Reversibility in Infinite Hamiltonian Systems with ... - CiteSeerX

Commun. Math. Phys. 189, 481 – 496 (1997)

Communications in

Mathematical Physics c Springer-Verlag 1997

Reversibility in Infinite Hamiltonian Systems with Conservative Noise? József Fritz1 , Carlangelo Liverani2 , Stefano Olla3 1

Department of Probability and Statistics, Eötvös Loránd University of Sciences, H-1088 Budapest, Múzeum krt. 6-8, Hungary. E-mail: [email protected] 2 II Universit´ a di Roma “Tor Vergata,” Dipartimento di Matematica, 00133 Roma, Italy. E-mail: [email protected] 3 Universit´ e de Cergy–Pontoise, Département de Mathématiques, 2 avenue Adolphe Chauvin, Pontoise 95302 Cergy–Pontoise Cedex, France and Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France. E-mail: [email protected] Received: 26 September 1996 / Accepted: 3 January 1997

In Memoriam Roland Dobrushin Abstract: The set of stationary measures of an infinite Hamiltonian system with noise is investigated. The model consists of particles moving in R3 with bounded velocities and subject to a noise that does not violate the classical laws of conservation, see [OVY]. Following [LO] we assume that the noise has also a finite radius of interaction, and prove that translation invariant stationary states of finite specific entropy are reversible with respect to the stochastic component of the evolution. Therefore the results of [LO] imply that such invariant measures are superpositions of Gibbs states. 0. Introduction Let denote the space of locally finite configurations ω = (qα , pa )α∈I indexed by a countable set I , that is qα , pα ∈ R3 are the position and momentum of particle α ∈ I ; the set {qα }α∈I has no limit points in R3 by assumption. The classical dynamics of the system is governed by a formal Hamiltonian H , H(ω) =

X

φ(pα ) +

α∈I

1 XX V (qα − qβ ) , 2 α∈I β6=α

where the kinetic energy φ : R3 7→ R is strictly convex with bounded derivatives, and V : R3 7→ R is a symmetric and superstable pair potential of finite range. The associated Liouville operator will be denoted by L , Lϕ =

X α∈I

?

h

∂H ∂ϕ ∂H ∂ϕ , i−h , i, ∂pα ∂qα ∂qα ∂pα

Work partially supported by grants CIPA-CT92-4016 and CHRX-CT94-0460 of the Commission of the European Community, and by grant T 16665 of the Hungarian NSF. Two of us (J.F. and C.L.) acknowledge hospitality of the Ervin Schrödinger Institute.

482

J. Fritz, C. Liverani, S. Olla

where h·, ·i denotes the usual scalar product in R3 . Almost nothing is known on the ergodic properties of such infinite systems. In fact, very few results are available even for finite systems of this type (e.g., [KSS, DL, BLPS, LW]). To ensure proper ergodic behavior of the system we add some noise, whereby obtaining stochastic equations of motion; these equations read dqα

=

dpα

=

φ0 (pα ) dt , d X X X θ θ − V 0 (qα − qβ ) dt + bα (ω) dt + σα,β (ω) dwα,β , β6=α

(0.1)

θ=1 β6=α

θ where wα,β is a family of independent one-dimensional Wiener processes for θ = θ θ 1, 2, ..., d and α 6= β such that wα,β = −wβ,α ; φ0 and V 0 denote the gradient of φ θ and V , respectively. The coefficients bα , σα,β : 7→ R3 are smooth local functions to be specified in the next section in such a way that total energy and momentum are both preserved by the randomized evolution (0.1). In addition, any Gibbs state IP with energy H will be a reversible measure for the stochastic part of the evolution: Z Z b b ϕ(ω)Lψ(ω) IP (dω) = ψ(ω)Lϕ(ω) IP (dω) (0.2)

for all smooth local functions ϕ, ψ : 7→ R , where b = Lψ

X α∈I

∂ψ 1 XXX θ 2 θ i+ hσα,β , (Dα,β ψ)σα,β i, ∂pα 4 d

hbα ,

(0.3)

θ=1 α∈I β6=α

2 ψ is the matrix of second derivatives obtained by applying Dα,β = ∂/∂pα − and Dα,β ∂/∂pβ twice to ψ . Since the Liouville operator is antisymmetric with respect to Gibbs e = L+L b also satisfies the stationary Kolmogorov distributions, the full generator, L e dIP = 0 for a wide class of test functions ψ and any Gibbs state IP . equation, ∫ Lψ The converse problem is much more complex. In our basic reference [LO] it is shown that if a translation invariant measure Q with finite specific entropy satisfies the stationary Kolmogorov equation and (0.2), together with some other technical conditions, then Q enjoys the Gibbs property. Let us remark that finiteness of specific entropy is a fairly natural and effective condition in the theory of hydrodynamics limits (see [OVY]). On the contrary, condition (0.2) looks rather restrictive and, at least in general, not particularly natural. The main purpose of this paper is to show that condition (0.2) of reversibility is superfluous (i.e., it follows from the stationarity of the measure).1 To obtain such a result we are forced to prove the existence of a semigroup defined by (0.1); its regularity (locality) will play a crucial role in the argument. This problem may not seem to be a very difficult one since φ0 (pα ) , the velocity of particle α, is bounded by assumption. However, ¯ ⊂ of initial configurations: we the evolution must be defined for a very large set ¯ need Q() = 1 for any probability measure Q of finite specific entropy. On the other hand, to obtain the necessary regularity properties of the dynamics we have to restrict the configuration space by excluding extremely high values of particle density. We shall see that the desired construction fails unless the dimension of the space is less than four, cf. [FD] and [S]. 1 Note that in applications to hydrodynamics the reversibility (0.2) is insured by construction (see [OVY], Lemma 4.4), hence the present paper does not add to hydrodynamics type problems for which the results in [LO] suffice. The focus here is on the classification of stationary translation invariant measures.

Reversibility in Infinite Hamiltonian Systems with Conservative Noise

483

1. Notations and Results Configurations can be interpreted as σ-finite integer valued measures on R3 × R3 ; sometimes we write ω = (q, p) with q = (qα )α∈I and p = (pα )α∈I , and if 3 ⊂ R3 , then ω3 denotes the restriction of ω to 3 , i.e. ω3 = (qα , pα )qa ∈3 , while |ω3 | is the cardinality of this set. The centered Pcubic box of side r > 0 will be denoted by 3r . Referring to functionals ω(ϕ) := α∈I ϕ(qα , pα ) , where ϕ : R3 × R3 7→ R is continuous with compact support, we equip with the associated weak topology and Borel structure, and C0 () denotes the space of cylinder (local) functions 9(ω) = f (ω(ϕ1 ), ω(ϕ2 ), ..., ω(ϕn )) such that f ∈ C(Rn ) . Since all sets Σ(δ), defined for an increasing sequence δ = (δ1 , δ2 , ...) such that δ1 ≥ 1 by Σ(δ) := {ω ∈ : |ω3n | ≤ δn and |pα | ≤ δn if qα ∈ 3n } are compact, we need not worry too much about topology. Indeed, in the forthcoming considerations we always do have an a priori bound allowing us to restrict calculations to some compact Σ(δ) . In these situations all reasonable topologies coincide, moreover any continuous function can be uniformly approximated by elements of C0 () in view of the Stone-Weierstrass Theorem. Interaction. We consider a repelling pair potential V of finite range such that V (x) = V (−x) is twice continuously differentiable, V (0) > 0 but V (x) = 0 if |x| > R0 , finally hx, V 0 (x)i ≤ 0 for all x ∈ R3 . These conditions imply that V is superstable, see [R]: for each cubic box or ball 3 ⊂ R3 there exist some constants A3 ≥ 0 and B3 > 0 such that, for any configuration, we have X X

V (qα − qβ ) ≥ B3 |ω3 |2 − A3 |ω3 |.

(1.1)

α:qα ∈3 β6=α

Kinetic energy. We assume that φ has bounded second derivatives, and velocities are also bounded, i.e. |φ0 (y)| ≤ c¯ < +∞ for all y ∈ R3 . To define Gibbs measures we need a lower bound: lim inf |y|→∞ φ(y)/|y| ≥ c > 0 . When results of [LO] are applied an extra technical condition on φ is needed. For simplicity one can consider the case in P3 which φ(y) = i=1 φ0 (yi ) for y = (y1 , y2 , y3 ) , where φ0 ∈ C ∞ (R) is strictly convex and 1 d2 00 2 iv 00 (φ (u))2 = φ000 0 (u) + φ0 (u)φ (u) 6= 0 2 du2 0 apart from, at most, finitely many points (see [LO], Sect. Two, “Condition on the Noise" d2 00 2 for the general condition that φ must satisfy). Notice that if du 2 (φ0 (u)) = 0 for each u and if we require the natural condition φ0 (u) = φ0 (−u), then φ000 is a constant, which is the classical case of a quadratic kinetic energy function. Stochastic perturbation of classical dynamics. There are several ways to select the coefficients of the stochastic perturbation. We set bα (ω) =

X β6=α

θ γα,β (q)F (pα , pβ ) and σα,β (ω) =

p γαβ (q)Gθ (pα , pβ ) ,

(1.2)

484


where, as in [LO],2 γα,β (q) = γβ,α (q) ≥ 0 is continuously differentiable, γα,β (q) > 0 if |qα − qβ | < R1 and it is zero for |qα − qβ | ≥ R1 , i.e. R1 > R0 is the radius of stochastic interaction. The functions F, Gθ : R6 7→ R3 are infinitely differentiable and bounded together with their derivatives; they are chosen in such a way that the stochastic interaction also preserves the total momentum and energy of an interacting couple of particles. Moreover, {Gθ }dθ=1 spans, at each point, all R3 . It is natural to assume that γα,β depends only on the interparticle distances, and it does not depend on a coordinate qδ if |qα − qδ | > R2 or |qβ − qδ | > R2 , where R2 > 2R1 is a constant. Therefore the stochastic interaction is also translation invariant and has a finite range R3 := R1 + R2 . A new feature of the present model is that γα,β vanishes when the number of particles near qα or qβ tends to infinity. For convenience, we set γα,β (q) = σ(qα − qβ )Θα,β (q) , where −1 X X χ(qα − qδ ) + χ(qβ − qδ ) . (1.3) Θα,β (q) := 1 + δ∈I

δ∈I

In (1.3) σ, χ : R 7→ [0, 1] are twice continuously differentiable, σ(x) = σ(−x) > 0 if |x| < R1 and it is zero for |x| ≥ R1 . Similarly, χ(x) = χ(−x) > 0 if |x| ≤ 2R1 and χ(x) = 0 if |x| > R2 with some R2 > 2R1 . A technical condition, |χ0 (x)| ≤ Kχ(x)1−κ , where 0 < κ < 1/9 , will be exploited in Lemma 2.4. θ θ = −wβ,α , the condition F (pα , pβ ) = −F (pβ , pα ) of antisymmetry of F Since wα,β clearly implies the conservation of total momentum. For convenience, we choose 3

1X hGθ (pα , pβ ), Dα,β iGθ (pα , pβ ) 2 d

F (pα , pβ )

:=

and

θ=1

θ ϕ Xα,β

:=

1 √ hGθ (pα , pβ ), Dα,β ϕi , 2

b of the random component of our process becomes3 then the formal generator L XXX θ θ b =1 γα,β (q)Xα,β ϕ . Xα,β Lϕ 2 d

(1.4)

θ=1 α∈I β6=α

In this case the orthogonality relations hGθ (pα , pβ ), φ0 (pα ) − φ0 (pβ )i = 0

(1.5)

imply the formal conservation of energy, see [LO]. To have conservation of phase volume it is also assumed that (1.6) hDα,β , Gθ (pα , pβ )i = 0 , θ θ Xα,β are symmetric with respect to Lebesgue measure i.e. the operators γα,β Xα,β dpα dpβ . As a consequence we shall see that the conservation laws imply the reversibilb . For an explicit example of F and Gθ , see the ity of Gibbs states with respect to L Appendix of [LO]. 2 In fact, in [LO], the functions γ αβ depend only on the variables qα and qβ ; yet all that is done there applies without changes to the situation described here. 3 The future requirements (1.5) and (1.6) imply that X θ ∗ = −X θ , where the adjoint is taken with α,β α,β respect to the measure defined by the kinetic energy, (see[LO]).


485

Gibbs measures. Let λ = (λ0 , λ1 , λ2 , λ3 , λ4 ) be a set of real parameters with λ4 > 0 and λ21 + λ22 + λ23 < c2 , and denote by Π the distribution of a Poisson process of unit intensity in R3 . A probability measure IP on is called a Gibbs state for H with parameters λ if its conditional distributions given the configuration outside of any cubic box 3 ⊂ R3 can be represented as " # 3 n X X 1 exp λ0 |ω3 | + λi piα − λ4 H3 (ω3 , ω3c ) Π(dq3 ) dp3 , IP [dω3 |ω3c ] = Z3 α=1 i=1

where Z3 is the normalization, and a natural decomposition ω3 = (q3 , p3 ) is used, see [D]. The local Hamiltonian, H3 is defined as   X X X φ(pα ) + 1 V (qα − qβ ) + V (qα − qβ ) , H3 (ω3 , ω3c ) = 2 q ∈ω q ∈ω c α

3

qβ ∈ω3 ; α6=β

β

3

the set of such measures will be denoted by Pλ , see [R] for the existence of Gibbs states for superstable interactions. Relative entropy. Let Q and P be probability measures on , and for any 3 ⊂ R3 denote F3 the set of continuous and bounded functions ψ : 7→ R such that ψ(ω) = ψ(ω3 ) for all ω ∈ . The entropy of Q in 3 , relative to P , is defined by (1.7) H3 (Q|P ) = sup IE Q (ψ) − log IE P (eψ ) , ψ∈F3

where IE Q denotes the expectation with respect to the probability measure Q . If 3 = R3 then the subscript 3 of H3 will be omitted; for properties of H3 , see, for example [OVY]. As a reference measure a distinguished, translation invariant, Gibbs state P = IP will be chosen. We say that Q has finite specific entropy if there exists a constant C such that H3 (Q|IP ) ≤ C(1 + |3|) for any cubic box 3 . If Q is translation invariant with finite specific entropy, then the particle density ρ = ρ(ω) is Q-a.s. defined as the following limit taken along any increasing sequence of cubic boxes, see [LO], ρ(ω) = lim |3|−1 |ω3 |. |3|→∞

Main results of [LO] for the system under consideration can be summarized as follows. Theorem 1.1. Suppose that Q is a translation invariant probability measure on with finite specific entropy, and let ρc := 3/(4πR13 ) . If Q[ρ(ω) > ρ0 ] = 1 for some ρ0 > ρc , e = L+L b in the sense that, for any smooth local (ii) Q is invariant with respect to L Q e function ψ we have IE (Lψ) = 0 , (i)

b , i.e. IE Q (ψ Lϕ) b b (iii) Q is reversible with respect to L = IE Q (ϕLψ) for any pair ϕ, ψ of smooth local functions, then Q is a convex combination of Gibbs states.

486


Statement of the result. Notice that the theorem above is stated without any reference to the existence of the infinite dynamics; properties (ii) and (iii) of invariance are purely formal. However, the extraction of such local information as reversibility is usually based on a method of Liapunov functions, namely entropy and its rate of change are compared, so the first step of our argument is intrinsically related to the evolution. Theorem 1.2. Under the conditions on the stochastic dynamics listed above, there exists ¯ ⊂ such that Q() ¯ = 1 for each Q with finite specific entropy. an explicitly defined set ¯ we have a unique strong solution ω(t) , t ≥ 0 to (0.1) such Moreover, for each ω0 ∈ ¯ a.s. The solution is a measurable function of the initial that ω(0) = ω0 and ω(t) ∈ configuration, and every Gibbs state IP ∈ Pλ with λ4 > 0 and λ1 = λ2 = λ3 = 0 is a stationary measure for the random evolution. This theorem is proven in the next section; solutions are defined by a limiting procedure starting from finite systems. The restriction on the parameters of a Gibbs measure in the last statement could have been removed by elaborating some technical details, but we do not need such a general assertion. Having constructed the infinite evolution we can consider stationary measures instead of simply measures formally invariant as in Theorem 1.1 (ii).4 Theorem 1.3. Every translation invariant stationary measure with finite specific enb of the generator; that is, condition tropy is reversible with respect to the stochastic part L (iii) in Theorem 1.1 holds. The proof of Theorem 1.3 is the content of Sect. 3. Combining the above results we get the final result of the paper: Theorem 1.4. Let Q be a translation invariant stationary measure with finite specific entropy, then condition (i) of Theorem 1.1 implies that Q is a superposition of Gibbs states.

2. Infinite Dynamics We start this section by describing the set of allowed initial configurations. Although the definition is a bit technical our choice boils down to configurations for which the energy in a box does not grow too fast with respect to the size of the box. The exact meaning of this construction will become more clear later on when the desired a priori bounds for a family of partial dynamics and the requirements for the existence of a unique limiting dynamics are discussed. Initial conditions. Let Hm (ω, r) denote the total energy of ω ∈ in a ball Bm (r) ⊂ R3 of center m and radius r ≤ ∞, i.e. Hm (ω, r) := H(ωBm (r) ); the number of points of q in Bqα (r) will be denoted as Nα (q, r) . For κ ∈ (0, 1/9) , see (1.3), and r ≥ R3 = R1 + R2 , define Hm (ω, R3 ) , H¯ κ,r (ω) := sup 3+2κ 1 |m|≤r + |m|

¯ κ,r (h) = {ω ∈ : H¯ κ,r (ω) ≤ h}.

4 Notice that, since the infinite dynamics satisfies Eqs. (0.1), if Q is stationary, then it satisfies (ii) of Theorem 1.1.


487

¯ κ,r (h) such that Q(Hm (ω, R3 )) ≤ Let Qr (k) be the set of Borel probability measures on k for all |m| ≤ r. Now the set of all allowed configurations is defined as [ ¯ κ,∞ (h) = {ω ∈ : H¯ κ,∞ (ω) < ∞}. ¯ κ,∞ = h>0

¯ (h) are compact in the weak topology of and, in Remember that the level sets √κ,∞3/2+κ ¯ κ,L (h) . view of (1.1), Nα (q, R3 ) = O( hL ) for qα ∈ Bm (L − R3 ) and (q, p) ∈ We shall see that the initial condition for the existence and uniqueness of the limiting dynamics could have been formulated in terms of Nα only, but a preservation of bounds on kinetic energy will be needed when we prove locality of the dynamics. Lemma 2.1. If κ > 0 then for any fixed k > 0 we have lim inf

inf

h→∞ r≥R3 Q∈Qr (k)

¯ κ,∞ ) = 1 if Q ∈ Q∞ := that is Q(

S k>0

¯ κ,r (h)) = 1, Q(

Q∞ (k).

Proof. This statement is a direct consequence of the Markov inequality. In fact we have some universal v > 0 such that (by vZ3 we denote the tridimensional cubic lattice of size v) X ¯ κ,r (h)c ) ≤ Q[Hm (ω, R3 ) > vh(1 + |m|3+2κ )] Q( m∈B0 (r)∩vZ3

X

≤

m∈B0 (r)∩vZ3

k X Q(Hm ) 1 ≤ , 3+2κ h(1 + |m| )v vh (1 + |m|3+2κ ) 3

which proves the statement for any κ > 0 .

m∈vZ

Observe that the entropy condition H3 (Q|IP ) ≤ C(1 + |3|) implies Q ∈ Q∞ via (1.7) and (1.1), see Lemma 3.1 of [LO]. Local dynamics. There are several ways to define a family of partial dynamics, the advantage of the following construction consists in its direct relation to Gibbs states. Let a : R3 7→ [0, 1] be twice continuously differentiable with compact support. We assume also |a0 (x)| ≤ 1 for all x ∈ R3 . We interpret a as a smooth version of the indicator function of a ball, its concrete shape is not very important. For every such cutoff a and inverse temperature λ4 > 0 we consider a system of stochastic differential equations, dqα dpα

1 λ4 H3 (ω3 ,ω3c ) ∂ e a(qα )e−λ4 H3 (ω3 ,ω3c ) dt , λ4 ∂pα 1 λ4 H3 (ω3 ,ω3c ) ∂ = e a(qα )e−λ4 H3 (ω3 ,ω3c ) dt λ4 ∂qα X γα,β (q)a(qβ )F (pα , pβ ) dt +a(qα ) =

−

β6=α d X X p p θ + a(qα ) a(qβ )γα,β (q)Gθ (pα , pβ ) dwα,β , θ=1 β6=α

(2.1)

488


where it is assumed that 3 ⊂ R3 is bounded and contains the support of a in its interior; in such a situation the equations above do not depend on the particular choice of 3 . Notice that in a region where a = 1 our particles follow the original equations of motion, while they are frozen outside of the support of a , i.e. q˙α = p˙α = 0 . Particles approaching the boundary of the support of a slow down, thus we have a smooth transition between moving and frozen particles, see [F1] for a similar construction. This means that we essentially have a finite dimensional diffusion. Let Pλt 4 ,a denote the Markov semigroup induced by partial dynamics (2.1), i.e. Pλt 4 ,a ψ(ω) := IE w (ψ(ω(t)) , where ω(t) is the solution with initial condition ω(0) = ω , ψ : 7→ R is continuous and bounded, while IE w denotes the expectation with respect to the joint distribution of our Wiener processes. By a direct application of the Ito lemma we see that the (formal) generator of Pλt 4 ,a e λ ,a = Lλ ,a + L b a , where decomposes as L 4 4 ∂ψ 1 X λ4 H3 (ω3 ,ω3c ) ∂ e a(qα )e−λ4 H3 (ω3 ,ω3c ) Lλ4 ,a ψ = − λ4 ∂pα ∂qα α∈I ∂ψ 1 X λ4 H3 (ω3 ,ω3c ) ∂ + e , (2.2) a(qα )e−λ4 H3 (ω3 ,ω3c ) λ4 ∂qα ∂pα α∈I

ba ψ L

=

1 2

d XX X

θ θ γα,β (q)a(qα )a(qβ )Xα,β (Xα,β ψ).

θ=1 α∈I β6=α

Since the coefficients of (2.1) are bounded smooth functions, we have a differentiable dependence of solutions on initial values. Therefore a class Da of twice continuously b a , e.g. in the space of condifferentiable functions forms a common core of Lλ4 ,a and L tinuous and bounded functions. An extension to the space L2 (IP λ ) of square integrable functions with respect to a distinguished Gibbs state IP λ follows by the next lemma. Lemma 2.2. Let λ1 = λ2 = λ3 = 0 while λ4 > 0 , then every Gibbs state IP ∈ Pλ satisfies b a ψ2 ) = IE IP (ψ2 L b a ψ1 ) IE IP (ψ1 Lλ4 ,a ψ2 ) = −IE IP (ψ2 Lλ4 ,a ψ1 ) and IE IP (ψ1 L for ψ1 , ψ2 ∈ Da , consequently IP is a stationary measure of the process Pλt 4 ,a for each cutoff a. Proof. Both symmetry relations follow from the definition of IP by integrating by parts. The property of reversibility is a direct consequence of (1.6). Integration by parts with respect to positions is possible because of the presence of the cutoff a . Since (2.1) violates the law of momentum conservation in regions where a is not a constant, Lemma 2.2 is not true for general Gibbs measures. Construction of the infinite dynamics. First we derive an a priori bound for local dynamics; we show that the set of initial conditions is preserved for all t > 0 and the related bound does not depend on the particular choice of the cutoff function a . Lemma 2.3. There exists a constant c1 depending only on λ4 and on the parameters of the infinite system (0.1) such that IE w (Hm (ωa (t), R3 )) ≤ (c1 + c1 t)(1 + Hm (ωa (0), R3 + c¯t) for all m, t and a , where ωa (t) is any solution to (2.1).


489

Proof. Since all velocities are bounded by c¯ , we have |qα (t) − qα (0)| ≤ c¯t , whence Nα (q(t), r) ≤ Nα (q(0), r + c¯t) , which yields an explicit deterministic bound for the potential energy via superstability (1.1). On the other hand, |bα (ω)| +

d X X

θ |σα,β (ω)|2 ≤ c01 Nα (q, R1 ) ,

θ=1 β6=α

and the same bound holds true for the corresponding coefficients of (2.1). From the stochastic equations by the Schwarz inequality we get (2.3) IE w |pα (t) − pα (0)|2 ≤ c001 t(1 + t)Nα (q(0), R1 + c¯t)2 , which completes the proof. Indeed, as φ(y) ≤ φ(0) + c¯|y| , taking the square root of both sides and summing for α ∈ I such that |qα (0) − m| ≤ R3 + c¯t , we get a bound for Hm (ω(t), R3 ) ; the square of the number of points at time zero is estimated again by superstability. To prove the existence of limiting solutions when the cutoff is removed we have to compare different partial solutions. Let AL denote the set of twice continuously differentiable a : R3 7→ [0, 1] with compact support such that |a0 (x)| ≤ 1 for all x and ¯ = (q(t), ¯ p(t)) ¯ denote a(x) = 1 if |x| ≤ L . For a, a¯ ∈ AL let ω(t) = (p(t), q(t)) and ω(t) the corresponding solutions to (2.1) with a common initial value ω(0) = ω(0) ¯ = (ξ, η) . Supposing |x| ≤ L − 2c¯t, for |x − ξα | ≤ R0 + 2c¯t we get ∂t |qα − q¯α | ≤ K0 |pα − p¯α | , whence Z t 2 |qα (s) − q¯α (s)||pα (s) − p¯α (s)| ds ; (2.4) |qα (t) − q¯α (t)| ≤ 2K0 0

K0 , K1 , K10 ...

denote constants depending only on the coeffihere and in what follows, cients of the infinite system. The case of the momentum variables is more complex. If a = a¯ = 1 can be assumed as before, then by Ito’s formula we get Z t Jα,1 (s) + Jα,2 (s) + Jα,3 (s) ds , IE w |pα (t) − p¯α (t)|2 = IE w 0

where

X hpα − p¯α , V 0 (qα − qβ ) − V 0 (q¯α − q¯β )i ,

=

−2

Jα,2

=

X 2 hpα − p¯α , γα,β (q)F (pα , pβ ) − γα,β (q)F ¯ (p¯α , p¯β )i ,

Jα,3

d X X p p 2 = | γα,β (q)Gθ (pα , pβ ) − γαβ (q)G ¯ θ (p¯α , p¯β )| .

Jα,1

α6=β

β6=α

θ=1 β6=α

Introduce now 0i for i = 1, 2 and r, t ≥ 0 by X |qα (t) − q¯a (t)|2 ; 01 (ξ, η, a, a¯ ; r, t) := α:|ξα |≤r

in the definition of 02 the variables qα and q¯a should be replaced by pα and p¯α , respectively. Our main tool is the following:

490


Lemma 2.4. Suppose that κ < 1/9 , 2c¯T < R1 . For all r ≥ R3 , t ≤ T , h > 0 and i = 1, 2 we have lim

sup

sup

L→∞ a,a∈A ¯ κ,L (h) ¯ L (ξ,η)∈

IE w (0i (ξ, η, a, a¯ ; r, t)) = 0 ,

and the convergence is uniform on the time interval [0, T ]. Proof. The idea of the proof is to define and to estimate a “distance" (based on 0i ) among different partial dynamics in boxes of radius r < L. This will lead us to Eq. (2.9) in which such a distance in a given box is related to the distance in a larger box, the result will easily follow. Let N¯ = N¯ L,T := max Nα (ξ, R3 + 2c¯T ) for all α ∈ I such that |ξa | + R3 + 2c¯T ≤ L . Suppose r +R3 +2c¯T < L , |ξα | ≤ r , and remember that |qδ (t)−ξδ | ≤ c¯t is always true. The uniform Lipschitz continuity of V 0 , σ , F and Gθ shall also be used without any further reference in the following calculations. Let γ˜ = γ˜ α,β (t) denote any matrix such that 0 ≤ γ˜ α,β (t) ≤ 1 if t ≤ T , moreover γ˜ α,β (t) = 0 whenever |ξα − ξβ | ≥ R3 + 2c¯t . For J1 we get X γ˜ α,β (t) |qα − q¯α | + |qβ − q¯β | =: K1 J˜α,1 (t). (2.5) Jα,1 (t) ≤ K1 |pα − p¯α | β∈I

In the case of J2 the pattern |ax − by| ≤ min{|a|, |b|}|x − y| + |a − b| max{|x|, |y|} is used several times to derive X γα,β (q) |pα − p¯α | + |pβ − p¯β | Jα,2 (t) ≤ K2 J˜α,1 (t) + K2 |pα − p¯α | β6=α

+K2 |pα − p¯α |σ(qα − qβ )

XX

|∂δ Θα,β (q˜α,β )||qδ − q¯δ | ,

β6=α δ∈I

where ∂δ := ∂/∂qδ and q˜α,β is an intermediate configuration on the line segment connecting q and q¯ . Observe that by Hölder’s inequality X X 2 |∂δ Θα,β (q)| ˜ ≤ KΘα,β (q) ˜ χ(q˜α − q˜δ )1−κ + χ(q˜β − q˜δ )1−κ δ∈I

δ∈I

≤

KΘα,β (q) ˜ Nα (q, ˜ R2 )κ + Nβ (q, ˜ R 2 )κ .

(2.6)

On the other hand, σ(qα − qβ ) > 0 implies |q˜αα,β − q˜βα,β | ≤ R1 + 2c¯t ≤ 2R1 , i.e. Nα (q, R1 ) ≤ Nα (q˜α,β , 2R1 ) . This means that Nα (q, R1 ) ≤ 1/Θα,β (q˜α,β ) , consequently κ Jα,2 (t) ≤ K20 N¯ L,T J˜α,1 (t) + K20 J˜α,2 (t) ; X J˜α,2 (t) := γα,β (q) |pα − p¯α |2 + |pβ − p¯β |2 . (2.7) β6=α

In a similar way we get Jα,3 (t)

≤

K3 J˜α,2 (t) + K3 J˜α,3 (t) 2 X σ(qα − qβ ) X |∂δ Θα,β (q˜α,β )||qδ − q¯δ | , +K3 α,β Θα,β (q˜ ) β6=α δ∈I X J˜α,3 (t) := γ˜ α,β (t) |qα − q¯α |2 + |pδ − p¯δ |2 , δ∈I


491

whence by the Cauchy inequality and (2.6) 2κ ˜ Jα,3 (t). Jα,3 (t) ≤ K30 J˜α,2 + K30 N¯ L,T

(2.8)

Introduce now d(r, t) := IE w (02 (ξ, η, a, a¯ ; r, t)) + N¯ L,T IE w (01 (ξ, η, a, a¯ ; r, t)) for t < T . Comparing (2.4), (2.5)–(2.8) and using the elementary inequality 2|pα − p¯α ||qδ − q¯δ | ≤ N¯ −1/2 |pα − p¯α |2 + N¯ 1/2 |qδ − q¯δ |2 , we obtain, by a direct calculation, Z 1/2+κ d(r, t) ≤ K4 N¯ L,T

t

d(r + R3 + 2c¯T, s) ds ,

(2.9)

0

which completes the proof by a standard iteration procedure. Indeed, we get d(r, t) ≤

T `+1 1/2+κ ` K4 N¯ L,T sup d(L, t) , `! t 0 depending only on R3 and T , while N¯ L,T = O( hL3/2+κ ) . Using |qα (t) − ξα | ≤ c¯t and the second a priori bound (2.3), we see that the right-hand side of (2.10) vanishes as L → +∞ because `! = O (`/e)` and (1/2 + κ)(3/2 + κ) < 1 by hypothesis. Now we are in a position to prove the existence and uniqueness of limiting solutions to (0.1). Let us consider a sequence of partial solutions ωn = ωn (t) , n ∈ N of (2.1) with ¯ κ,∞ ; the corresponding cutoff an : R3 7→ R a common initial value ωn (0) = (ξ, η) ∈ is assumed to be a decreasing smooth function of |x| such that an (x) = 1 if |x| ≤ n and an (x) = 0 if |x| > n + 1 . In view of Lemma 2.4 ωn converges in probability to some limit ω(t) for each t < T = R1 /2c¯ . It is easy to verify that the limit satisfies the infinite system (0.1); the uniqueness of limiting solutions follows again by Lemma 2.4. Since ¯ κ,∞ , the construction extends T does not depend on the initial configuration (ξ, η) ∈ to all times. Properties of the infinite dynamics. Let Pnt denote the Markov semigroup induced by the partial dynamics ωn . Since ωn (t) is a continuous function of the initial data, it is well defined by Pnt ψ(ξ, η) := IE w (ψ(ωn (t))) if ωn (0) = (ξ, η) for any measurable and ¯ κ,∞ 7→ R . As a limit of measurable functions, the limiting solution ω(t) bounded ψ : is again a jointly measurable function of (ξ, η) and the random element representing the θ , the limiting semigroup, P t can be defined in the same way. Wiener processes wα,β If the initial configuration is distributed by Q ∈ Q∞ , then QPnt and QP t denote the evolved measure at time t > 0 . In view of Lemma 2.1 and Lemma 2.3 we know that QP t ∈ Q∞ , too. While Pnt has fairly good regularity properties, semigroup theory does not apply directly to the limiting case. Nevertheless, all we need in the next section is summarized as follows. ¯ κ,∞ 7→ R is a continuous and bounded local function, Lemma 2.5. Suppose that ψ : i.e. ψ(ω) := ψ(ωB0 (r) ) for some r > 0 , then lim sup

sup

`→∞ n>`+r Q∈Qn (k)

|QPnt ψ − QP t ψ| = 0

for all t, k > 0 , and the convergence is uniform on compact time intervals.

492


Proof. The a priori bound of Lemma 2.3 extends immediately to the limiting dynamics, ¯ κ,r (h)) ≥ 1 − ε , thus for any ε, T > 0 we have some k¯ > k and h > k¯ such that Q( ¯ κ,r (h)) ≥ 1 − ε , and QP t ( ¯ κ,r (h)) ≥ 1 − ε whenever t < T , n > r + R3 + 2c¯T QPnt ( ¯ is compact, there exists also an ε0 > 0 such that ¯ κ,r (k) and Q ∈ Qn (k) . Since 0 ¯ ≤ε |qα − q¯α | + |pα − p¯α | < ε , for all α ∈ I with |qα |, |q¯α | ≤ r, implies |ψ(ω) − ψ(ω)| ¯ . Therefore the statement follows from Lemma 2.4 and Chebishev ¯ κ,r (k) for ω, ω¯ ∈ inequality by a 3ε argument. The final statement of Theorem 1.2 on stationarity of certain Gibbs states is now a direct consequence of Lemma 2.2. 3. An Entropy Argument In this section we extend a familiar argument by Holley [H] to the present more complex situation. For a probability measure Q on , let H(Q|IP λ ) denote the entropy relative to a distinguished Gibbs state IP λ with λ1 = λ2 = λ3 = 0, as defined by (1.7) with 3 = R3 . The family of partial dynamics (2.1) has been chosen such that IP λ is a common stationary measure of each local dynamics Pnt = Pλ4 ,an introduced in Sect. 2. Therefore Pnt is a strongly continuous semigroup in L2 (IP λ ) , and smooth cylinder functions e n = Ln + L b n , see (2.2). Remember that Ln := Lλ ,a , form a core for its generator L 4 n the Hamiltonian part, is antisymmetric in L2 (IP λ ) while the symmetric (reversible) ba . b n := L component is just L n If G is a generator in L2 (IP λ ), then the corresponding Donsker-Varadhan rate function is defined as o n Z Gψ dQ : ψ ∈ Dom G, inf ψ > 0 . D(Q|G) = sup − ψ ψ If G is self-adjoint and G < 0, then we can apply the following result due to Donsker and Varadhan (cf. [DV], Theorem 5). p Theorem √ 3.1. D(Q|G) < +∞ if and only if Q