ ′ defining a (formal)dynamical system given by the differential equation, ∂ t u ( t ) = X ( u ( t )) , (2.6)where t ∈ R u ( t ) ∈ Φ ′ is a curve with prescribed initial condition u (0) = u ∈ Φ ′ . Theabove equation may not make sense and usually additional assumptions on the solutions u ( · )or the vector field X are required. One can look at the field equation (2.6) from a statisticalpoint of view. So, instead of studying the initial value problem (2.6) for each fixed data u ,one can consider the dynamical evolution of an ensemble of initial datum given by a probabilitydistribution. It turns that the statistical dynamics related to the vector field equation (2.6) aredescribed by the Liouville (transport) equation, ddt Z Φ ′ F ( u ) dµ t ( u ) = Z Φ ′ h∇ F ( u ) , X ( u ) i dµ t ( u ) , (2.7)for all F ∈ C ∞ c,cyl (Φ ′ ) and where t ∈ R µ t ∈ P (Φ ′ ) is a curve of statistical solutions with aprescribed initial condition µ ∈ P (Φ ′ ). Note that the definition of the Liouville equation requiresneither a symplectic nor a Poisson structure. We remark also that the Liouville equation (2.7)may not make sense without further requirements on the vector field X or on the solutions µ t .Existence of solutions for the above Liouville equation (2.7) and its relationship with the original ZIED AMMARI AND VEDRAN SOHINGER field equation (2.6) is studied in [4]. In this article, we focus only on stationary solutions of theLiouville equation that represent dynamical equilibrium. A Borel probability measure µ on Φ ′ is a stationary solution of the Liouville equation (2.7) if and only if for all F ∈ C ∞ c,cyl (Φ ′ ) thefunction h∇ F ( · ) , X ( · ) i is µ -integrable and Z Φ ′ h∇ F ( u ) , X ( u ) i dµ = 0 . (2.8)Not all stationary solutions correspond to a statistical equilibrium of the dynamical system. TheKMS condition that we shall define below is a widely accepted criterion that defines the notionof statistical equilibrium and stability. Kubo-Martin-Schwinger condition:
The Kubo-Martin-Schwinger (KMS) condition, given belowin (2.9), is a dynamical characterization of equilibrium measures of the Liouville equation (2.7)at an inverse positive temperature β >
0. These equilibrium measures will be called KMS statesand their definition is rigourously given below.
Definition 2.3 (KMS states) . Let X : Φ ′ → Φ ′ be a Borel vector field and β >
0. We say that µ ∈ P (Φ ′ ) is a ( β, X )-KMS state if and only if the function h f, X ( · ) i is µ -integrable for all f ∈ Φand for all
F, G ∈ C ∞ c,cyl (Φ ′ ), we have Z Φ ′ { F, G } ( u ) dµ = β Z Φ ′ h∇ F ( u ) , X ( u ) i G ( u ) dµ , (2.9)with the Poisson bracket {· , ·} is defined in (2.5).The existence and uniqueness of such KMS equilibrium states is in general a non trivial questionas one can see for instance in [28]. Despite this fact, it is useful to underline the general propertiesof these measures. Lemma 2.4. If µ is a ( β, X ) -KMS state then the identity (2.9) is true for all F, G ∈ C ∞ b,cyl (Φ ′ ) .Proof. Using a standard pointwise approximation argument of functions ϕ ∈ C ∞ b ( R n ) by se-quences of functions in C ∞ c ( R n ), the equality (2.9) extends by dominated convergence to all F, G ∈ C ∞ b,cyl (Φ ′ ). (cid:3) Not all stationary solutions of the Liouville equation (2.7) are KMS equilibrium states, but theconverse is true. Remark that here the time invariance is formulated without appealing to a flowfor the field equation (2.6).
Proposition 2.5.
Any ( β, X ) -KMS state is a stationary solution of the Liouville equation (2.7) .Proof. Thanks to Lemma 2.4, it is enough to take in (2.9) the function G ( · ) = 1 which belongsto C ∞ b,cyl (Φ ′ ) and hence obtaining (2.8). (cid:3) One important geometric feature of the set of KMS states is convexity.
Proposition 2.6.
The set of ( β, X ) -KMS states is a convex subset of P (Φ ′ ) .Proof. Let α ∈ [0 ,
1] and µ, ν two ( β, X )-KMS states. Then one easily checks that the integrabilityof the functions h f, X ( · ) i and the identity (2.9) are satisfied with respect to the probabilitymeasure αµ + (1 − α ) ν . (cid:3) A simple identification of the KMS states in terms of their characteristic functions is providedbelow. We note that, in [20], the identity (2.10) is regarded as the definition of KMS states.
Theorem 2.7.
Let µ ∈ P (Φ ′ ) , X a Borel vector field on Φ ′ and β > be given. Then the twofollowing assertions are equivalent.(i) µ is a ( β, X ) -KMS state. IBBS MEASURES AS KMS EQUILIBRIUM STATES 9 (ii) For all ϕ , ϕ ∈ Φ , the function h ϕ , X ( · ) i is µ -integrable and h ϕ , J ϕ i Z Φ ′ e i h u,ϕ i dµ + iβ Z Φ ′ h ϕ , X ( u ) i e i h u,ϕ i dµ = 0 . (2.10) Proof.
Assume (i) true and take f, g ∈ Φ such that f = ϕ and f + g = ϕ . Then h J ϕ , ϕ i e i h u,ϕ i = −h f, J g i e i h u,f i e i h u,g i = −{ sin( h f, u i ) , sin( h g, u i ) } + { cos( h f, u i ) , cos( h g, u i ) } + i { sin( h f, u i ) , cos( h g, u i ) } + i { cos( h f, u i ) , sin( h g, u i ) } . Hence, integrating the above equality with respect to µ and using for each bracket the KMScondition (2.9) and Lemma 2.4, one shows h J ϕ , ϕ i Z Φ ′ e i h u,ϕ i dµ = β Z Φ ′ h f, X ( u ) i (cid:18) − cos h f, u i sin h g, u i − sin h f, u i cos h g, u i + i cos h f, u i cos h g, u i − i sin h f, u i sin h g, u i (cid:19) dµ = β Z Φ ′ h ϕ , X ( u ) i (cid:0) − sin h f + g, u i + i cos h f + g, u i (cid:1) dµ = iβ Z Φ ′ h ϕ , X ( u ) i e i h u,ϕ i dµ. Thus, (ii) is proved. Conversely, suppose that (ii) holds true, then as before the equation (2.10)gives for all f, g ∈ Φ, h f, J g i Z Φ ′ e i h u,f + g i dµ = − iβ Z Φ ′ h f, X ( u ) i e i h u,f + g i dµ . (2.11)For F, G ∈ C ∞ c,cyl (Φ ′ ) there exists n, m ∈ N and ϕ ∈ C ∞ c ( R n ), ψ ∈ C ∞ c ( R m ) satisfying (2.1)respectively. The inverse Fourier transform gives F ( u ) = (2 π ) − n/ Z R n ˆ ϕ ( t , . . . , t n ) e i P nj =1 t j h u,e j i dt ,G ( u ) = (2 π ) − m/ Z R m ˆ ψ ( s , . . . , s m ) e i P mj =1 s j h u,e j i ds , where ˆ ϕ and ˆ ψ are respectively the Fourier transform of ϕ and ψ . By Lemma 2.2, one has { F, G } ( u ) = − (2 π ) − ( n + m )2 Z R n × R m ˆ ϕ ( t , . . . , t n ) ˆ ψ ( s , . . . , s m ) (cid:10) n X j =1 t j e j , J m X j =1 s j e j (cid:11) e i P nj =1 t j h u,e j i + i P mj =1 s j h u,e j i dtds . Hence, writing the identity (2.11) with f = P nj =1 t j e j and g = P mj =1 s j e j and multiplying it byˆ ϕ ( t , . . . , t n ) × ˆ ψ ( s , . . . , s m ) and then integrating with respect to t j and s j , one obtains Z Φ ′ { F, G } ( u ) dµ = β Z Φ ′ R ( u ) G ( u ) dµ , (2.12) where R ( · ) is a real-valued function given by R ( u ) = i (2 π ) − n Z R n (cid:10) n X j =1 t j e j , X ( u ) (cid:11) ˆ ϕ ( t , . . . , t n ) e i P nj =1 t j h u,e j i dt = − (2 π ) − n Z R n (cid:10) n X j =1 t j e j , X ( u ) (cid:11) (cid:18) ℜ e( ˆ ϕ )( t , . . . , t n ) sin h n X j =1 t j e j , u i + ℑ m( ˆ ϕ )( t , . . . , t n ) cos h n X j =1 t j e j , u i (cid:19) dt = h∇ F ( u ) , X ( u ) i . The last equality follows by Lemma 2.1. Thus, the identity (2.12) yields the KMS condition(2.9). (cid:3)
The following statement may be interpreted as the passivity of the dynamical system at equi-librium which means that the system is unable to perform mechanical work in a cyclic process(see e.g. [13, Section 5.4.4] for an analogy with quantum KMS states).
Corollary 2.8. If µ is a ( β, X ) -KMS state then for all ϕ ∈ Φ and all F ∈ C ∞ b,cyl (Φ ′ ) , Z Φ ′ h ϕ, X ( u ) i dµ = 0 , and Z Φ ′ h∇ F ( u ) , X ( u ) i F ( u ) dµ = 0 . Proof.
In order to obtain the first identity, we take ϕ = 0 in (2.10). In order to obtain the secondidentity, we take G = F in the KMS condition (2.9) and apply Lemma 2.4. (cid:3) Stationary and equilibrium hierarchies.
In the recent article [5], a duality is establishedbetween the Liouville equation (2.7) and a Bose-Einstein hierarchy equation generalizing theGross-Pitaevskii and Hartree hierarchies studied for instance in [18, 19, 36, 51] and the referencestherein. See also [30, 33] for similarity with the BBGKY hierarchy of classical mechanics. In thisparagraph, we extend the above duality to stationary and equilibrium solutions. Throughout, weassume that X is a Borel vector field on Φ ′ . Lemma 2.9.
Let µ ∈ P (Φ ′ ) be such that h ϕ, X ( · ) i is µ -integrable for any ϕ ∈ Φ . Then µ is astationary solution of the Liouville equation, i.e. it solves (2.8) , if and only if for all ϕ ∈ Φ , Z Φ ′ h ϕ, X ( u ) i e i h u,ϕ i dµ = 0 . (2.13) Proof.
Suppose that µ is a stationary solution, then the identity (2.8) extends to all F ∈ C ∞ b,cyl (Φ ′ ).In particular, taking F ( · ) = cos h· , ϕ i and F ( · ) = sin h· , ϕ i in C ∞ b,cyl (Φ ′ ) one obtains for any ϕ ∈ span { e j , j ∈ N } , Z Φ ′ cos h u, ϕ i h ϕ, X ( u ) i dµ = Z Φ ′ sin h u, ϕ i h ϕ, X ( u ) i dµ = 0 . (2.14)Hence, the equality (2.13) is proved for all ϕ ∈ Φ by a density argument. Conversely, if theidentity (2.13) holds true then taking real and imaginary parts one gets (2.14). Hence, usingLemma 2.1 one recovers (2.8). (cid:3)
Lemma 2.10.
Let µ ∈ P (Φ ′ ) and assume that for any ϕ ∈ Φ there exists < C < such thatfor all k ∈ N , Z Φ ′ (cid:12)(cid:12) h u, ϕ i k h ϕ, X ( u ) i (cid:12)(cid:12) dµ ≤ C k +1 k ! . (2.15) IBBS MEASURES AS KMS EQUILIBRIUM STATES 11
Then µ is a stationary solution of the Liouville equation (2.8) if and only if for all k ∈ N and ϕ ∈ Φ , Z Φ ′ h u, ϕ i k h ϕ, X ( u ) i dµ = 0 . Proof.
Thanks to the hypothesis (2.15), the function m : λ Z Φ ′ e iλ h u,ϕ i h ϕ, X ( u ) i dµ , is analytic on the disc D = { λ ∈ C : | λ | < C − } . Hence, using Lemma 2.9 the measure µ is astationary solution of the Liouville equation if and only if d k mdλ k (0) = i k Z Φ ′ h u, ϕ i k h ϕ, X ( u ) i dµ = 0 . (cid:3) Similarly, one proves the following result concerning the equilibrium KMS solutions of theLiouville equation.
Lemma 2.11.
Let µ ∈ P (Φ ′ ) and assume that for any ϕ , ϕ ∈ Φ there exists < C < suchthat for all k ∈ N , Z Φ ′ (cid:12)(cid:12) h u, ϕ i (cid:12)(cid:12) k dµ ≤ C k +1 k ! , Z Φ ′ (cid:12)(cid:12) h u, ϕ i k h ϕ , X ( u ) i (cid:12)(cid:12) dµ ≤ C k +1 k ! . (2.16) Then µ is a ( β, X ) -KMS state if and only if for all k ∈ N and ϕ , ϕ ∈ Φ , β Z Φ ′ h u, ϕ i k h ϕ , X ( u ) i dµ = k h ϕ , J ϕ i Z Φ ′ h u, ϕ i k − dµ ; (2.17) and Z Φ ′ h ϕ , X ( u ) i dµ = 0 . (2.18) Proof.
Since the functions m : λ Z Φ ′ e iλ h u,ϕ i h ϕ , X ( u ) i dµ , m : λ Z Φ ′ e iλ h u,ϕ i dµ , are analytic on the disc D = { λ ∈ C : | λ | < C − } , one can replace ϕ j by λϕ j for j = 1 , λ -power series on D to deduce a relation on the coefficients.Indeed, one has ∞ X k =0 i k k ! λ k +1 (cid:18) h ϕ , J ϕ i Z Φ ′ h u, ϕ i k dµ (cid:19) + iβ ∞ X k =0 i k k ! λ k (cid:18) Z Φ ′ h u, ϕ i k h ϕ , X ( u ) i dµ (cid:19) = 0 . Such an equation yields the claimed equilibrium moment relations. (cid:3)
Assume further that the operator J defines a compatible complex structure over the Hilbertspace H , i.e.: For all u, v ∈ H • J = − ; • σ ( J u, u ) ≥ • σ ( J u, J v ) = σ ( u, v ). In particular, H can be considered as a complex Hilbert space,( η + iζ ) u := ηu − ζJ u , (2.19)endowed with the inner product h u, v i H C = h u, v i + iσ ( u, v ) . (2.20)We now state an equivalence between stationary solutions of the Liouville equation and sta-tionary hierarchy equations, given by (2.21) below. Proposition 2.12.
Assume that the Hilbert space H is endowed with a complex structure asabove and suppose that X ( e iθ u ) = e iθ X ( u ) for all θ ∈ R and u ∈ Φ ′ . Consider µ ∈ P (Φ ′ ) which is U (1) -invariant and satisfies the estimates (2.15) . Then µ solves (2.8) if and only if thesymmetric hierarchy equation k X j =1 Z Φ ′ (cid:18)(cid:12)(cid:12) u ( j − ⊗ X ( u ) ⊗ u ( k − j ) (cid:11)(cid:10) u ⊗ k (cid:12)(cid:12) + (cid:12)(cid:12) u ⊗ k (cid:11)(cid:10) u ( j − ⊗ X ( u ) ⊗ u ( k − j ) (cid:12)(cid:12)(cid:19) dµ = 0 , (2.21) holds for all k ∈ N .Remark . We recall that µ ∈ P (Φ ′ ) is said to be U (1)-invariant if for any B ∈ B (Φ ′ ) andany θ ∈ R we have, µ ( { e iθ u, u ∈ B } ) = µ ( B ) . The identity (2.21) shall be understood in a weak sense, i.e.: For all ψ , ψ ∈ Sym k (Φ), theintegrals Z Φ ′ (cid:10) ψ , u ⊗ · · · ⊗ X ( u ) ⊗ · · · ⊗ u (cid:11)(cid:10) u ⊗ k , ψ (cid:11) dµ , and Z Φ ′ (cid:10) ψ , u ⊗ k (cid:11)(cid:10) u ⊗ · · · ⊗ X ( u ) ⊗ · · · ⊗ u, ψ (cid:11) dµ , are well-defined where Sym k (Φ) is the k -fold algebraic symmetric tensor product of Φ. Moreover,the hierarchy equation (2.21) can be interpreted as a system of infinite coupled equations for the k -densities { γ ( k ) } k ∈ N defined in the weak sense by γ ( k ) = Z Φ ′ | u ⊗ k ih u ⊗ k | dµ , as sesquilinear maps on Sym k (Φ) × Sym k (Φ). For more details on this formulation and relationshipwith Gross-Pitaevskii hierarchies, we refer the reader to [5, Section 3]. Proof.
Suppose that µ is a stationary solution of the Liouville equation (2.8). According toLemma 2.10, one has Z Φ ′ ( ℜ e h u, ϕ i H C ) k ℜ e h ϕ, X ( u ) i H C dµ = 0 . Hence, the U (1)-invariance of µ and a polynomial expansion yield,0 = k X j =0 Z Φ ′ (cid:18) kj (cid:19) h ϕ, e iθ u i jH C h e iθ u, ϕ i k − jH C (cid:18) h ϕ, e iθ X ( u ) i H C + h e iθ X ( u ) , ϕ i H C (cid:19) dµ , All the terms in the above sum are zero because of the U (1)-invariance, except the ones obtainedby taking 2 j + 1 = k and 2 j − k ( k should be odd), which can be seen by taking averages in θ ∈ [0 , π ]. Therefore, one obtains that for all p ∈ N , Z Φ ′ h ϕ, u i p − H C h u, ϕ i pH C h ϕ, X ( u ) i H C dµ + Z Φ ′ h ϕ, u i pH C h u, ϕ i p − H C h X ( u ) , ϕ i H C dµ = 0 , IBBS MEASURES AS KMS EQUILIBRIUM STATES 13 so 0 = p X j =1 Z Φ ′ h ϕ ⊗ p , u ⊗ ( j − ⊗ X ( u ) ⊗ u ⊗ ( p − j ) i H C h u ⊗ p , ϕ ⊗ p i H C + h ϕ ⊗ p , u ⊗ p i H C h u ⊗ ( j − ⊗ X ( u ) ⊗ u ⊗ ( p − j ) , ϕ ⊗ p i H C dµ = (cid:10) ϕ ⊗ p , p X j =1 (cid:18) Z Φ ′ (cid:12)(cid:12) u ⊗ ( j − ⊗ X ( u ) ⊗ u ⊗ ( p − j ) i H C h u ⊗ p (cid:12)(cid:12) + (cid:12)(cid:12) u ⊗ p i H C h u ⊗ ( j − ⊗ X ( u ) ⊗ u ⊗ ( p − j ) (cid:12)(cid:12) dµ (cid:19) ϕ ⊗ p (cid:11) H C = (cid:10) ϕ ⊗ p , Q p ϕ ⊗ p (cid:11) H C , where Q p denotes the sum in the previous line, interpreted as a quadratic form on the p -foldalgebraic symmetric tensor product space Sym p (Φ). Thanks to the polarization formula, h η ⊗ p , Q p ξ ⊗ p i H C = Z Z h ( e iπθ η + e iπϕ ξ ) ⊗ p , Q p ( e iπθ η + e iπϕ ξ ) ⊗ p i H C e iπ ( pθ − pϕ ) dθdϕ , and the fact that any element in Sym p (Φ) can be written as combination of { η ⊗ p , η ∈ Φ } , oneobtains the claimed hierarchy equation (2.21). The converse statement follows by reversing theabove arguments. (cid:3) We end up this section with an equivalence result between KMS equilibrium states and equi-librium hierarchies.
Theorem 2.14.
Assume that the Hilbert space H is endowed with a complex structure as aboveand suppose that X ( e iθ u ) = e iθ X ( u ) for all θ ∈ R and u ∈ Φ ′ . Consider a U (1) -invariant µ ∈ P (Φ ′ ) satisfying the estimates (2.16) . Then µ is a ( β, X ) -KMS state if and only if for all ϕ , ϕ ∈ Φ and p ∈ N we have that βp + 1 Z Φ ′ (cid:10) ϕ , u (cid:11) p +1 H C (cid:10) u, ϕ (cid:11) pH C (cid:10) X ( u ) , ϕ (cid:11) H C dµ = 2 i (cid:10) ϕ , ϕ (cid:11) H C Z Φ ′ (cid:10) ϕ , u (cid:11) pH C (cid:10) u, ϕ (cid:11) pH C dµ . (2.22) Proof.
Suppose that µ is a ( β, X )-KMS state. By Lemma 2.11, we have that (2.17) holds. Byusing the U (1) invariance of X and µ , we can rewrite this identity as β k +1 Z Φ ′ (cid:16) e − iθ (cid:10) u, ϕ (cid:11) H C + e iθ (cid:10) ϕ , u (cid:11) H C (cid:17) k (cid:16) e iθ (cid:10) ϕ , X ( u ) (cid:11) H C + e − iθ (cid:10) X ( u ) , ϕ (cid:11) H C (cid:17) dµ = k k − ℜ e (cid:10) ϕ , − iϕ (cid:11) H C Z Φ ′ (cid:16) e − iθ (cid:10) u, ϕ (cid:11) H C + e iθ (cid:10) ϕ , u (cid:11) H C (cid:17) k − dµ, (2.23)for θ ∈ [0 , π ]. We then take the average over θ ∈ [0 , π ] in (2.23) to deduce that both sidesvanish if k is even and for k = 2 p + 1 odd, by using the Newton binomial formula, the aboveidentity is equivalent to βp + 1 (cid:20) Z Φ ′ (cid:10) ϕ , u (cid:11) p +1 H C (cid:10) u, ϕ (cid:11) pH C (cid:10) X ( u ) , ϕ (cid:11) H C dµ + Z Φ ′ (cid:10) ϕ , u (cid:11) pH C (cid:10) u, ϕ (cid:11) p +1 H C (cid:10) ϕ , X ( u ) (cid:11) H C dµ (cid:21) = 2 (cid:16) i h ϕ , ϕ i H C − i (cid:10) ϕ , ϕ (cid:11) H C (cid:17) Z Φ ′ (cid:12)(cid:12) h ϕ , u (cid:11) H C | p dµ . (2.24)The identity (2.24) holds for all ϕ , ϕ ∈ Φ. In particular, it holds if we replace ϕ e iθ ϕ forany θ ∈ [0 , π ]. We hence deduce (2.22). The converse follows by analogous arguments. Notethat the identity (2.18) is true thanks to the U (1) invariance of the measure µ and the vectorfield X . (cid:3) Gaussian measures and KMS states
We show in this section that Gaussian measures in infinite dimensional spaces are fundamentalexamples of KMS equilibrium states. It is possible to study Gaussian measures from differentpoints of view. Here, we consider Gaussian measures on dual nuclear spaces, abstract Wienerspaces and Gaussian probability spaces. Our aim is to outline the fundamental aspects of KMSstates and the emphasize their applicability in various contexts.3.1.
Gaussian measures on countably Hilbert nuclear spaces.
The general setting givenin Subsection 2.1 will be restricted here since we are going to consider Gaussian measures onthe dual space Φ ′ . Therefore it is useful to require that Φ is a suitable nuclear space. Beforeproceeding further, we give the precise assumptions on the spaces. We recall that H is alwaysassumed to be a separable real Hilbert space endowed with a symplectic structure σ induced bythe operator J (see Subsection 2.1) satisfying J Φ ⊆ Φ and that H is endowed with a Hilbertrigging Φ ⊆ H ⊆ Φ ′ , such that Φ is dense in H . Assume furthermore that Φ is a countably Hilbert nuclear space . Thismeans that Φ is a Fr´echet space whose topology is given by an increasing sequence of compatibleHilbertian norms {k · k n , n ∈ N } and such that taking H n to be the completion of Φ with respectto the norm k · k n , one has the chain of embeddings,Φ ⊆ · · · ⊆ H n ⊆ H n − · · · ⊆ H , satisfying for all n ∈ N the existence of m ∈ N , m ≥ n , such that the embedding i m,n : ( H m , k · || m ) → ( H n , || · || n ) , defines a trace-class operator. Recall that the norms are said to be compatible if for any sequence( x k ) k in Φ that is Cauchy for both || · || n and || · || m , one has(lim k x k = 0 in H n ) ⇔ (lim k x k = 0 in H m ) . This shows in particular that i m,n is a well defined embedding and hence H m can be identifiedwith a subset of H n whenever n ≤ m . Moreover, the space Φ is identified with the topologicalprojective limit associated to the projective system ( H n , i n,m ) such thatΦ = \ n ∈ N H n = lim ←− H n . For more details on nuclear spaces see e.g. [29]. The main example for such a setting is givenby the rigging S ( R d ) ⊆ L ( R d ) ⊆ S ′ ( R d ) where S ( R d ) is a nuclear space endowed for instancewith the sequence of norms: k ϕ k n = (cid:18) X | α |≤ n (cid:13)(cid:13) (1 + | x | ) n/ D α ϕ ( x ) (cid:13)(cid:13) L ( R d ) (cid:19) / . In this framework it is known that the Minlos theorem provides an elegant generalization of theBochner theorem. The point is that the (canonical) Gaussian measures on infinite dimensionalHilbert spaces are not σ -additive measures on H but only additive cylindrical set measures.However, such cylindrical set measures extend to probability measures by means of a radonifyingembedding on a larger space. A convenient statement of the Minlos theorem is given below.Recall that a normalized positive-definite functional G : Φ → C is a map satisfying:(i) G (0) = 1; IBBS MEASURES AS KMS EQUILIBRIUM STATES 15 (ii) For all n ∈ N , λ j ∈ C , u j ∈ Φ for j = 1 , . . . , n , n X j,k =1 ¯ λ j λ k G ( u j − u k ) ≥ . Theorem 3.1 (Minlos’ theorem) . Assume that Φ is a countably Hilbert nuclear space. Thenany continuous normalized positive definite functional G on Φ is the characteristic function of aunique µ ∈ P (Φ ′ ) such that for all w ∈ Φ , G ( w ) = Z Φ ′ e i h u,w i dµ . For more details on the above theorem, we refer to [35, Thm. 4.7] and [29, Chapter IV].Consider a positive symmetric (bounded or unbounded) operator A : D ( A ) ⊆ H → H suchthat A ≥ c for some constant c > D ( A ) ⊃ Φ. In particular, A is invertible with A − being a bounded operator on H . Corollary 3.2.
Let β > be given. There exists a unique µ β, ∈ P (Φ ′ ) such that its characteristicfunction is given for all v ∈ H by ˆ µ β, ( v ) = Z Φ ′ e i h v,u i dµ β, = e − β h v,A − v i . Recall the spaces of cylindrical smooth functions (2.1), the gradient (2.3) and Poisson structureon C b,cyl (Φ ′ ) (2.5), as well as the definition of KMS states in Definition 2.3 from Subsection 2.1. Theorem 3.3.
The Gaussian measure µ β, provided by Corollary 3.2 is a ( β, X ) -KMS state forthe linear dynamical system given by the vector field X = J A .Proof.
In order to prove that µ β, is a KMS state, we will use Theorem 2.7. Let ϕ , ϕ ∈ Φ begiven. Then, using the Cauchy-Schwarz inequality one easily checks that the function h ϕ , X ( · ) i is µ β, -integrable, Z Φ ′ (cid:12)(cid:12) h AJ ϕ , u i (cid:12)(cid:12) dµ β, ≤ (cid:18) Z Φ ′ h AJ ϕ , u i dµ β, (cid:19) / < ∞ , since AJ ϕ ∈ H and all the second moments of the Gaussian measure µ β, are finite (i.e.: h f, ·i ∈ L ( µ β, ) for all f ∈ H , see Theorem 4.4 and Remark 4.5). Using Corollary 3.2, observethat i Z Φ ′ h ϕ , X ( u ) i e i h u,ϕ i dµ β, = dds (cid:18) Z Φ ′ e i h u, − sAJϕ + ϕ i dµ β, (cid:19) (cid:12)(cid:12) s =0 = dds (cid:18) e − β h− sAJϕ + ϕ ,A − ( − sAJϕ + ϕ ) i (cid:19) (cid:12)(cid:12) s =0 = 1 β h J ϕ , ϕ i e − β h ϕ ,A − ϕ i = 1 β h J ϕ , ϕ i Z Φ ′ e i h u,ϕ i dµ β, . This proves the identity (2.10) and hence µ β, is a ( β, X )-KMS state. (cid:3) Remark . An interesting example for the above Theorem 3.3 is the so-called white noise measure . According to Minlos’ Theorem 3.1, there exists a unique probability measure µ wn on S ′ ( R ) having the characteristic functionalˆ µ wn ( u ) = Z S ′ ( R ) e i h u,w i dµ wn ( w ) = e − k u k L R ) , named the canonical Gaussian measure or white noise measure on S ′ ( R ) corresponding to thechoice β = 1, Φ = S ( R ), A = and J is any operator inducing a non-degenerate symplecticstructure on L ( R ) such that J S ( R ) ⊆ S ( R ).3.2. Wiener and Gaussian probability spaces.
The result in Theorem 3.3 extends to ab-stract Wiener spaces and Gaussian probability spaces. Indeed, one can prove that the canonicalGaussian measure in both cases is a ( β, X )-KMS state for β = 1 and for a given linear dynamicalsystem. Abstract Wiener space:
Let B be a separable Banach space such that the Hilbert space H isembedded into B through an injective continuous linear map i : H → B . Assume that the map i radonifies the canonical Gaussian cylinder set measure on H . Then ( i, H, B ) is called an abstractWiener space (see e.g. [32, 35]). This means that there exists a Borel probability measure µ ws on B such that its characteristic function is given for all u ∈ H byˆ µ ws ( u ) = Z B e i h u,w i dµ ws ( w ) = e − k u k . As in Theorem 3.3, one shows that the canonical Gaussian measure µ ws on B is a ( β, X )-KMSstate for the dynamical system induced by the vector field X = J with β = 1 and J is anyoperator implementing a symplectic structure on H . Gaussian probability space: is a complete probability space (Ω , Σ , P ) with a family of centeredGaussian random variables W ( f ) : Ω → R indexed by a separable Hilbert space H such that forall f, g ∈ H , E ( W ( f ) W ( g )) = h f, g i . (3.1)In particular, the map f ∈ H W ( f ) ∈ L (Ω , P ) is a linear isometry. For more details onGaussian probability spaces, we refer the reader to the book [42]. As before we are going to provethat the probability measure P is a ( β, X )-KMS state for a certain dynamical system with aninverse temperature β = 1. Let { e j } be an orthonormal basis of H and define the linear operator J as, J e j − = e j , J e j = − e j − , ∀ j ∈ N , (3.2)Then, J induces a symplectic structure on H . Furthermore, consider ( α j ) j ∈ N a sequence ofpositive real numbers such that ∞ X j =1 α − j < ∞ . (3.3)Using this sequence, one can define a Hilbert rigging H + ⊆ H ⊆ H − by taking H + = (cid:8) u ∈ H | ∞ X j =1 α j h u, e j i < ∞ (cid:9) , as a Hilbert space endowed with the inner product given for any u, v ∈ H + by, h u, v i H + = ∞ X j =1 α j h u, e j ih e j , v i ;and considering H − as the dual of H + with respect to the inner product of H . Remark that thenorm on H − is given by, k u k H − = (cid:18) ∞ X j =1 α − j h u, e j i (cid:19) / . We note the following analogue of Theorem 3.3 in the context of Gaussian spaces.
IBBS MEASURES AS KMS EQUILIBRIUM STATES 17
Lemma 3.5.
For all f, g ∈ H + , we have h g, J f i E (cid:16) e iW ( g ) (cid:17) + i E (cid:16) W ( − J f ) e iW ( g ) (cid:17) = 0 . (3.4) Proof.
The idea of the proof is similar to that of Theorem 3.3, except that now we do not have avector field at our disposal. Instead, we use the Gaussian structure. We start by observing thatfor all f ∈ H , we have E (cid:16) e iW ( f ) (cid:17) = e − k f k . (3.5)In order to deduce identity (3.5), we note that by Wick’s rule and (3.1), we have that for all k ∈ N E (cid:16) ( W ( f )) k (cid:17) = ( (2 n )! n ! 2 n k f k n if k = 2 n is even0 if k is odd.For f, g , we compute dds E (cid:16) e iW ( − sJf + g ) (cid:17) (cid:12)(cid:12) s =0 = i E (cid:16) W ( − J f ) e iW ( g ) (cid:17) . (3.6)On the other hand, by using (3.5), we can rewrite (3.6) as dds e − k− sJf + g k (cid:12)(cid:12) s =0 = −h g, J f i E (cid:16) e iW ( g ) (cid:17) . (3.7)The identity (3.4) follows from (3.6) and (3.7). (cid:3) In order to see the above identity (3.4) as a KMS condition similar to (2.10), one needs tointroduce a vector field X that is interpreted as an element of the space L (Ω , P ; H − ) of squareintegrable H − -valued functions. Lemma 3.6.
Let X n = P nj =1 W ( e j ) J e j ∈ L (Ω , P ; H ) . Then the sequence ( X n ) n ∈ N convergesto an element X ∈ L (Ω; H − ) , i.e.: X = ∞ X j =1 W ( e j ) J e j ∈ L (Ω , P ; H − ) . Proof.
It is enough to show that ( X n ) n ∈ N is a Cauchy sequence in L (Ω , P ; H − ). Indeed, one has k X n − X m k L (Ω , P ; H − ) = Z Ω k X n − X m k H − d P = Z Ω ∞ X j =1 α − j h X n − X m , e j i d P Using the definition of X n and applying (3.2), one notices that for j ∈ N , h X n − X m , e j i = 1 [ m +2 ,n +1] ∩ N ( j ) W ( e j − ) − [ m,n − ∩ N +1 ( j ) W ( e j +1 ) . Thus, one concludes by (3.3) k X n − X m k L (Ω , P ; H − ) = n X j = m +1 α − j E (cid:0) W ( e j ) (cid:1) −→ n,m →∞ . (cid:3) As in Subsection 2.1, one defines the class of smooth compactly supported cylindrical functions F ∈ C ∞ c,cyl (Ω) as all the functions F : Ω → R satisfying F = ϕ ( W ( e ) , . . . , W ( e n )) , for some n ∈ N and ϕ ∈ C ∞ c ( R n ). Similarly, one can introduce a gradient for these functionsgiven as below, ∇ F = n X j =1 ∂ j ϕ ( W ( e ) , . . . , W ( e n )) W ( e j ) e j ∈ L (Ω , P ; H ) . Hence, one can also introduce a Poisson bracket for any
F, G ∈ C ∞ c,cyl (Ω) as, { F, G } = (cid:10) ∇ F, J ∇ G (cid:11) ∈ L (Ω , P ) . Proposition 3.7.
Let (Ω , Σ , P ) be a Gaussian probability space with J and X defined as before.Then P is a (1 , X ) -KMS state in the following sense: For all F, G ∈ C (Ω) , E (cid:0) { F, G } (cid:1) = E (cid:0) h∇ F, X i G (cid:1) . (3.8) Proof.
Since the map f W ( f ) is a linear isometry from H to L (Ω , P ), one checks that for f ∈ H + h f, X i = lim n n X j =1 h f, J e j i W ( e j ) = lim n W (cid:18) n X j =1 h− J f, e j i e j (cid:19) = W ( − J f ) . Therefore, the equality (3.4) reads, h g, J f i E (cid:16) e iW ( g ) (cid:17) + i E (cid:16) h f, X i e iW ( g ) (cid:17) = 0 . Following the same lines of the proof of Theorem 2.7, one proves the KMS condition (3.8). (cid:3)
Remark . The identity (3.8) can be regarded as a generalization of the KMS condition (2.9)to Gaussian probability spaces or more generally to stochastic processes.4.
The Gibbs-KMS equivalence
In this section, we address the problem of equivalence between Gibbs measures and KMSstates. It is quite instructive to first consider finite dimensional dynamical systems since theyprovide significant insight into the problem. Afterwards, we consider in Subsection 4.2 the case ofcomplex linear infinite dimensional dynamical systems; while nonlinear infinite dynamical systemsare treated in the last Subsection 4.3.4.1.
Finite dimensional dynamical systems.
Let E be a Hermitian space of dimension n endowed with a scalar product h· , ·i which is anti-linear with respect to the left component. Fixan orthonormal basis { e , . . . , e n } . One can consider E as a Euclidean vector space with respectto the scalar product h· , ·i E, R := ℜ e h· , ·i . For convenience, we simply denote by E R the Euclidean vector space ( E, h· , ·i E, R ). Notice that ifwe set f j = ie j , for j = 1 , · · · , n , then { e , . . . , e n , f , . . . , f n } is an orthonormal basis of E R andwe have the decomposition E R = K ⊕ iK , (4.1)where K = span R { e , . . . , e n } . Moreover, E R is isomorphic to the direct sum K ⊕ K through thecanonical R -linear mapping: ∀ x, y ∈ K, E R ∋ x + iy ⇋ x ⊕ y ∈ K ⊕ K . (4.2)
IBBS MEASURES AS KMS EQUILIBRIUM STATES 19
Within this isomorphism the complex structure in E is implemented in K ⊕ K by the linearoperator J = (cid:20) − (cid:21) such that J : K ⊕ K → K ⊕ K and J u ⊕ v = v ⊕ − u . In particular, J = − and J corresponds,via the above isomorphism, to the multiplication by the complex − i on E . In the sequel, we willsometimes use the identification E R ≃ K ⊕ K without making reference to the isomorphism (4.2). Symplectic structure:
The Hermitian space E is naturally equipped with a canonical non-degenerate symplectic form: σ ( · , · ) := ℑ m h· , ·i . In particular, the following relation holds true for all u, v ∈ E , σ ( u, v ) = h iu, v i E, R = h u, J v i K ⊕ K . (4.3)Moreover, since K = { u ∈ E | σ ( u, v ) = 0 for all v ∈ K } then K is a Lagrangian subspace andthe isomorphism (4.2) provides a polarization of the phase-space E into canonical position andmomentum coordinates. Poisson structure:
Consider two smooth real-valued functions
F, G ∈ C ∞ ( E ). The Poissonbracket is defined by, { F, G } ( u ) := n X j =1 ∂F∂e j ( u ) ∂G∂f j ( u ) − ∂G∂e j ( u ) ∂F∂f j ( u ) , (4.4)where the partial derivatives are given by ∂F∂e j ( u ) = lim λ → ,λ ∈ R F ( u + λe j ) − F ( u ) λ , ∂G∂f j ( u ) = lim λ → ,λ ∈ R G ( u + λf j ) − G ( u ) λ . Such a bracket is skew symmetric and satisfies both the Leibniz rule and the Jacobi identity. Itis sometimes useful to use the derivatives with respect to the complex coordinates. For this, wedefine the Wirtinger derivatives by ∂F∂z j ( u ) := ∂F∂e j ( u ) − i ∂F∂f j ( u ) , ∂F∂ ¯ z j ( u ) := ∂F∂e j ( u ) + i ∂F∂f j ( u ) . (4.5)Hence, one can write the Poisson bracket as { F, G } ( u ) = 12 i n X j =1 ∂F∂z j ( u ) ∂G∂ ¯ z j ( u ) − ∂G∂z j ( u ) ∂F∂ ¯ z j ( u ) . One can also write the Poisson bracket using the symplectic form σ in (4.3). In fact, consider aFr´echet differentiable function F : E → R . Then its real differential is a R -linear form given forall v ∈ K ⊕ K , such that v = P nj =1 v j e j ⊕ P nj =1 w j e j , by ∇ F ( u )[ v ] = n X j =1 v j ∂F∂e j ( u ) + w j ∂F∂f j ( u ) . Hence, it can be identified with the following element of K ⊕ K ≃ E R , ∇ F ( u ) = n X j =1 ∂F∂e j ( u ) e j ⊕ ∂F∂f j ( u ) e j . (4.6) The standard definition has 1 / Thus, one checks that for all Fr´echet differentiable functions
F, G : E → R , { F, G } ( u ) = σ ( ∇ F ( u ) , ∇ G ( u )) . (4.7) Hamiltonian system:
Consider a function h : E R ≃ K ⊕ K → R of class C . Then as above,the differential of h is given by, ∇ h ( u ) = n X j =1 ∂h∂e j ( u ) e j + ∂h∂f j ( u ) f j ≡ n X j =1 ∂h∂e j ( u ) e j ⊕ ∂h∂f j ( u ) e j . (4.8)Define the ∂ ¯ z operator as ∂ ¯ z F ( u ) = n X j =1 ∂F∂ ¯ z j ( u ) e j , then using the Wirtinger’s derivatives in (4.5), one remarks − i∂ ¯ z h ( u ) ≡ J ∇ h ( u ) . A Hamiltonian dynamical system on the phase-space E is then defined by means of the energyfunctional h and the associated continuous vector field X : E → E given for all u ∈ E by X ( u ) = − i∂ ¯ z h ( u ) ≡ J ∇ h ( u ) . (4.9)Indeed, the Hamiltonian system is governed by the vector field equation,˙ u ( t ) = X ( u ( t )) , (4.10)where u : I ⊆ R → E is a C curve and I is a time interval. The differential equation (4.10)is complemented by an initial condition u ( t ) = u ∈ E at a fixed initial time t ∈ I . Sincethe vector field X is only continuous, one cannot apply the Cauchy-Lipschitz theorem and theexistence of a smooth flow is not guaranteed. Nevertheless, the Peano existence theorem providesat least the existence of local solutions for the equation (4.10). Gibbs measure:
In order to define the Gibbs measure for the above Hamiltonian system, weassume that z β := Z E e − βh ( u ) dL < + ∞ , (4.11)for some β > dL is the Lebesgue measure on E . In this case, we define the Gibbsmeasure of the Hamiltonian system (4.10), at inverse temperature β >
0, as the Borel probabilitymeasure given by µ β = e − βh ( · ) dL R E e − βh ( u ) dL ≡ z β e − βh ( · ) dL . (4.12)Notice that z β >
0. When the Hamiltonian system (4.10) admits a smooth global flow, we knowby the classical Liouville theorem that the Lebesgue and the Gibbs measures are invariant withrespect to this flow.
KMS states:
The general framework presented in Section 2.1 is applicable in the finite dimen-sional setting. We henceforth consider Φ = E R = Φ ′ with the vector field X : E → E derived fromthe Hamiltonian functional h : E → R as in (4.9) and consider the KMS states as in Definition2.3. Specifically, we say that µ ∈ P ( E ) is a ( β, X )-KMS state if and only if: Z E { F, G } ( u ) dµ = β Z E ℜ e h∇ F ( u ) , X ( u ) i G ( u ) dµ , (4.13)for any compactly supported smooth functions F, G ∈ C ∞ c ( E ). The following lemma is useful toexpress the above KMS condition in terms of the Poisson bracket. IBBS MEASURES AS KMS EQUILIBRIUM STATES 21
Lemma 4.1.
For any F ∈ C ∞ c ( E ) and u ∈ E , we have ℜ e h∇ F ( u ) , X ( u ) i = { F, h } ( u ) . Proof.
Using the isomorphism (4.2) and the identity (4.8), one can write the vector field (4.9) as X ( u ) = J n X j =1 ∂h∂e j ( u ) e j ⊕ ∂h∂f j ( u ) e j = n X j =1 ∂h∂f j ( u ) e j ⊕ − ∂h∂e j ( u ) e j . Similarly, using (4.6), one obtains ℜ e h X ( u ) , ∇ F ( u ) i = n X j =1 ∂F∂e j ( u ) ∂h∂f j ( u ) − ∂h∂e j ( u ) ∂F∂f j ( u ) = { F, h } ( u ) . (cid:3) Thus, by Lemma 4.1, the KMS condition (4.13) is equivalent to the identity Z E { F, G } ( u ) dµ = β Z E { F, h } ( u ) G ( u ) dµ , (4.14)for any compactly supported smooth functions F, G ∈ C ∞ c ( E ). Theorem 4.2.
Consider a function h : E → R of class C on the phase-space E and assumethat (4.11) holds for some β > . Then µ ∈ P ( E ) satisfies the KMS condition (4.14) if and onlyif µ is the Gibbs measure µ β in (4.12) .Proof. Let us check that the Gibbs measure µ β satisfies the KMS condition (4.14). Indeed, usingthe Fubini theorem and an integration by parts one shows, Z E ∂F∂e j ( u ) ∂G∂f j ( u ) dµ β = − z β Z E G ( u ) ∂∂f j (cid:18) ∂F∂e j ( u ) e − βh ( u ) (cid:19) dL = − Z E G ( u ) ∂ F∂f j ∂e j ( u ) dµ β + β Z E G ( u ) ∂F∂e j ( u ) ∂h∂f j ( u ) dµ β , and Z E ∂G∂e j ( u ) ∂F∂f j ( u ) dµ β = − z β Z E G ( u ) ∂∂e j (cid:18) ∂F∂f j ( u ) e − βh ( u ) (cid:19) dL = − Z E G ( u ) ∂ F∂e j ∂f j ( u ) dµ β + β Z E G ( u ) ∂F∂f j ( u ) ∂h∂e j ( u ) dµ β . Hence, by (4.4), we have Z E { F, G } ( u ) dµ = β Z E { F, h } ( u ) G ( u ) dµ β . Conversely, consider a Borel probability measure µ such that the KMS condition (4.14) is satisfied.Then remark that for any F, G ∈ C ∞ c ( E ), we have by the Leibniz rule that (cid:8) F, Ge − βh ( u ) (cid:9) = (cid:8) F, G (cid:9) e − βh ( u ) − β (cid:8) F, h (cid:9) G ( u ) e − βh ( u ) . Therefore, (cid:8)
F, Ge − βh ( u ) (cid:9) e βh ( u ) = (cid:8) F, G (cid:9) − β (cid:8) F, h (cid:9) G ( u ) , and notice that the above right hand side is integrable with respect to the measure µ . Hence, theKMS condition (4.14) gives Z E (cid:8) F e − βh ( u ) , G (cid:9) e βh ( u ) dµ = 0 . Since e βh ( · ) is a positive Borel function, the map B ν ( B ) := Z B e βh ( u ) dµ , defined for all Borel sets B of E , gives a Borel measure on E . So, one obtains that for any F, G ∈ C ∞ c ( E ), ν (cid:16)(cid:8) F e − βh ( u ) , G (cid:9)(cid:17) = Z E (cid:8) F e − βh ( u ) , G (cid:9) ( u ) dν = 0 . But since the classical Hamiltonian h is a C -function, one obtains for all F ∈ C c ( E ) and G ∈ C ∞ c ( E ), ν (cid:0)(cid:8) F, G (cid:9)(cid:1) = 0 . This condition implies that ν is a multiple of the Lebesgue measure. Indeed, take G ( · ) = ℜ e h e j , ·i ϕ ( · ) or G ( · ) = ℜ e h f j , ·i ϕ ( · ) with ϕ ∈ C ∞ ( E ) being equal to 1 on an open set con-taining the support of F . Then the Poisson brackets give, { F, G } = − ∂F∂f j ( u ) , or { F, G } = ∂F∂e j ( u ) . So, in a distributional sense the derivatives in all the directions of the measure ν are zero andtherefore dν = c dL for some constant c >
0. Using the normalization condition for µ , oneconcludes that c − = Z E e − βh ( u ) dL = z β , and consequently µ = 1 z β e − βh ( · ) dL = µ β . (cid:3) Remark . Later on we will see that the above Theorem 4.2 can be extended to non-smoothvector fields. Indeed, one notes that Theorem 4.14 below applies with minor modifications to thefinite dimensional setting.4.2.
Linear infinite dimensional dynamical systems.
For applications in PDEs it is conve-nient to work in a more concrete setting than the one from Section 3. In particular, we supposethat H is a separable complex Hilbert space. Hence, H is naturally equipped with a naturalsymplectic structure σ ( · , · ) = ℑ m h· , ·i , a real scalar product h· , ·i H, R := ℜ e h· , ·i and a compatiblecomplex structure. Note that H as a real Hilbert space will be denoted by H R . Complex linear Hamiltonian system:
Consider a positive operator A : D ( A ) ⊆ H → H suchthat, ∃ c > , A ≥ c . (4.15)The linear Hamiltonian dynamical system is given by the quadratic energy functional, h : D ( A / ) → R , h ( u ) = 12 h u, Au i . (4.16)So, the vector field in this case is the linear operator X : D ( A ) → H , X ( u ) = − iAu, leading to the linear differential equation governing the dynamics of the system,˙ u ( t ) = X ( u ( t )) = − iAu ( t ) . (4.17) IBBS MEASURES AS KMS EQUILIBRIUM STATES 23
Compact resolvent:
We suppose additionally that the operator A admits a compact resol-vent. Therefore, there exists an orthonormal basis of H composed of eigenvectors { e j } j ∈ N of A associated respectively to their eigenvalues { λ j } j ∈ N such that for all j ∈ N , Ae j = λ j e j . (4.18)Furthermore, assume the following assumption: ∃ s ≥ ∞ X j =1 λ sj < + ∞ . (4.19)Remark that if we set f j = i e j for all j ∈ N , then { e j , f j } j ∈ N is an O.N.B of H R . Weighted Sobolev spaces:
One can introduce weighted Sobolev spaces using the operator A asfollows. For any r ∈ R , define the inner product: ∀ x, y ∈ D ( A r ) , h x, y i H r := h A r/ x, A r/ y i . Let H s denote the Hilbert space ( D ( A s/ ) , h· , ·i H s ) where s ≥ H − s denotes the completion of the pre-Hilbert space ( D ( A − s/ ) , h· , ·i H − s ). Hence, one has thecanonical continuous and dense embeddings (Hilbert rigging), H s ⊆ H ⊆ H − s . (4.20)Remark that H − s identifies also with the dual space of H s relatively to the inner product of H . Cylindrical smooth functions:
Using the O.N.B. { e j , f j } j ∈ N , one considers the spaces of smoothcylindrical functions as in Subsection 2.1. More specifically, consider for n ∈ N the followingmapping π n : H − s → R n given by π n ( x ) = ( h x, e i H, R , . . . , h x, e n i H, R ; h x, f i H, R , . . . , h x, f n i H, R ) . (4.21)Then we define C ∞ c,cyl ( H − s ), respectively C ∞ b,cyl ( H − s ), as the set of all functions F : H − s → R such that F = ϕ ◦ π n (4.22)for some n ∈ N and ϕ ∈ C ∞ c ( R n ), respectively ϕ ∈ C ∞ b ( R n ). In particular, the gradient of F at the point u ∈ H − s is given by ∇ F ( u ) = n X j =1 ∂ (1) j ϕ ( π n ( u )) e j + ∂ (2) j ϕ ( π n ( u )) f j ∈ H − s . (4.23)where ∂ (1)1 ϕ, . . . , ∂ (1) n ϕ and ∂ (2)1 ϕ, . . . , ∂ (2) n ϕ are the partial derivatives of ϕ with respect to the n first and n second coordinates respectively. It also useful to introduce the following mapping, P n : H − s → E n u n X j =1 h e j , u i e j , (4.24)where E n = span C { e , . . . , e n } a finite dimensional subspace of H − s . The Euclidean structure of E n is the canonical one such that { e j , f j } j =1 ...,n is an O.N.B. Poisson structure:
We now precisely describe the Poisson structure over the algebra of smoothcylindrical functions C ∞ b,cyl ( H − s ). Consider F, G ∈ C ∞ b,cyl ( H − s ) such that for all u ∈ H − s , F ( u ) = ϕ ◦ π n ( u ) , G ( u ) = ψ ◦ π m ( u ) , (4.25) where ϕ ∈ C ∞ b ( R n ) and ψ ∈ C ∞ b ( R m ) for some n, m ∈ N . Then, for all such F, G ∈ C ∞ b,cyl ( H − s ), { F, G } ( u ) := min( n,m ) X j =1 ∂ (1) j ϕ ( π n ( u )) ∂ (2) j ψ ( π m ( u )) − ∂ (1) j ψ ( π m ( u )) ∂ (2) j ϕ ( π n ( u )) . (4.26) Gibbs measure:
The Hamiltonian system (4.16)-(4.17) admits a Gibbs measure at inversetemperature β >
0, formally given by µ β, ≡ e − βh ( · ) du R e − βh ( u ) du , and rigourously defined as a Gaussian measure on the Hilbert space H − s for the exponent s ≥ m ∈ H − s is the mean-vector of µ ∈ P ( H − s ) if for any f ∈ H − s the function u
7→ h f, u i H − s , R is µ -integrable and h f, m i H − s , R = Z H − s h f, u i H − s , R dµ . When m = 0, one says that µ is a zero-mean or centered measure. Additionally, the covarianceoperator of the Borel probability measure µ on H − s is a linear operator Q : H − s R → H − s R suchthat for any f, g ∈ H − s the function u
7→ h f, u i H − s , R h u, g i H − s , R is µ -integrable and h f, Q g i H − s , R = Z H − s h f, u − m i H − s , R h u − m, g i H − s , R dµ . For more details on Gaussian measures over Hilbert spaces, we refer the reader to the book byBogachev [8, Chapter 2]. In particular, the following result is well-known.
Theorem 4.4.
Let β > and assume that the assumptions (4.15) and (4.19) are satisfied. Thenthere exists a unique zero-mean Gaussian measure on H − s , denoted µ β, , such that its covarianceoperator is β − A − (1+ s ) , i.e.: for all f, g ∈ H − s β h f, A − (1+ s ) g i H − s , R = Z H − s h f, u i H − s , R h u, g i H − s , R dµ β, , (4.27) or equivalently for all f, g ∈ H s , β h f, A − g i H, R = Z H − s h f, u i H, R h u, g i H, R dµ β, . (4.28) Moreover, the characteristic function of µ β, is given for any w ∈ H s by, ˆ µ β, ( w ) = Z H − s e i h w,u i H, R dµ β, = e − β h w,A − w i H , (4.29) or equivalently for any v ∈ H − s we have Z H − s e i h v,u i H − s, R dµ β, = e − β h v,A − (1+ s ) v i H − s . (4.30) Remark . The following observations are useful.(i) The Gaussian measure µ β, given above coincides with the one provided by Corollary 3.2if one considers Φ = ∩ r> D ( A r ). In particular, it is not difficult to prove that Φ is acountably Hilbert nuclear space and µ β, ( H − s ) = 1.(ii) Remark that in particular, one hasTr H R [ β − A − (1+ s ) ] = ∞ X j =1 βλ sj = Z H − s k u k H − s dµ β, . (4.31) IBBS MEASURES AS KMS EQUILIBRIUM STATES 25 (iii) Note that, according to (4.28), the random variable h f, ·i H, R ∈ L ( µ β, ) for all f ∈ H − inthe sense that h f, ·i H, R := lim n h f n , ·i H, R in L ( µ β, ) with ( f n ) n ∈ N any norm approximatingsequence in H s of f ∈ H − .It is convenient to characterize µ β, using a position and momentum coordinates system. So,we define two sequences of image measures given by ν nβ, := ( π n ) ♯ µ β, , and µ nβ, := ( P n ) ♯ µ β, respectively on R n and E n . Here, ( · ) ♯ denotes the pushforward. We can explicitly compute thesemeasures. Lemma 4.6.
The Gaussian measure µ β, satisfies the following relations for all n ∈ N . ν nβ, = ( π n ) ♯ µ β, = n Y j =1 βλ j π e − β λj ( x j + y j ) dx j dy j , (4.32) µ nβ, = ( P n ) ♯ µ β, = e − β h· ,A ·i dL n R E n e − β h u,Au i dL n , (4.33) where L n is the Lebesgue measure on the Euclidean space E n of dimension n .Proof. Let κ n denote the measure in the right hand side of (4.32). One easily computes thecharacteristic function of κ n ,ˆ κ n ( ζ , . . . , ζ n ; η , . . . , η n ) = Z R n e i P nj =1 ( x j ζ j + y j η j ) dκ n = n Y j =1 e − λjβ ( ζ j + η j ) . On the other hand, by Theorem 4.4, one checksˆ ν nβ, ( ζ , . . . , ζ n ; η , . . . , η n ) = Z H − s e i P nj =1 ( h u,e j i H, R ζ j + h u,f j i H, R η j ) dµ β, = ˆ κ n ( ζ , . . . , ζ n ; η , . . . , η n ) . So, this shows that κ n = ν nβ, . Similarly, to prove the second relation, it is enough to note thatthe characteristic function of the right hand side of (4.33) is given by e − h w, ( βA ) − w i H, R = ˆ µ nβ, ( w ) , ∀ w ∈ E n , where the equality follows from (4.29). (cid:3) Let i n : E n → H − s , i n ( u ) = u for all u ∈ E n , be the canonical embedding of E n into H − s . Itis useful to introduce for any µ ∈ P ( H − s ) the image measures on H − s given by µ n = ( i n ◦ P n ) ♯ µ . (4.34) Lemma 4.7.
For any µ ∈ P ( H − s ) , the sequence ( µ n ) n ∈ N converges narrowly to µ on P ( H − s ) .Proof. We note that the maps i n ◦ P n : H − s → H − s are linear and continuous. One checks thatfor all u ∈ H − s , || i n ◦ P n ( u ) − u || H − s = ∞ X j = n +1 λ − sj ( h u, e j i H, R + h u, f j i H, R ) . Hence, this proves that the sequence ( i n ◦ P n ( u )) n ∈ N converges towards u in H − s . Consequently,one shows that by dominated convergence, for any continuous bounded function F : H − s → R ,we have lim n Z H − s F ( u ) dµ n ( u ) = lim n Z H − s F ( i n ◦ P n ( u )) dµ = Z H − s F ( u ) dµ ( u ) . (cid:3) Within the framework of this subsection, we prove a KMS-Gibbs equivalence result. Beforedoing this, we remark that for all F ∈ C ∞ c,cyl ( H − s ) satisfying (4.22), ℜ e h X ( u ) , ∇ F ( u ) i = n X j =1 ∂ (1) j ϕ ( π n ( u )) h− iAu, e j i H, R + ∂ (2) j ϕ ( π n ( u )) h− iAu, f j i H, R , is a well-defined continuous bounded function on H − s . Therefore, the KMS condition (2.9) inDefinition 2.3 makes sense. Theorem 4.8.
Suppose that assumptions (4.15) and (4.19) hold. Let X be the vector field givenby X ( u ) = − iAu and β > . Then µ ∈ P ( H − s ) is a ( β, X ) -KMS state if and only if µ is theGaussian measure µ β, provided by Theorem 4.4.Proof. Let µ ∈ P ( H − s ) satisfy the KMS condition (2.9). Consider the image measure ˜ µ n =( P n ) ♯ µ ∈ P ( E n ). For F, G ∈ C ∞ c,cyl ( H − s ) as in (4.25) with n = m , one remarks that for any u ∈ H − s , { F, G } ( P n ( u )) = { F, G } ( u ) , ℜ e h∇ F ( P n ( u )) , X ( P n ( u )) i = ℜ e h∇ F ( u ) , X ( u ) i . Hence, the KMS condition (2.9) reads as, Z H − s { F, G } ( P n u ) dµ = β Z H − s ℜ e h∇ F ( P n ( u )) , X ( P n ( u )) i G ( P n ( u )) dµ , and consequently Z E n { F, G } ( w ) d ˜ µ n ( w ) = β Z E n ℜ e h∇ F ( w ) , X ( w ) i G ( w ) d ˜ µ n ( w ) . This means that the Borel probability measure ˜ µ n on E n satisfies the KMS condition (4.14) infinite dimensions with the continuous vector field X ,n ( u ) = − iAu and the C energy functional h ,n ( u ) = h u, Au i for u ∈ E n . So, by Theorem 4.2 one concludes that˜ µ n = e − β h· ,A ·i dL n ( · ) R E n e − β h u,Au i dL n ( u ) . Now, using Lemma 4.6 one obtains ˜ µ n = µ nβ, = ( P n ) ♯ µ β, . Moreover, by applying Lemma 4.7 for the Borel probability measure µ , recalling (4.34) andapplying Lemma 4.6 again one obtains µ n = ( i n ◦ P n ) ♯ µ = ( i n ) ♯ ˜ µ n = ( i n ) ♯ µ nβ, = ( i n ◦ P n ) ♯ µ β, −→ n →∞ µ β, . Since by Lemma 4.7, µ n converges narrowly to µ , we deduce that µ = µ β, . Conversely, Theorem 3.3 and Remark 4.5-(i) show that µ β, is a ( β, X )-KMS state. Note thatthanks to the complex structure on H , one has that J ≡ − i and X = − iA ≡ J A . (cid:3) IBBS MEASURES AS KMS EQUILIBRIUM STATES 27
Nonlinear infinite dimensional dynamical systems.
In this part we address the ques-tion of equilibrium (KMS) states for nonlinear Hamiltonian PDEs and their equivalence to Gibbsmeasures. We consider the same setting and notation as in Subsection 4.2 above. In particular, µ β, is the Gaussian measure provided by Theorem 4.4.In order to explicitly define an abstract nonlinear dynamical system that encloses the mostimportant examples of PDEs that we wish to explore, we use the framework of Malliavin calculusand Gross-Sobolev spaces. First, we explain the main ideas behind these concepts and refer thereader to the book [42] for further details. Note that the spaces used here are slightly differentfrom the ones in the above reference. Lemma 4.9 (Malliavin derivative) . For p ∈ [1 , ∞ ) the linear operator ∇ with domain D = C ∞ c,cyl ( H − s ) and ∇ : D ⊂ L p ( µ β, ) −→ L p ( µ β, ; H − s ) ,F F where ∇ F is given by (4.23) , is closable.Proof. Let F n ∈ C ∞ c,cyl ( H − s ), n ∈ N , be a sequence such that F n → L p ( µ β, ) and ∇ F n → Y in L p ( µ β, ; H − s ). In order to prove that the operator ∇ is closable, one needs to show that Y = 0. Indeed, using Proposition A.1 one proves for any G ∈ C ∞ b,cyl ( H − s ), ϕ ∈ span R { e j , f j ; j =1 , . . . , k } and ε > Z H − s e G ( u ) h∇ F n ( u ) , ϕ i dµ β, = Z H − s F n ( u ) (cid:18) − h∇ e G ( u ) , ϕ i + β e G ( u ) h u, Aϕ i (cid:19) dµ β, , where e G = G e − ε h· ,Aϕ i ∈ C ∞ b,cyl ( H − s ) satisfies e G h· , Aϕ i ∈ C ∞ b,cyl ( H − s ) ⊆ L q ( µ β, ) where q is theH¨older conjugate of p . Taking the limit n → ∞ of both sides and using the H¨older inequality,one obtains Z H − s e G ( u ) h Y, ϕ i dµ β, = 0 . Letting ε → C ∞ b,cyl ( H − s ) in L p ( µ β, ), one shows for any ϕ ∈ span R { e j , f j ; j = 1 , . . . , k } , h Y, ϕ i = 0 , µ β, − a.s. Hence, using the separability of H s and a density argument one proves that Y = 0 almost surelywith respect to µ β, . (cid:3) In light of Lemma 4.9, one can introduce the following
Gross-Sobolev spaces.
Definition 4.10 (Gross-Sobolev spaces) . For p ∈ [1 , ∞ ), we denote the closure domain of thelinear operator ∇ from Lemma 4.9 by D ,p ( µ β, ). On D ,p ( µ β, ), we consider the norm k F k p D ,p ( µ β, ) := k F k pL p ( µ β, ) + k∇ F k pL p ( µ β, ; H − s ) . (4.35)By Lemma 4.9, we obtain that D ,p ( µ β, ) endowed with the above graph norm (4.35) is aBanach space. Furthermore, if p = 2, it is a Hilbert space with the inner product h F, G i D , ( µ β, ) = h F, G i L ( µ β, ) + h∇ F, ∇ G i L ( µ β, ; H − s ) . Here, we are applying Proposition A.1 for functions in C ∞ c,cyl ( H − s ) and C ∞ b,cyl ( H − s ) for which the norm (4.35)is finite. At this step, we do not need to apply the full strength of Proposition A.1. The abstract nonlinear dynamical system that we shall consider is defined as a pair consistingof a linear operator A satisfying (4.15), (4.18)-(4.19) and a Borel nonlinear energy functional h I : H − s → R satisfying for some β > e − βh I ( · ) ∈ L ( µ β, ) and h I ∈ D , ( µ β, ) . (4.36)More specifically, the vector field of the system is given by X ( u ) = − iAu − i ∇ h I ( u ) , (4.37)defining a field equation in the interaction representation given through the non-autonomousdifferential equation for v ( t ) := e itA u ( t ),˙ v ( t ) = e itA X I ( e − itA v ( t )) , where X I = − i ∇ h I : H − s → H − s is a Borel vector field belonging to L ( µ β, ; H − s ) and t ∈ R → v ( t ) ∈ H − s is a stochastic process solution. Remark that by the Cauchy-Schwarz inequality theassumption (4.36) implies that X I ∈ L ( µ β, ; H − s ).According to Definition 2.9, a Borel probability measure µ on H − s is a ( β, X )-KMS state ofthe dynamical system induced by the vector field X = − iA + X I , at inverse temperature β > F, G ∈ C ∞ c,cyl ( H − s ), Z H − s { F, G } ( u ) dµ = β Z H − s ℜ e h∇ F ( u ) , − iAu + X I ( u ) i G ( u ) dµ . (4.38) Theorem 4.11.
Assume that the assumption (4.36) is true and furthermore for any ϕ ∈ H s thefunction h∇ h I , ϕ i e − βh I ∈ L ( µ β, ) . Then the Gibbs measure µ β = e − βh I µ β, R H − s e − βh I dµ β, , (4.39) is a ( β, X ) -KMS state satisfying (4.38) where X is the vector field given in (4.37) .Proof. Let
F, G ∈ C ∞ c,cyl ( H − s ) be such that for some p, q ∈ N and ϕ ∈ C ∞ c ( R p ) , ψ ∈ C ∞ c ( R q ),we have F ( u ) = ϕ ◦ π p ( u ) , G ( u ) = ψ ◦ π q ( u ) , where π p , π q are the mappings in (4.21). Hence, according to (4.26) one writes Z H − s { F, G } dµ β = p X j =1 z β Z H − s (cid:0) ∂ e j F ∂ f j G − ∂ e j G ∂ f j F (cid:1) e − βh I dµ β, , where ∂ e j , ∂ f j are directional derivatives and z β = Z H − s e − βh I dµ β, (4.40)is the normalization constant in (4.39). We note that there exists a sequence of functions θ k ∈ C b ( R ) such that θ k ( x ) → e − βx and θ ′ k ( x ) → − βe − βx pointwise for all x ∈ R with0 ≤ θ k ( x ) ≤ e − βx and | θ ′ k ( x ) | ≤ ce − βx , (4.41)for some constant c > k large enough. Indeed, we can take θ k ( x ) = e − βx if x ≥ − k and θ k ( x ) = arctan( − βe βk ( x + k )) + e βk if x < − k . By applying the dominated convergence IBBS MEASURES AS KMS EQUILIBRIUM STATES 29 theorem and Proposition A.1 with ϕ = f j and ϕ = e j respectively, we have Z H − s { F, G } dµ β = lim k p X j =1 z β Z H − s (cid:0) ∂ e j F ∂ f j G − ∂ e j G ∂ f j F (cid:1) θ k ( h I ) dµ β, = lim k p X j =1 z β Z H − s G (cid:18) − ∂ f j (cid:0) ∂ e j F θ k ( h I ) (cid:1) + β h u, Af j ih∇ F, e j i θ k ( h I ) (cid:19) + G (cid:18) ∂ e j (cid:0) ∂ f j F θ k ( h I ) (cid:1) − β h u, Ae j ih∇ F, f j i θ k ( h I ) (cid:19) dµ β, . Note that we also used Lemma A.2 in order to deduce that θ k ( h I ), ∂ e j F θ k ( h I ) and ∂ f j F θ k ( h I )belong to D , ( µ β, ). Moreover, one observes that p X j =1 ∂ e j F ∂ f j h I − ∂ f j F ∂ e j h I = ℜ e h∇ F, − i ∇ h I i , p X j =1 h u, Af j ih∇ F, e j i − h u, Ae j ih∇ F, f j i = ℜ e h∇ F, − iAu i , Thus, using the assumptions of the Theorem and dominated convergence, one obtains Z H − s { F, G } dµ β = lim k z β Z H − s G (cid:18) ℜ e h∇ F, i ∇ h I i θ ′ k ( h I ) + β ℜ e h∇ F, − iAu i θ k ( h I ) (cid:19) dµ β, = β z β Z H − s G (cid:18) ℜ e h∇ F, − i ∇ h I i + ℜ e h∇ F, − iAu i (cid:19) e − βh I dµ β, . So, this proves the KMS condition (4.38) for the Gibbs measure µ β . (cid:3) Our next main result shows that the dynamical system at hand admits a unique KMS statewhich is the Gibbs measure µ β . But before stating such a result, we need to prove some prelimi-nary results. Proposition 4.12.
Assume (4.36) is true. Let µ be a Borel probability measure on H − s whichis absolutely continuous with respect to µ β, , i.e. there exists a non-negative density ̺ ∈ L ( µ β, ) such that for all Borel sets, we have µ ( B ) = Z B ̺ ( u ) dµ β, . Assume further that the density ̺ ∈ D , ( µ β, ) . If µ is a ( β, X ) -KMS state satisfying (4.38) thenthe density ̺ satisfies the equation, ∇ ̺ + β̺ ∇ h I = 0 , (4.42) in L ( µ β, ; H − s ) with ∇ is the Malliavin derivative from Lemma 4.9.Proof. Consider µ ∈ P ( H − s ) satisfying the KMS condition (4.38) and the above hypothesis.There exists a sequence ̺ n ∈ C ∞ b,cyl ( H − s ) such that ̺ n → ̺ , ∂ e j ̺ n → ∂ e j ̺ and ∂ f j ̺ n → ∂ f j ̺ in L ( µ β, ) for all j ∈ N . Then using the Leibniz rule, one proveslim n Z H − s { F, G̺ n } dµ β, = lim n Z H − s { F, G } ̺ n dµ β, + Z H − s { F, ̺ n } G dµ β, = Z H − s { F, G } dµ + Z H − s ℜ e h∇ F, − i ∇ ̺ i G dµ β, . On the other hand, the KMS condition satisfied by the measure µ β, yieldslim n Z H − s { F, G̺ n } dµ β, = β lim n Z H − s ℜ e h∇ F, X i G̺ n dµ β, = β Z H − s ℜ e h∇ F, X i G dµ , where X ( u ) = − iAu . Hence, the two above equalities give β Z H − s ℜ e h∇ F, X i G dµ = Z H − s { F, G } dµ + Z H − s ℜ e h∇ F, − i ∇ ̺ i G dµ β, . Since µ satisfies the KMS condition with the vector field X = X + X I , then one concludes forall G ∈ C ∞ c,cyl ( H − s ), Z H − s ℜ e h∇ F, β̺ X I − i ∇ ̺ i G dµ β, = 0 . We recall (4.37) and apply a standard density argument to deduce that ∇ ̺ + β̺ ∇ h I = 0 ,µ β, -almost surely and as an element of L ( µ β, ; H − s ). (cid:3) Lemma 4.13.
Assume (4.36) is true and e − βh I ∈ L ( µ β, ) . Let ̺ be the density in Proposition4.12 and take any convex combination, ˜ ̺ = α̺ + (1 − α ) k e − βh I k L ( µ β, ) e − βh I , (4.43) with α ∈ (0 , . Then log(˜ ̺ ) ∈ D , ( µ β, ) and ∇ log(˜ ̺ ) = ∇ ˜ ̺ ˜ ̺ . Proof.
One checks that e − βh I ∈ D , ( µ β, ) and ∇ e − βh I + βe − βh I ∇ h I = 0. Indeed, take θ k thesame function as in the proof of Theorem 4.11. Then by dominated convergence one haslim k k θ k ( h I ) − e − βh I k L ( µ β, ) = 0 , lim k,k ′ k∇ θ k ( h I ) − ∇ θ k ′ ( h I ) k L ( µ β, ; H − s ) = 0 . Hence, the sequence θ k ( h I ) converges to e − βh I in D , ( µ β, ). Thus, one concludes as a consequenceof Proposition 4.12 that ∇ ˜ ̺ + β ˜ ̺ ∇ h I = 0 . Since ˜ ̺ > ∇ ˜ ̺ ˜ ̺ = − β ∇ h I ∈ L ( µ β, ; H − s ) . There exists a sequence of functions κ k ∈ C b ( R ) such that κ n ( x ) → log( x ) and κ ′ n ( x ) → /x pointwise for x > | κ n ( x ) | ≤ | log( x ) | , | κ ′ n ( x ) | ≤ c | x | , for some constant c > x >
0. Indeed, we can take κ n ( x ) = log( x ) for 1 /n ≤ x ≤ n , κ n ( x ) = − n/x + 1 + log( n ) for x > n and κ n ( x ) = n arctan( x − /n ) − log( n ) for x < /n .By Lemma A.2 one knows that κ n (˜ ̺ ) ∈ D , ( µ β, ) and ∇ κ n (˜ ̺ ) = κ ′ n (˜ ̺ ) ∇ ˜ ̺ . Hence, dominatedconvergence yieldslim n Z H − s (cid:12)(cid:12)(cid:12)(cid:12) κ n (˜ ̺ ) − log(˜ ̺ ) (cid:12)(cid:12)(cid:12)(cid:12) dµ β, = lim n Z H − s (cid:13)(cid:13)(cid:13)(cid:13) κ ′ n (˜ ̺ ) ∇ ˜ ̺ − ∇ ˜ ̺ ˜ ̺ (cid:13)(cid:13)(cid:13)(cid:13) H − s dµ β, = 0 . IBBS MEASURES AS KMS EQUILIBRIUM STATES 31
Therefore, one concludes that log(˜ ̺ ) ∈ D , ( µ β, ) and ∇ log(˜ ̺ ) = ∇ ˜ ̺ ˜ ̺ . (cid:3) Theorem 4.14.
Assume that (4.36) is true and e − βh I ∈ L ( µ β, ) . Let µ be a Borel probabilitymeasure on H − s which is absolutely continuous with respect to µ β, , i.e. there exists a non-negative density ̺ ∈ L ( µ β, ) such that for all Borel sets, µ ( B ) = Z B ̺ ( u ) dµ β, . Assume further that ̺ ≡ dµdµ β, ∈ D , ( µ β, ) . Then µ is a ( β, X ) -KMS state for the vector field X in (4.37) if and only if µ is equal to the Gibbs measure µ β = e − βh I µ β, R H − s e − βh I ( u ) dµ β, . Proof.
Sufficiency follows from Theorem 4.11. Take ˜ ̺ the convex combination density in Lemma4.13 and remark that ˜ ̺ >
0. By Lemma 4.13 and the assumption (4.36), one knows that thefunction F = log(˜ ̺ ) + βh I ∈ D , ( µ β, ) . Moreover, one has ∇ (cid:0) log(˜ ̺ ) + βh I (cid:1) = ∇ ˜ ̺ ˜ ̺ + β ∇ h I = 0 . Hence, using Proposition A.4 one concludes that for some constant c ∈ R log(˜ ̺ ) + βh I = c ,µ β, -almost surely. Finally, using the normalization of the density ̺ one shows that ̺ = e − βh I k e − βh I k L ( µ β, ) . (cid:3) Remark . In statistical mechanics it is common to characterize the Gibbsmeasure µ β by means of relative entropy functional. So, it is not surprising to find a link betweenour analysis based on KMS states and the concept of entropy. In particular, note that thefunctional F ( ̺ ) = log( ̺ ) + βh I used in the proof of Theorem 4.14 is similar to the integrand thatone found in the formula of the relative entropy , E µ β, ( µ ) = Z H − s ̺ log( ̺ ) dµ β, + β Z H − s h I dµ = Z H − s (cid:0) log( ̺ ) + βh I (cid:1) ̺ dµ β, , where ̺ is the density satisfying µ = ̺µ β, . We note that E µ β, ( µ ) = E µ β ( µ ) − log( z β ) , where z β is given by (4.40) and one knows that E µ β ( µ ) is non-negative with E µ β ( µ ) = 0 if andonly if µ = µ β . In particular, this means that µ β is the unique minimizer of the relative entropy.5. Nonlinear PDEs
In this section we apply the concept of KMS states to various examples of nonlinear PDEs,namely to the nonlinear Schr¨odinger, Hartree, and wave (Klein-Gordon) equation. The construc-tion of invariant Gibbs measures for such equations is well understood. In particular, the analysisis based on probabilistic tools, truncation to a finite number of Fourier modes, and nonlinearstability estimates (see e.g. [9, 16, 23, 37, 41, 53] and the references therein). Here we emphasizethat the nonlinearities appearing in the above equations belong to the Gross-Sobolev spaces.Thus, it is possible to appeal to the Malliavin calculus and to apply our results.
Nonlinear Schr¨odinger equations.
Gibbs measures for NLS equations are well-studied,due to the fact that they are useful tools for establishing existence of global solutions and wellposedness for rough datum, see e.g. [10, 11, 12, 16, 43, 44] and the references therein.Consider the Hilbert space H = L ( T d ) where T d = R d / (2 π Z d ) is the flat d -dimensional torusand define the Sobolev weighted spaces H − s , as in Subsection 4.2, by means of the positiveself-adjoint operator A = − ∆ + , (5.1)where ∆ is the Laplacian on T d . So, the family { e k = e ikx } k ∈ Z d forms an O.N.B of eigenvectorsfor the operator A which admits a compact resolvent. Throughout this section, we consider s > d − . (5.2)Note that (4.18)-(4.19) are satisfied for s as in (5.2). In particular, in the one dimensional casewe can take s = 0. Therefore, according to Subsection 4.2, the Gaussian measure µ β, given byTheorem 4.4 is a well defined Borel probability measure on H − s and it is the unique ( β, X )-KMSstate for the vector field X = − iA and for any inverse temperature β >
0. We now analyze theKMS-condition in the context of various nonlinear Schr¨odinger-type equations, which we describein detail below.In the sequel, we write h x i = p | x | for the Japanese bracket. Furthermore, we write A . B if there exists C > A ≤ CB . If C depends on the parameters a , . . . , a k , wewrite A . a ,...,a k B . We write A & B if B . A . Finally, if A . B and B . A , we write A ∼ B .
1. The Hartree equation on T . When d = 1, we consider V : T ≡ T → R a pointwisenonnegative even L function. The Hartree nonlinear functional is given as h I ( u ) = 14 Z T Z T | u ( x ) | V ( x − y ) | u ( y ) | dx dy ≥ . (5.3)
2. The Hartree equation on T d , d = 2 ,
3. When d = 2 ,
3, we need to renormalise theinteraction by means of
Wick-ordering (see e.g. [11, 24, 50]). We summarise the constructionhere. Given n ∈ N , we recall the projection map in (4.24) that we take in our case to be P n = X | k |≤ n | e k ih e k | , and define for x ∈ T d σ n,β := Z H − s | P n u ( x ) | dµ β, = X | k |≤ n β ( | k | + 1) ∼ ( log nβ , if d = 2 nβ , if d = 3 . (5.4)Note that σ n,β is independent of x . Let us henceforth use the shorthand u n := P n u (5.5)and consider the Wick ordering with respect to µ β, : | u n | : = | u n | − σ n,β . (5.6)We observe that the above construction depends on β , but we suppress this in the notation. Welet h In,β ( u ) := 14 Z T d Z T d : | u n ( x ) | : V ( x − y ) : | u n ( y ) | : dx dy . (5.7) IBBS MEASURES AS KMS EQUILIBRIUM STATES 33
Here and in the sequel, we write : | u n ( x ) | : instead of : | u n | : ( x ) for (5.6) evaluated at x . Wework with even V ∈ L ( T d ) such that there exist ǫ > C > k ∈ Z d the following estimates hold. ( ≤ ˆ V ( k ) ≤ C h k i ǫ if d = 20 ≤ ˆ V ( k ) ≤ C h k i ǫ if d = 3 . (5.8)In particular, V is assumed to be of positive type (i.e. ˆ V is pointwise nonnegative). Under theassumptions (5.8), the arguments in [11] show that (5.7) converges in L p ( µ β, ), for all p ≥
1, to h I ( u ) = lim n h In,β ( u ) ≡ Z T d Z T d : | u ( x ) | : V ( x − y ) : | u ( y ) | : dx dy. (5.9)Let us note that, in the recent work [21], the authors extend the result of [11] for d = 3 topotentials satisfying 0 ≤ ˆ V ( k ) ≤ C h k i − ǫ . We do not consider this extension in our current paper.We recall the details of the proof of (5.9) in Appendix B. We refer to (5.9) as the Wick-orderedHartree nonlinear functional . Since V is of positive type, we have that h I ( u ) ∈ [0 , ∞ ) µ β, - almost surely . (5.10)Note that (5.3) and (5.10) imply that e − βh I ∈ L ( µ β, ) in dimension d = 1 , ,
3. The NLS equation on T . In the one dimensional case, the assumption (4.19) is satisfied for s = 0 and the nonlinear functional is given by h I ( u ) = 1 q Z T | u ( x ) | q dx ≥ . (5.11)for q = 2 r with r ∈ N , r ≥
4. The NLS equation on T . On T , we consider the general Wick-ordered nonlinearity. Given r ∈ N , and recalling (5.4) we define: | u n | r : = ( − r r ! σ rn,β L r (cid:18) | u n | σ n,β (cid:19) , (5.12)where L r is the r -th Laguerre polynomial. Note that this is a generalization of (5.6) since L ( x ) = − x + 1. For a given s >
0, one can consider the nonlinear Borel functional h I : H − s → R defined as the following limit in L ( µ β, ), h I ( u ) = lim n h In ( u ) = lim n r Z T : | u n | r : dx ≡ r Z T : | u | r : dx . (5.13)We refer the reader to [43] for a self-contained proof of (5.13).For the nonlinear functionals introduced above, the following statement holds true. Proposition 5.1.
The nonlinear Borel functionals h I ( u ) given by (5.3) , (5.9) , (5.11) and (5.13) belong to the Gross-Sobolev spaces D ,p ( µ β, ) for all ≤ p < ∞ .Proof. We prove each case separately. (i) The Hartree equation on T . It is well-known that for V ∈ L ( T ), we have that h I ( u ) ∈ L p ( µ β, ) . (5.14)We note that it suffices to prove (5.14) when p ≥ p ∈ [1 ,
2) then follows fromH¨older’s inequality. More precisely, by using the Cauchy-Schwarz inequality, Young’s inequality,and the Sobolev embedding H ζ ( T ) ֒ → L ( T ) for ζ ∈ ( , ) in (5.3), we get that0 ≤ h I ( u ) . ζ k V k L k u k H ζ , and we deduce (5.14) by arguing similarly as for (4.31) above. A direct calculation shows that ∇ h I ( u ) = ( V ∗ | u | ) u . (5.15)For s ∈ ( − , ζ := − s ∈ (0 , ). By using (5.15) and by applying Lemma B.1 twice forsufficiently small α , we deduce that for some ζ ′ ∈ ( ζ, ) k∇ h I ( u ) k L p ( µ β, ; H − s ) = (cid:13)(cid:13) k ( V ∗ | u | ) u k H ζ (cid:13)(cid:13) L p ( µ β, ) . ζ,V,p (cid:13)(cid:13) k u k H ζ ′ (cid:13)(cid:13) L p ( µ β, ) = (cid:18) Z k u k pH ζ ′ dµ β, (cid:19) /p < ∞ . (5.16)In (5.16), we used the observation that k V ∗ f k H θ ≤ k ˆ V k ℓ ∞ k f k H θ ≤ k V k L k f k H θ . (ii) The Hartree equation on T d , d = 2 ,
3. We note that (5.9) implies that h I ( u ) ∈ L p ( µ β, ).As was noted earlier, (5.9) can be deduced from the arguments of [11] under the assumptionsgiven by (5.8). When d = 3, a detailed proof of this fact is given in [50, Lemma 1.4 (i)]. Notethat, here, one assumes that V ∈ L q ( T ) for q >
3, which follows from (5.8) (see [11, (29)] and[50, (1.44)-(1.45)]). When d = 2, this fact is shown in detail in [50, Lemma 1.4 (ii)] if, in additionto satisfying, one assumes that V is pointwise nonnegative. In Appendix B, we present the proofof (5.9) from [11] which does not require pointwise nonnegativity of V .A direct calculation shows that ∇ h In ( u ) = P n (cid:20)(cid:18) Z V ( · − y ) : | u n ( y ) | : dy (cid:19) u n (cid:21) = P n (cid:2) ( V ∗ : | u n | : ) u n (cid:3) . As was noted earlier, it suffices to consider p ≥
2. For s as in (5.2), we want to show that ( h In ( u ))is a Cauchy sequence in L p ( µ β, ; H − s ).By Minkowski’s inequality, we have k∇ h In ( u ) k L p ( µ β, ; H − s ) ∼ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ∈ Z d , | k |≤ n h k i − s (cid:12)(cid:12)(cid:12)(cid:2) ( V ∗ : | u n | : ) u n (cid:3)b ( k ) (cid:12)(cid:12)(cid:12) ! / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( µ β, ) ≤ X k ∈ Z d , | k |≤ n h k i − s (cid:13)(cid:13)(cid:2) ( V ∗ : | u n | : ) u n (cid:3)b ( k ) (cid:13)(cid:13) L p ( µ β, ) ! / . Likewise, for n ≥ m , we have k∇ h In ( u ) − ∇ h Im ( u ) k L p ( µ β, ; H − s ) . X k ∈ Z d , | k |≤ m h k i − s (cid:13)(cid:13)(cid:2) ( V ∗ : | u n | : ) u n (cid:3)b ( k ) − (cid:2) ( V ∗ : | u m | : ) u m (cid:3)b ( k ) (cid:13)(cid:13) L p ( µ β, ) ! / + X k ∈ Z d ,m< | k |≤ n h k i − s (cid:13)(cid:13)(cid:2) ( V ∗ : | u n | : ) u n (cid:3)b ( k ) (cid:13)(cid:13) L p ( µ β, ) ! / . (5.17)In what follows, we view the Gaussian measure µ β, as the probability measure induced by themap ω ∈ Ω φ ( x ) ≡ φ ωβ ( x ) = 1 √ β X k ∈ Z d g k ( ω ) h k i e ik · x , (5.18)where ( g k ) k ∈ Z d is a sequence of independent standard complex Gaussian random variables (centredwith variance equal to 1) on a probability space (Ω , Σ , P ). Recalling (5.17) and (5.18), we considerfor fixed k ∈ Z d the expression k [( V ∗ : | u n | : ) u n ] b ( k ) k L p ( µ β, ) = k [( V ∗ : | φ n | : ) φ n ] b ( k ) k L p (Ω) , (5.19) IBBS MEASURES AS KMS EQUILIBRIUM STATES 35 where φ n ≡ φ ωn,β := P n φ ωβ .We recall the following estimate from [49, Theorem I.22]. Lemma 5.2.
Let the random variable ψ be a polynomial in ( g j ) j ∈ Z d of degree m ∈ N . Then, forall p ≥ , we have that k ψ k L p (Ω) ≤ ( p − m/ k ψ k L (Ω) . We note that ( V ∗ : | φ n | : ) φ n ] b ( k ) is a polynomial in ( g j ) j ∈ Z d of degree three. Therefore, byLemma 5.2, we have that(5.19) ≤ ( p − k [( V ∗ : | φ n | : ) φ n ] b ( k ) k L (Ω) . (5.20)Recalling (5.6) and (5.18), we have( : | φ n | : ) b ( k ) = β X | ℓ |≤ n | g ℓ ( ω ) | − h ℓ i ! k =0 + β X | ℓ |≤ n, | ℓ |≤ n,ℓ − ℓ = k g ℓ ( ω ) g ℓ ( ω ) h ℓ i h ℓ i ! k =0 . (5.21)Therefore, by (5.21), we get that for k ∈ Z d with | k | ≤ n , we have[( V ∗ : | φ n | : ) φ n ] b ( k ) = b V (0) β / X | ℓ |≤ n | g ℓ ( ω ) | − h ℓ i ! g k ( ω ) h k i + 1 β / X | ℓ |≤ n, | ℓ |≤ n, | ℓ |≤ nℓ − ℓ + ℓ = k,ℓ = ℓ b V ( ℓ − ℓ ) g ℓ ( ω ) g ℓ ( ω ) g ℓ ( ω ) h ℓ i h ℓ i h ℓ i =: I n ( k ) + II n ( k ) . (5.22)We now analyse each of the terms I n ( k ) and II n ( k ) separately. Analysis of I n ( k ). By using H¨older’s inequality, and Lemma 5.2, we have that k I n ( k ) k L (Ω) . β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | ℓ |≤ n | g ℓ ( ω ) | − h ℓ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g k ( ω ) h k i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | ℓ |≤ n | g ℓ ( ω ) | − h ℓ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω) (cid:13)(cid:13)(cid:13)(cid:13) g k ( ω ) h k i (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) , which by using the fact that | g n | − . X | ℓ |≤ n h ℓ i ! / h k i . h k i . (5.23)By analogous arguments as for (5.23), we deduce that for n ≥ m , we have k I n ( k ) − I m ( k ) k L (Ω) . β X m< | ℓ |≤ n h ℓ i ! / h k i . h m i θ h k i , (5.24)for some θ > Analysis of II n ( k ). Let us first compute k II n ( k ) k L (Ω) = 1 β Z X | ℓ |≤ n, | ℓ |≤ n, | ℓ |≤ nℓ − ℓ + ℓ = k,ℓ = ℓ X | ℓ ′ |≤ n, | ℓ ′ |≤ n, | ℓ ′ |≤ nℓ ′ − ℓ ′ + ℓ ′ = k,ℓ ′ = ℓ ′ b V ( ℓ − ℓ ) b V ( ℓ ′ − ℓ ′ ) × g ℓ ( ω ) g ℓ ( ω ) g ℓ ( ω ) h ℓ i h ℓ i h ℓ i g ℓ ′ ( ω ) g ℓ ′ ( ω ) g ℓ ′ ( ω ) h ℓ ′ i h ℓ ′ i h ℓ ′ i dω . (5.25)We can use Wick’s theorem to deduce that(5.25) ≤ A n ( k ) + B n ( k ) + C n ( k ) , (5.26) where A n ( k ) := 1 β X | ℓ |≤ n, | ℓ |≤ n, | ℓ |≤ nℓ − ℓ + ℓ = k,ℓ = ℓ (cid:0) b V ( ℓ − ℓ ) (cid:1) h ℓ i h ℓ i h ℓ i (5.27) B n ( k ) := 1 β X | ℓ |≤ n, | ℓ |≤ n, | ℓ |≤ nℓ − ℓ + ℓ = k,ℓ = ℓ b V ( ℓ − ℓ ) b V ( ℓ − ℓ ) 1 h ℓ i h ℓ i h ℓ i (5.28) C n ( k ) := 1 β X | ℓ |≤ n, | ℓ ′ |≤ n b V ( k − ℓ ) b V ( k − ℓ ′ ) 1 h k i h ℓ i h ℓ ′ i . (5.29)We now analyse the cases d = 2 and d = 3 separately. Analysis of II n ( k ) when d = 2. We have A n ( k ) . β X ℓ X ℓ ,ℓ ℓ − ℓ = k − ℓ h ℓ i h ℓ i ! (cid:0) b V ( k − ℓ ) (cid:1) h ℓ i . X ℓ log h k − ℓ ih k − ℓ i (cid:0) b V ( k − ℓ ) (cid:1) h ℓ i . X ℓ h k − ℓ i h ℓ i . log h k ih k i . (5.30)Here, we used Lemma B.2 with δ = 2 and M = ρ = 0 twice and we recalled (5.8).Using b V ( ℓ − ℓ ) b V ( ℓ − ℓ ) ≤ h(cid:0) b V ( ℓ − ℓ ) (cid:1) + (cid:0) b V ( ℓ − ℓ ) (cid:1) i (5.31)in (5.28) and arguing analogously as for (5.30), we get that B n ( k ) . β log h k ih k i . (5.32)Finally, by (5.8) and Lemma B.2 with δ = ǫ and M = ρ = 0, we have C n ( k ) . β h k i X ℓ b V ( k − ℓ ) 1 h ℓ i ! . h k i (cid:18) log h k ih k i ǫ (cid:19) . h k i . (5.33)Using (5.26), (5.30), (5.32), and (5.33), we deduce that k II n ( k ) k L (Ω) . log / h k ih k i . (5.34)Let n ≥ m be given. We recall (5.22) and argue analogously as for (5.25) to write k II n ( k ) − II m ( k ) k L (Ω) = 1 β Z X | ℓ |≤ n, | ℓ |≤ n, | ℓ |≤ n max {| ℓ | , | ℓ | , | ℓ |} >mℓ − ℓ + ℓ = k,ℓ = ℓ X | ℓ ′ |≤ n, | ℓ ′ |≤ n, | ℓ ′ |≤ n max {| ℓ ′ | , | ℓ ′ | , | ℓ ′ |} >mℓ ′ − ℓ ′ + ℓ ′ = k,ℓ ′ = ℓ ′ b V ( ℓ − ℓ ) b V ( ℓ ′ − ℓ ′ ) × g ℓ ( ω ) g ℓ ( ω ) g ℓ ( ω ) h ℓ i h ℓ i h ℓ i g ℓ ′ ( ω ) g ℓ ′ ( ω ) g ℓ ′ ( ω ) h ℓ ′ i h ℓ ′ i h ℓ ′ i dω . (5.35)As in (5.26), we have (5.35) ≤ A n,m ( k ) + B n,m ( k ) + C n,m ( k ) , (5.36) IBBS MEASURES AS KMS EQUILIBRIUM STATES 37 where we modify (5.27), (5.28), and (5.29) as A n,m ( k ) := 1 β X | ℓ |≤ n, | ℓ |≤ n, | ℓ |≤ n max {| ℓ | , | ℓ | , | ℓ |} >mℓ − ℓ + ℓ = k,ℓ = ℓ (cid:0) b V ( ℓ − ℓ ) (cid:1) h ℓ i h ℓ i h ℓ i (5.37) B n,m ( k ) := 1 β X | ℓ |≤ n, | ℓ |≤ n, | ℓ |≤ n max {| ℓ | , | ℓ | , | ℓ |} >mℓ − ℓ + ℓ = k,ℓ = ℓ b V ( ℓ − ℓ ) b V ( ℓ − ℓ ) 1 h ℓ i h ℓ i h ℓ i (5.38) C n,m ( k ) := 1 β X | ℓ |≤ n, | ℓ ′ |≤ n max {| k | , | ℓ | , | ℓ ′ |} >m b V ( k − ℓ ) b V ( k − ℓ ′ ) 1 h k i h ℓ i h ℓ ′ i . (5.39)We observe that for any ρ ∈ (0 , A n,m ( k ) . β,ρ log h k ih k i − ρ h m i ρ . (5.40)In order to obtain (5.40), we argue similarly as for (5.30). If max {| ℓ | , | ℓ |} > m , in our firstapplication of Lemma B.2, we take M = m . We then use (5.8) and argue as for (5.30). If | ℓ | > m , we take M = 0 in the first application and M = m in the second application of LemmaB.2.Similarly, for any ρ ∈ (0 , B n,m ( k ) . β,ρ log h k ih k i − ρ h m i ρ . (5.41)Finally, for any ρ ∈ (0 , ǫ ), we have C n,m ( k ) . β,ρ log h k ih k i − ρ h m i ρ . (5.42)In order to deduce (5.42), we need to consider the contributions max {| ℓ | , | ℓ ′ |} > m and | k | > m separately. If max {| ℓ | , | ℓ ′ |} > m , we argue as for (5.33), but in one of the applications of LemmaB.2 we take M = m . If | k | > m , then (5.42) follows from (5.33).Using (5.36), (5.40), (5.41), and (5.42), it follows that for θ > k II n ( k ) − II m ( k ) k L (Ω) . β,θ log / h k ih k i − θ h m i θ . (5.43)Recalling (5.22) and using (5.18) followed by Lemma 5.2 in (5.17), we have that for θ > k∇ h In ( u ) − ∇ h Im ( u ) k L p ( µ β, ; H − s ) . p X k, | k |≤ m h k i − s k I n ( k ) − I m ( k ) k L (Ω) + X k, | k |≤ m h k i − s k II n ( k ) − II m ( k ) k L (Ω) ! / + X k,m< | k |≤ n h k i − s k I n ( k ) k L (Ω) + X k,m< | k |≤ n h k i − s k II n ( k ) k L (Ω) ! / . s,β,θ h m i θ . (5.44)Here, we used (5.23), (5.24), (5.34), (5.43), and the assumption that s >
0. Therefore, ( h In ( u )) isa Cauchy sequence in L p ( µ β, ; H − s ). Analysis of II n ( k ) when d = 3. We now show that for d = 3, (5.34) and (5.43) get replaced by k II n ( k ) k L (Ω) . β h k i ǫ (5.45)and k II n ( k ) − II m ( k ) k L (Ω) . β,θ h k i ǫ − θ h m i θ , (5.46)for θ ∈ [0 , ), whenever n ≥ m . Using (5.23), (5.24), (5.45), (5.46), the fact that s > , andarguing as in (5.44), we indeed deduce that ( h In ( u )) is a Cauchy sequence in L p ( µ β, ; H − s ).Our goal is now to show (5.45) and (5.46). With A n ( k ) , B n ( k ) , C n ( k ) defined as in (5.27),(5.28), (5.29), we have the following estimates. By arguing similarly as in (5.30), we have A n ( k ) . β X ℓ h k − ℓ i (cid:0) b V ( k − ℓ ) (cid:1) h ℓ i . X ℓ h k − ℓ i ǫ h ℓ i . h k i ǫ . (5.47)We again use (5.31) in (5.28) and argue as in the proof of (5.47) to deduce that B n ( k ) . β h k i ǫ . (5.48)Here, we first used Lemma B.3 with δ = 0 and M = ρ = 0. Then, we recalled (5.8) and weused Lemma B.3 with δ = 2 + ǫ and M = ρ = 0.Similarly, when d = 3, (5.8) and Lemma B.3 with δ = ǫ and M = ρ = 0 imply that X ℓ b V ( k − ℓ ) 1 h ℓ i . X ℓ h k − ℓ i ǫ h ℓ i . h k i ǫ . (5.49)Using (5.49) and arguing as in (5.33), we get that C n ( k ) . β h k i ǫ . (5.50)We hence obtain (5.45) from (5.47), (5.48), and (5.50). We now show (5.46). With A n,m ( k ), B n,m ( k ), C n,m ( k ) defined as in (5.37), (5.38), (5.39), we have the following estimates.We observe that for any ρ ∈ [0 , A n,m ( k ) . β,ρ h k i ǫ − ρ h m i ρ . (5.51)Here, we argue as for (5.51). By analogous arguments, we have B n,m ( k ) . β,ρ h k i ǫ − ρ h m i ρ . (5.52)Finally, modifying the proof of (5.50) analogously as in (5.42), we have that for any ρ ∈ [0 , C n,m ( k ) . β,ρ h k i ǫ − ρ h m i ρ . (5.53)We hence obtain (5.46) from (5.51), (5.52), (5.53). (iii) The NLS equation on T . We use the Sobolev embedding H ζ ( T ) ֒ → L q ( T ) for ζ ∈ ( − q , )in (5.11) to deduce that h I ( u ) . ζ k u k qH ζ . We then deduce that h I ( u ) ∈ L p ( µ β, ) as in part (i). A direct calculation shows that ∇ h I ( u ) = | u | q − u . IBBS MEASURES AS KMS EQUILIBRIUM STATES 39
By using Lemma B.1 q − α , and by arguing as in (5.16), we get thatfor some ζ ′ ∈ ( − s, ) k∇ h I ( u ) k L p ( µ β, ; H − s ) . s,q,ζ ′ (cid:18) Z k u k ( q − pH ζ ′ dµ β, (cid:19) /p < ∞ . We conclude that h I ∈ D ,p ( µ β, ). (iv) The NLS equation on T . In case h I is given by (5.13), the result is a consequence of [43,Proposition 1.1] and [43, Proposition 1.3]. Indeed, it is proved there that ( h In ) n ∈ N and ( ∇ h In ) n ∈ N are Cauchy sequences in L p ( µ β, ) and L p ( µ β, ; H − s ) respectively. We omit the details (cid:3) Thus, as a consequence of Theorems 4.11 and 4.14, one concludes that the above NLS dynamicalsystems on the torus T d admit each a unique KMS state given by the Gibbs measure µ β = z − β e − βh I µ β, with z β an appropriate normalization constant. Note that uniqueness here is in thesense of Theorem 4.14 and it is among measures that are absolutely continuous with respect to µ β, with a density ̺ ∈ D , ( µ β, ). Finally, one remarks that such a result suggests the study ofgeneral dynamical systems satisfying the condition h I ∈ D , ( µ β, ) without relying on the preciseform of h I . The above discussion is summarized below. Corollary 5.3.
Let β > and s > − / satisfying (5.2) . Consider h I to be one of the nonlinearBorel functionals h I : H − s → R of the Hartree or NLS equations given respectively by (5.3) , (5.9) , (5.11) and (5.13) . Then:(i) The Gaussian measure µ β, is the unique ( β, X ) -KMS state of the vector field X = − iA .(ii) The nonlinear functionals h I ∈ D ,p ( µ β, ) and e − βh I ∈ L p ( µ β, ) for all ≤ p < ∞ .(iii) The Gibbs measure µ β = e − βh I R H − s e − βh I dµ β, µ β, is a stationary solution of the Liouville equation (2.8) .(iv) The Gibbs measure µ β is the unique ( β, X ) -KMS state, for the vector field X = − i ( A + ∇ h I ) , among all the absolutely continuous measures µ with respect to µ β, such that dµdµ β, ∈ D , ( H − s ) .Remark . The previous corollary extends also to the NLS nonlinearity (5.13) on 2-dimensionalcompact Riemannian manifolds without boundary or on bounded domains in R following [43,Proposition 4.3, 4.5 and 4.6]. We omit the details.5.2. Nonlinear wave (Klein-Gordon) equations.
One can study Gibbs measures for non-linear wave (Klein-Gordon) equations by means of probabilistic and PDE methods, see e.g.[14, 15, 16, 17, 23, 40, 44] and the references therein. The nonlinearities that are usually consid-ered are similar to the ones recalled in the above Subsection 5.1. The main difference comes fromthe use of the real structure of fields rather than the complex one. Specifically, we consider thenonlinear wave (Klein-Gordon) equation on the torus T d , d = 1 , , ( ∂ t u + u − ∆ u + ∇ h I ( u ) = 0 u | t =0 = f , ∂ t u | t =0 = f (5.54)where ∇ h I is the Malliavin derivative of some functional h I that we will specify below. Beforeproceeding, we explain how the nonlinear wave equation (5.54) fits within the general frameworkof Subsection 4.3. Framework for wave equations:
Consider the Hilbert space H = L R ( T d ) ⊕ L R ( T d ) where L R standsfor real-valued square integrable functions and define the Sobolev spaces for γ ∈ R as H γ = H γ R ( T d ) ⊕ H γ − R ( T d ) . (5.55)The nonlinear wave equation (5.54) takes the form ∂ t (cid:18) uv (cid:19) = X (cid:18) uv (cid:19) = X (cid:18) uv (cid:19) + (cid:18) −∇ h I ( u ) (cid:19) with the vector field X given by X ≡ (cid:20) ∆ − (cid:21) = J z }| {(cid:20) − (cid:21) A z }| {(cid:20) − ∆ + (cid:21) (5.56)with J is a compatible complex structure on H and A is a positive linear operator. Remark that H is endowed with a canonical symplectic structure induced by J and given by σ ( u ⊕ v, u ′ ⊕ v ′ ) = h u ⊕ v, J u ′ ⊕ v ′ i L R ⊕ L R . Moreover, H can be considered as a complex Hilbert space according to (2.19)-(2.20) and thecouple ( J, A ) defines a complex linear Hamiltonian system as in (4.16)-(4.17). Now, the vectorfield X can be written as X (cid:18) uv (cid:19) = J (cid:18) A (cid:18) uv (cid:19) + (cid:18) ∇ h I ( u )0 (cid:19)(cid:19) . (5.57)Although the operator A does not have a compact resolvent, it still possible to do the sameanalysis as before. Indeed, one uses the Sobolev spaces in (5.55) instead of the definition givenin Subsection 4.2. Results for wave equations:
The Gaussian Gibbs measure µ β, in this case is defined as a productmeasure such that dµ β, ( u, v ) = dµ β, ( u ) dµ β, ( v ) , with µ β, and µ β, are the Gaussian measures on the distribution space D ′ ( T d ) with covarianceoperators β − ( − ∆ + ) − and β − respectively. The existence and uniqueness of such measuresfollow from Corollary 3.2. According to Theorem 4.4, for s > − / µ β, coincides with the centered Gaussian measure with covariance operator β − (cid:20) ( − ∆ + ) − (1+ s )
00 ( − ∆ + ) − (1+ s ) (cid:21) on the Sobolev space H − s given in (5.55). In particular, µ β, is a Borel probability measure over H − s . Therefore, Theorem 3.3 shows that µ β, is a ( β, X )-KMS state with the vector field X in (5.56). Moreover, using the arguments of Subsection 5.1 one can define rigourously the Gibbsmeasure of the nonlinear wave equation as µ β = 1 z β e − βh I µ β, ⊗ µ β, , where z β is a normalization constant and h I is one of the following possibilities h I = (5.3) or (5.11) if d = 1 , (5.9) or (5.13) if d = 2 , (5.9) if d = 3 . (5.58) IBBS MEASURES AS KMS EQUILIBRIUM STATES 41
Note that since the fields are real, the Wick-ordered power nonlinearity : u rn : in (5.12) is definedin this case through Hermite polynomials H r instead of Laguerre polynomials,: u rn : = σ r/ n,β H r (cid:18) σ − / n,β u n (cid:19) . (5.59)In particular, the nonlinearity h I depends only on the u variable. Furthermore, in (5.18), we addthe condition g − k = g k . (5.60)By arguing analogously as in the proof of Proposition 5.1, we deduce that h I belongs to theGross-Sobolev spaces D ,p ( µ β, ) or D ,p ( µ β, ) for all 1 ≤ p < ∞ (note that additional condition(5.60) does not increase the L p (Ω) norms of the relevant quantities). Consequently the Malliavinderivatives ∇ h I are well-defined. Thus, we have at hand all the ingredients to apply Theorem4.11 and 4.14. So, all the statements (i)-(iv) of Corollary 5.3 with the appropriate modificationshold true for the nonlinear wave equation (5.54) with the nonlinearities (5.58). In particular, weemphasize the following result where H − s is defined according to (5.55) and s satisfying (5.2). Corollary 5.5.
The Gibbs measure µ β = z β e − βh I µ β, ⊗ µ β, is the unique ( β, X ) -KMS state,for the vector field X = J A + J ( ∇ h I ⊕ , among all the absolutely continuous measures µ withrespect to µ β, such that dµdµ β, ∈ D , ( H − s ) . As in Remark 5.4, the above Corollary extends to wave equations with the nonlinearity (5.13)defined on 2-dimensional compact Riemannian manifolds without boundary or on bounded do-mains in R following the discussion in [44, Section 1.2].5.3. The focusing NLS and the local KMS condition.
Consider the focusing NLS equationon the one-dimensional torus T , i∂ t u = − ∆ u + u − | u | p − u , for 4 ≤ p ≤ h I ( u ) = − p Z T | u ( x ) | p dx , which is similar to (5.11) with a negative sign corresponding to a focusing nonlinearity. Recallthat here one has the same framework as in Subsection 5.1 with s = 0. Although it is notpossible to define a global Gibbs measure in this case because of the negative sign in the front ofthe nonlinear term h I , it is proved in [9, 37] that the Gibbsian local measure µ β = 1 z β,R e − βh I ( · ) [0 ,R ] (cid:0) k · k L ( T ) (cid:1) µ β, , z β,R := Z L ( T ) e − βh I ( · ) [0 ,R ] (cid:0) k · k L ( T ) (cid:1) dµ β, (5.61)is well-defined for some arbitrary constant R > ≤ p < R > p = 6. Lemma 5.6.
Let β > be given and χ ∈ C ∞ c ( R ) such that χ ( x ) = 0 for all | x | ≥ R where R is as in (5.61) . Then the Radon-Nikodym derivative of the measure µ β in (5.61) with respect to µ β, satisfies dµ β dµ β, ∈ D , ( µ β, ) . Proof.
Note that | χ ( · ) | ≤ c [0 ,R ] ( · ) for some constant c >
0. Hence, the statement G ( · ) = e − βh I ( · ) χ ( k · k L ( T ) ) ∈ L q ( µ β, ) for all q ≥
1, is essentially proved in [9, 37]. For an expositorysummary of the construction, we refer the reader to [47]. Using Lemma 5.1 and the approximationidea in Lemma 4.13, one shows ∇ G ( u ) = − β ∇ h I ( u ) G ( u ) + 2 e − βh I ( · ) χ ′ ( k · k L ( T ) ) u ∈ L ( µ β, ; L ( T )) . (5.62) In fact, take θ k the same sequence of functions as in the proof of Theorem 4.11 and G k ( · ) = θ k ( h I ( · )) χ ( k · k L ( T ) ) . Since one knows by Lemma 5.1 that the functionals k · k L ( T ) and h I belong to D ,p ( µ β, ), thenusing Lemma A.2 and the chain rule (2.4) one proves in L ( µ β, ; L ( T )), ∇ G k ( u ) = θ ′ k ( h I ( u )) ∇ h I ( u ) χ ( k u k L ( T ) ) + 2 θ k ( h I ( u )) χ ′ ( k u k L ( T ) ) u . (5.63)Hence, dominated convergence with the estimates (4.41) give the following limits in L ( µ β, ) and L ( µ β, ; L ( T )) respectively,lim k θ ′ k ( h I ( · )) χ ( k · k L ( T ) ) = − βe − βh I ( · ) χ ( k · k L ( T ) ) , and lim k θ k ( h I ( u )) χ ′ ( k u k L ( T ) ) u = e − βh I ( · ) χ ′ ( k · k L ( T ) ) u . Thus, the above limits with the H¨older inequality yield the claimed identity (5.62) when carrying k → ∞ in the equality (5.63). (cid:3) We show here that such a measure µ β satisfies a local form of the KMS condition. Proposition 5.7.
Let β > be given and let µ β be the Borel measure defined in (5.61) . Let χ ∈ C ∞ c ( R ) be such that χ ( x ) = 0 for all | x | ≥ R with R the radius given in (5.61) . Then, taking F ( · ) = χ ( k · k L ( T ) ) , we have that for all G ∈ C ∞ c,cyl ( L ( T )) Z L ( T ) { F, G } dµ β = β Z L ( T ) ℜ e (cid:10) ∇ F ( u ) , i ∆ u − iu − i ∇ h I ( u ) (cid:11) L ( T ) G ( u ) dµ β . (5.64) Proof.
It follows by applying the integration by parts formula in Proposition A.1. Since G ∈ C ∞ c,cyl ( L ( T )) there exists n ∈ N and ψ ∈ C ∞ c ( R n ) such that G ( u ) = ψ ( π n u ). So, using thePoisson bracket formula (4.26) and the fact that ∂ e j F ( u ) = ∂ f j F ( u ) = 0 if k u k L ( T ) > R , oneshows Z L ( T ) { F, G } dµ β = 1 z β,R n X j =1 Z L ( T ) (cid:18) ∂ e j F ( u ) ∂ f j G ( u ) − ∂ e j G ( u ) ∂ f j F ( u ) (cid:19) e − βh I ( u ) dµ β, , Thanks to Lemma 5.6, one knows that ∂ e j F ( · ) e − βh I ( · ) ∈ D , ( µ β, ). Therefore, applying Propo-sition A.1 with the function G in (A.1) replaced by e F j = ∂ e j F e − βh I ∈ D , ( µ β, ) and F by G ∈ C ∞ c,cyl ( L ( T )), one obtains Z L ( T ) ∂ e j F ( u ) e − βh I ( u ) h∇ G ( u ) , f j i dµ β, = Z L ( T ) G ( u ) (cid:18) − ∂ f j e F ( u ) + β e F j ( u ) h u, Af j i (cid:19) dµ β, , and similarly Z L ( T ) ∂ f j F ( u ) e − βh I ( u ) h∇ G ( u ) , e j i dµ β, = Z L ( T ) G ( u ) (cid:18) − ∂ e j ⌣ F j ( u ) + β ⌣ F j ( u ) h u, Ae j i (cid:19) dµ β, , where ⌣ F j = ∂ f j F e − βh I ∈ D , ( µ β, ) and A is given by (5.1). Remark that the proof of Lemma5.6 yields ∂ f j e F j ( u ) = − β∂ f j h I ( u ) ∂ e j F ( u ) e − βh I ( u ) + ∂ f j ∂ e j F ( u ) e − βh I ( u ) ,∂ e j ⌣ F j ( u ) = − β∂ e j h I ( u ) ∂ f j F ( u ) e − βh I ( u ) + ∂ e j ∂ f j F ( u ) e − βh I ( u ) . IBBS MEASURES AS KMS EQUILIBRIUM STATES 43
Hence, one concludes Z L ( T ) { F, G } dµ β = βz β,R n X j =1 Z L ( T ) G ( u ) (cid:18) ∂ e j F ( u ) ∂ f j h I ( u ) − ∂ e j h I ( u ) ∂ f j F ( u )+ ∂ e j F ( u ) h Au, f j i − ∂ f j F ( u ) h Au, e j i (cid:19) e − βh I ( u ) dµ β, = βz β,R Z L ( T ) G ( u ) (cid:18)(cid:10) ∇ F ( u ) , − i ∇ h I ( u ) (cid:11) + (cid:10) − iAu, ∇ F ( u ) (cid:11)(cid:19) e − βh I ( u ) dµ β, . Thus, recalling that h· , ·i = ℜ e h· , ·i L ( T ) , one proves the local KMS condition. (cid:3) Remark . It is not difficult to see that one can replace the assumption on F in Proposition 5.7with F ∈ C b ( L ( T )) such that F ( u ) = 0 for all u with k u k L ( T ) > R ′ and R ′ ∈ (0 , R ) arbitrary.Namely, taking χ ∈ C ∞ c ( R ) such that χ ( x ) = 1 for all | x | ≤ R ′ and χ ( x ) = 0 for all | x | > R , thenwe have F e − βh I = F χ ( k · k L ( T ) ) e − βh I . So, the boundedness of F and Lemma 5.6 shows that F e − βh I ∈ L ( µ β, ). Moreover, the product rule yields ∇ [ F e − βh I ] = ∇ F χ ( k · k L ( T ) ) e − βh I + F ∇ [ χ ( k · k L ( T ) ) e − βh I ] ∈ L ( µ β, ; L ( T )) . Thus, one concludes that
F e − βh I ∈ D , ( µ β, ) which is the main point in the proof of Proposition5.7. Appendix A. Malliavin calculus
For completeness, we give a short overview of some useful tools from Malliavin calculus. Inparticular, the following integration by parts formula is useful.
Proposition A.1.
Let F ∈ C ∞ b,cyl ( H − s ) and G ∈ D , ( µ β, ) or F ∈ D , ( µ β, ) and G ∈ C ∞ b,cyl ( H − s ) . Then for any ϕ ∈ H , Z H − s G ( u ) h∇ F ( u ) , ϕ i dµ β, = Z H − s F ( u ) (cid:18) − h∇ G ( u ) , ϕ i + β G ( u ) h u, Aϕ i (cid:19) dµ β, . (A.1) Proof.
First, one proves that for any R ∈ C ∞ b,cyl ( H − s ) and ϕ ∈ span R { e j , f j ; j = 1 , . . . , k } forsome k ∈ N , we have Z H − s h∇ R ( u ) , ϕ i dµ β, = β Z H − s h u, Aϕ i R ( u ) dµ β, . (A.2)Indeed, the above equality (A.2) follows from equation (4.32) in Lemma 4.6 and a standardintegration by parts on R k . On the one hand taking R = ψ ◦ π n for some n ∈ N with n ≥ k and ψ ∈ C ∞ b ( R n ) and integrating by parts again, we have Z H − s h∇ R ( u ) , ϕ i dµ β, = n X j =1 Z R n (cid:18) h e j , ϕ i ∂ (1) j ψ + h f j , ϕ i ∂ (2) j ψ (cid:19) dν nβ, = β Z R n (cid:28) n X j =1 λ j ( x j e j + y j f j ) , ϕ (cid:29) ψ ( x , . . . , x n ; y , . . . , y n ) dν nβ, , (A.3) where ν nβ, is the Gaussian measure in Lemma 4.6. On the other hand, we have β Z H − s h u, Aϕ i R ( u ) dµ β, = β k X j =1 Z R n λ j (cid:18) h u, e j ih e j , ϕ i + h u, f j ih f j , ϕ i (cid:19) ψ ( π n u ) dµ β, = β Z R n (cid:28) k X j =1 λ j ( x j e j + y j f j ) , ϕ (cid:29) ψ ( x , . . . , x n ; y , . . . , y n ) dν nβ, , (A.4)where π n is the mapping in (4.21). Since n ≥ k , we have that (cid:28) k X j =1 λ j ( x j e j + y j f j ) , ϕ (cid:29) = (cid:28) k X j =1 λ j ( x j e j + y j f j ) , ϕ (cid:29) . (A.5)We hence deduce (A.2) from (A.3), (A.4), and (A.5). Now, the identity (A.2) extends to all ϕ ∈ H thanks to a standard approximation argument and Remark 4.5-(iii). The integration byparts formula (A.1), for any F, G ∈ C ∞ b,cyl ( H − s ), is a straightforward consequence of (A.2) with R = F G ∈ C ∞ b,cyl ( H − s ) and the product rule (2.4). Finally, (A.1) extends to all G ∈ D , ( µ β, )(resp. F ∈ D , ( µ β, )) by the density of C ∞ b,cyl ( H − s ) in D , ( µ β, ) with respect to its graph norm(4.35). (cid:3) Lemma A.2.
Let χ ∈ C b ( R ) and F ∈ D ,p ( µ β, ) , for p ∈ [1 , ∞ ) , then χ ( F ) ∈ D ,p ( µ β, ) and ∇ χ ( F ) = χ ′ ( F ) ∇ F .
Proof.
Suppose that we are given F n ∈ C ∞ c,cyl ( H − s ), n ∈ N , a sequence such that F n → F in D ,p ( µ β, ). Then the chain rule yields ∇ χ ( F n ) = χ ′ ( F n ) ∇ F n . Since F n → F in L p ( µ β, ), there exists a subsequence ( F n k ) k such that F n k → F µ β, -almosteverywhere. Therefore, one obtainslim k χ ′ ( F n k ) ∇ F n k = χ ′ ( F ) ∇ F , in L p ( µ β, ) and ( χ ( F n k )) k is a Cauchy sequence in D ,p ( µ β, ) which is a Banach space. (cid:3) The space P ( H − s ) is defined as the set of smooth cylindrical functions F : H − s → R such thatthere exists n ∈ N and F ( u ) = ϕ ( h u, e i , . . . , h u, e n i ; h u, f i , . . . , h u, f n i ) for all u ∈ H − s , where ϕ ∈ C ∞ ( R n ) is such that for all multi-indices α ∈ N n , there exists constants m α , c α ≥ | ∂ α ϕ ( x ) | ≤ c α (1 + | x | ) m α for all x ∈ R n . Lemma A.3.
The following inclusions hold true for all p ∈ [1 , ∞ ) , C ∞ b,cyl ( H − s ) ⊆ P ( H − s ) ⊆ D ,p ( µ β, ) . Proof.
Recall the explicit form of the centered Gaussian measures ν nβ, = ( π n ) ♯ µ β, defined over R n and given in Lemma 4.6. Since all the moments of such measures are finite, one concludesfor all m ≥ Z H − s (cid:0) n X j =1 h u, e j i + h u, f j i (cid:1) m dµ β, = Z R n (cid:0) n X j =1 x j + y j (cid:1) m dν nβ, < ∞ . (A.6)Hence, P ( H − s ) is included in all the spaces L p ( µ β, ) for all p ∈ [1 , ∞ ). Since the gradient of F ∈ P ( H − s ) is also given by the identity (4.23), one obtains using the estimates (A.6) that ∇ F ∈ L p ( µ β, ; H − s ) for all p ∈ [1 , ∞ ). (cid:3) IBBS MEASURES AS KMS EQUILIBRIUM STATES 45
The following well-known result asserts that a random variable whose Malliavin derivative iszero, is almost surely constant.
Proposition A.4.
Let F ∈ D , ( µ β, ) such that ∇ F = 0 for µ β, -almost surely. Then F isconstant µ β, -almost surely.Proof. It is a straightforward consequence of the Wiener chaos decomposition (see e.g. [42,Proposition 1.2.2 and 1.2.5]). In fact, consider H n ( · ) to be the n th Hermite polynomial andΨ a,b ( · ) = √ a ! b ! ∞ Y j =1 H a j (cid:0) h· , ˜ e j i (cid:1) ∞ Y j =1 H b j (cid:0) h· , ˜ f j i (cid:1) , with ˜ e j = p βλ j e j , ˜ f j = p βλ j f j and a = ( a j ) j ∈ N , b = ( b j ) j ∈ N such that a j , b j are non-negativeintegers with a j = b j = 0 except for a finite number of indices. Then such a family { Ψ a,b } formsan orthonormal basis of the space L ( µ β, ). Furthermore, a standard computation yields (cid:10) ∇ Ψ a,b , ∇ Ψ a ′ ,b ′ (cid:11) L ( µ β, ; H − s ) = β δ a,a ′ δ b,b ′ ∞ X j =1 λ − sj ( a j + b j ) . (A.7)So, using the orthogonal decomposition with respect to the basis { Ψ a,b } and (A.7), one proves k∇ F k L ( µ β, ; H − s ) = X a,b h F, Ψ a,b i (cid:18) ∞ X j =1 λ − sj ( a j + b j ) (cid:19) , (A.8)for all F in the algebraic vector space spanned by { Ψ a,b } . Then a density argument extends (A.8)to all F ∈ D , ( µ β, ). Hence, ∇ F = 0 almost surely implies that h F, Ψ a,b i = 0 for all a = 0 or b = 0. Thus, one concludes that F ( · ) = h F, Ψ , i Ψ , ( · ) = c for some real constant c . (cid:3) Appendix B. Proofs of auxiliary facts from Section 5
We note a product estimate in one-dimension. This is used in part (i) of the proof of Proposition5.1.
Lemma B.1.
Let ζ ∈ (0 , / and α ∈ (0 , be given. The following estimate holds on T . k f g k H ζ . ζ,α k f k H ζ + α k g k H − α + k f k H − α k g k H ζ + α . Proof.
Let D ζ denote the Fourier multiplier with symbol h k i ζ and let F − denote the inverseFourier transform. We note that k f g k H ζ ∼ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i ζ X k ′ ∈ Z ˆ f ( k − k ′ ) ˆ g ( k ′ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h k i ζ X k ′ ∈ Z | ˆ f ( k − k ′ ) | | ˆ g ( k ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ k . ζ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ′ ∈ Z h k − k ′ i ζ | ˆ f ( k − k ′ ) | | ˆ g ( k ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k ′ ∈ Z | ˆ f ( k − k ′ ) | h k ′ i ζ | ˆ g ( k ′ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ k ∼ ζ (cid:13)(cid:13)(cid:13)(cid:16) D ζ F − | ˆ f | (cid:17) F − | ˆ g | (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13) F − | ˆ f | (cid:16) D ζ F − | ˆ g | (cid:17) (cid:13)(cid:13)(cid:13) L , which by H¨older’s inequality is ≤ (cid:13)(cid:13) D ζ F − | ˆ f | (cid:13)(cid:13) L − α (cid:13)(cid:13) F − | ˆ g | (cid:13)(cid:13) L α + (cid:13)(cid:13) F − | ˆ f | (cid:13)(cid:13) L α (cid:13)(cid:13) D ζ F − | ˆ g | (cid:13)(cid:13) L − α . (B.1)By using Sobolev embedding, we have H α ( T ) ֒ → L − α ( T ) , H − α ( T ) ֒ → L α ( T ) . (B.2)Substituting (B.2) into (B.1) and using the fact that L -based Sobolev norms are invariant undertaking absolute values of the Fourier transform, we deduce the claim. (cid:3) Let us now prove two useful discrete convolution estimates. In the estimates below, we aresumming over elements in Z d (with appropriate constraints). Lemma B.2.
Let d = 2 . Let δ ∈ (0 , and M ≥ be given. For all n ∈ Z and all ρ ∈ [0 , δ ) , wehave X k + ℓ = n max {| k | , | ℓ |}≥ M h k i δ h ℓ i . δ,ρ log h n ih n i δ − ρ h M i ρ . (B.3) Lemma B.3.
Let d = 3 . Let δ ≥ and M ≥ be given. For all n ∈ Z and all ρ ∈ [0 , , wehave X k + ℓ = n max {| k | , | ℓ |}≥ M h k i δ h ℓ i . δ,ρ h n i δ − ρ h M i ρ . Proof of Lemma B.2.
We need to consider two cases, depending on the relative sizes of | k | and | ℓ | . Case A: | k | ≥ | ℓ | .In this case, we are estimating X ℓ, | n − ℓ |≥ M h n − ℓ i δ h ℓ i . (B.4)We now need to consider three subcases, depending on the size of | ℓ | . Subcase A1: | ℓ | ≤ | n | .Note that then by the triangle inequality | n − ℓ | ≥ | n | . In particular, since | n − ℓ | ≥ M , we havethat 1 h n − ℓ i δ . δ,ρ h n i δ − ρ h M i ρ . (B.5)Furthermore, we have X | ℓ |≤ | n | h ℓ i . log h n i . (B.6)Combining (B.5) and (B.6), we deduce that the contribution to (B.4) from this subcase satisfiesthe bound in (B.3). Subcase A2: | n | < | ℓ | < | n | .In this subcase, we have | ℓ | ∼ | n | . Furthermore, we have M ≤ | n − ℓ | ≤ | n | . Putting everythingtogether, we get that if δ ∈ (0 , ≤ X ℓ, | n − ℓ |≤ | n | h n − ℓ i δ ! h n i . δ h n i − δ h n i . δ,ρ h n i δ − ρ h M i ρ . (B.7)If δ = 2, the upper bound gets modified tolog h n ih n i . ρ log h n ih n i − ρ h M i ρ . (B.8)Note that (B.7) and (B.8) are acceptable upper bounds. Subcase A3: | ℓ | ≥ | n | .We now have | n − ℓ | ∼ | ℓ | & M . Therefore, the contribution to (B.4) is . δ X ℓ, | ℓ | & max { M, | n |} h ℓ i δ . δ,ρ h n i δ − ρ h M i ρ , IBBS MEASURES AS KMS EQUILIBRIUM STATES 47 which is an acceptable upper bound.
Case B: | k | ≤ | ℓ | .Since k + ℓ = n , we have that | ℓ | ≥ | n | . Hence in this case, we are estimating X ℓ, | ℓ |≥ max { M, | n | } h n − ℓ i δ h ℓ i . (B.9)We now need to consider two subcases, depending on the size of | ℓ | . Subcase B1: | ℓ | ≤ | n | .In this case, we have that | ℓ | ∼ | n | & M and | n − ℓ | ≤ | n | . Therefore, the contribution to (B.9)is . X ℓ, | n − ℓ |≤ | n | h n − ℓ i δ ! h n i . δ,ρ log h n ih n i δ − ρ h M i ρ . Here, we argued as in (B.7) and (B.8).
Subcase B2: | ℓ | > | n | .In this case, we have that | n − ℓ | ∼ | ℓ | & M . Hence, the contribution to (B.9) is . δ X ℓ, | ℓ |≥ max { M, | n | } h ℓ i δ . δ,ρ h n i δ − ρ h M i ρ . (cid:3) Proof of Lemma B.3.
The proof is similar to that of Lemma B.2. We just outline the maindifferences.
Case A: | k | ≥ | ℓ | .In this case, (B.4) gets replaced by X ℓ, | n − ℓ |≥ M h n − ℓ i δ h ℓ i . (B.10)We consider three subcases as earlier. Subcase A1: | ℓ | ≤ | n | .Instead of (B.5) and (B.6), we use 1 h n − ℓ i δ . δ,ρ h n i δ − ρ h M i ρ . and X | ℓ |≤ | n | h ℓ i . h n i , which give us the desired bound. Subcase A2: | n | < | ℓ | < | n | .Here, we note that X ℓ, | n − ℓ |≤ | n | h n − ℓ i δ . δ h n i − δ (B.11)and we argue similarly as in (B.7). Subcase A3: | ℓ | ≥ | n | .We argue as in Subcase A3 in the proof of Lemma B.2 and obtain that the contribution to (B.10)is . δ X ℓ, | ℓ | & max { M, | n |} h ℓ i δ . δ,ρ h n i δ − ρ h M i ρ . Case B: | k | ≤ | ℓ | .Instead of (B.9), we need to estimate X ℓ, | ℓ |≥ max { M, | n | } h n − ℓ i δ h ℓ i . (B.12)We consider two subcases as earlier. Subcase B1: | ℓ | ≤ | n | .The contribution to (B.12) is . X ℓ, | n − ℓ |≤ | n | h n − ℓ i δ ! h n i . δ,ρ h n i δ − ρ h M i ρ . Here, we recalled (B.11).
Subcase B2: | ℓ | > | n | .The contribution to (B.9) is . δ X ℓ, | ℓ |≥ max { M, | n | } h ℓ i δ . δ,ρ h n i δ − ρ h M i ρ . (cid:3) We present the proof of (5.9) for d = 2 and V as in (5.8). Let us note that this proof can bededuced from [11] and we just present it here for the convenience of the reader. Proof of (5.9) . We recall (5.6) and rewrite (5.7) h In ( u ) ≡ h In,β ( u ) = 14 b V (0) (cid:2) (: | u n | :) b (0) (cid:3) + 14 X k =0 (cid:0) | u n | (cid:1)b ( k ) (cid:0) | u n | (cid:1)b ( − k ) b V ( k )=: h I, n ( u ) + h I, n ( u ) . (B.13)We show that the sequences ( h I, n ( u )) and ( h I, n ( u )) are bounded in L p ( µ β, ). By appropriatelymodifying the proof, using Lemma B.2 and the same arguments as in part (ii) of the proof ofProposition 5.1, we get that these sequences are Cauchy in L p ( µ β, ). We omit the details of thisstep.Recalling (5.18), (5.21), and using and H¨older’s inequality, we get that k h I, n k L p ( µ β, ) . p,β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | ℓ |≤ n | g ℓ ( ω ) | − h ℓ i ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | ℓ |≤ n | g ℓ ( ω ) | − h ℓ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω) . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | ℓ |≤ n | g ℓ ( ω ) | − h ℓ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω) . X | ℓ |≤ n h ℓ i . . (B.14)Similarly, k h I, n k L p ( µ β, ) . p,β Z Ω X | ℓ |≤ n, | ℓ |≤ n, | ℓ |≤ n, | ℓ |≤ nℓ − ℓ + ℓ − ℓ =0 ,ℓ = ℓ X | ℓ ′ |≤ n, | ℓ ′ |≤ n, | ℓ ′ |≤ n, | ℓ ′ |≤ nℓ ′ − ℓ ′ + ℓ ′ − ℓ ′ =0 ,ℓ ′ = ℓ ′ b V ( ℓ − ℓ ) b V ( ℓ ′ − ℓ ′ ) × g ℓ ( ω ) g ℓ ( ω ) g ℓ ( ω ) g ℓ ( ω ) h ℓ i h ℓ i h ℓ i h ℓ i g ℓ ′ ( ω ) g ℓ ′ ( ω ) g ℓ ′ ( ω ) g ℓ ′ ( ω ) h ℓ ′ i h ℓ ′ i h ℓ ′ i h ℓ ′ i dω . X + Y , (B.15)
IBBS MEASURES AS KMS EQUILIBRIUM STATES 49 where X := X ℓ ,ℓ b V ( ℓ − ℓ ) 1 h ℓ i h ℓ i ! . (B.16)and Y := X ℓ − ℓ + ℓ − ℓ =0 b V ( ℓ − ℓ ) 1 h ℓ i h ℓ i h ℓ i h ℓ i . (B.17)In the last step, we used Wick’s theorem and we used (5.8) to bound b V . Y . We nowestimate the expressions (B.16) and (B.17).By (5.8), we have X . X ℓ h ℓ i X ℓ h ℓ − ℓ i ǫ h ℓ i ! . X ℓ log h ℓ ih ℓ i ǫ ! . . (B.18)Above, we used Lemma B.2 with δ = ǫ and M = ρ = 0 to estimate the sum in ℓ .Furthermore, we have Y = X ℓ ,ℓ b V ( ℓ − ℓ ) 1 h ℓ i h ℓ i X ℓ ,ℓ ℓ − ℓ = − ℓ + ℓ h ℓ i h ℓ i . X ℓ ,ℓ b V ( ℓ − ℓ ) 1 h ℓ i h ℓ i log h ℓ − ℓ ih ℓ − ℓ i . X ℓ ,ℓ h ℓ i h ℓ i h ℓ − ℓ i . X ℓ log h ℓ ih ℓ i . . (B.19)Above, we used Lemma B.2 with δ = 2 and M = ρ = 0 to estimate the sum in ℓ , ℓ and in ℓ .We also used (5.8). The boundedness claim now follows from (B.13), (B.14), (B.15), (B.18), and(B.19). (cid:3) References [1] Michael Aizenman, Sheldon Goldstein, Christian Gruber, Joel L. Lebowitz, and Philippe A. Martin. On theequivalence between KMS-states and equilibrium states for classical systems.
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Univ Rennes, [UR1], CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
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