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Large deviations for a stochastic model of heat flow 5 0 0 1 2 3 2 Lorenzo Bertini , Davide Gabrielli , Joel L. Lebowitz n a 1 Dipartimentodi Matematica, Universitàdi Roma La Sapienza J P.le A.Moro 2, 00185 Roma, Italy 7 2 Dipartimentodi Matematica, Universitàdell’Aquila 2 67100 Coppito, L’Aquila, Italy 3 Departmentof Mathematics and Physics, RutgersUniversity ] h NewBrunswick, NJ08903, USA c e Dedicated to Mitchell Feigenbaum on the Occasion of His Sixtieth Birthday m - t Abstract a t We investigate a one dimensional chain of 2N harmonic oscillatorsin which neigh- s . boring sites have their energies redistributed randomly. The sites −N and N are t a in contact with thermal reservoirs at different temperature τ and τ . Kipnis, − + m Marchioro,andPresutti[18]provedthatthis modelsatisfiesFourier’slawandthat - in the hydrodynamical scaling limit, when N →∞, the stationary state has a lin- d ear energy density profile θ¯(u), u∈[−1,1]. We derive the large deviation function n o S(θ(u))fortheprobabilityoffinding,inthestationarystate,aprofileθ(u)different c from θ¯(u). The function S(θ) has striking similarities to, but also large differences [ from,thecorrespondingoneofthesymmetricexclusionprocess. Likethelatteritis 1 nonlocal and satisfies a variationalequation. Unlike the latter it is not convex and v the Gaussian normal fluctuations are enhanced rather than suppressed compared 1 to the local equilibrium state. We also briefly discuss more generalmodel and find 8 the features common in these two and other models whose S(θ) is known. 6 1 0 5 0 / t a m - d n o c : v i X r a Keywords: Stationary non reversible states, Large deviations, Boundary driven stochastic systems. 2000 MSC: 82C22,82C35, 60F10. 1 1. Introduction Thepropertiesofsystemsmaintainedinstationarynonequilibriumstates(SNS) by contacts with very large (formally infinite) thermal reservoirs in different equi- libriumstatesareofgreattheoreticalandpracticalimportance. Thesearearguably the simplest examples of nonequilibrium systems to which the elegant, universal, and successful formalism of equilibrium statistical mechanics might hopefully be extended. A striking universal feature of equilibrium systems is the Boltzmann– Einsteinrelationaccordingtowhichfluctuationsinmacroscopicobservables,arising from the grainy microscopic structure of matter, can be described fully in terms of the macroscopic thermodynamic functions (entropy, free energy) without any recourse to the microscopic theory. In trying to develop a similar formalism for SNS we have to start with the fluctuations. There has therefore been much ef- fort devoted to developing a mathematically rigorousfluctuation theory for simple modelSNS.Thishasledto someinterestingrecentresultsforconservativesystems in contact with particle reservoirsat different chemical potentials [4–7,10–12]. In particular it has been possible to obtain explicitly the large deviation functionals (LDF) for some one dimensional lattice systems. The internal dynamics of these systems is governed by simple exclusion processes, symmetric (SEP) or asymmetric(ASEP),while the entranceandexitofparticlesatthe twoboundaries are prescribed by the chemical potentials, λ , of the right and left reservoirs. The ± LDFgivesthe logarithmofthe probabilitiesoffinding macroscopicdensity profiles ρ(u), where u is the macroscopic space variable, different from the typical values ρ¯(u);namelywehaveProb(ρ(u))∼exp{−NF(ρ)}whereN isthenumberoflattice sites. In the symmetric case, the situation we shall be primarily concerned with here, the typical profile ρ¯(u) is givenby the stationary solution of the diffusion equation ∂ ρ(t,u)=(1/2)∂ D∂ ρ(t,u) ,u∈[−1,1]with boundaryconditionsρ¯(±1)=ρ . t u u ± The values ρ correspond to the densities in an equilibrium system with chemical ± (cid:0) (cid:1) potentialsλ . Thelattercanbeobtainedbysettingthechemicalpotentialofboth ± endreservoirsequaltoeachother,λ =λ . Wenotethatinthisequilibriumcase, + − the functionF is simply relatedto the free energy ofthe system. For ρ 6=ρ and + − constantdiffusioncoefficientD (thatisdensityindependent andspatiallyuniform) the profile ρ¯(u) is linear; this is the only case solvedso far for the SEP.Theresults for the LDF of the SEP for this SNS contained some surprises. The most striking of these is non–locality: the probability of density profiles ρ (u)andρ (u)indisjointmacroscopicregionsAandB isnotgivenbyaproduct A B oftheseparateprobabilities,i.e.theLDFisnotadditive. Thisisverydifferentfrom theequilibriumcasewheretheLDFisgiven(essentially)byanintegralofthelocal free energydensity forthe specifiedprofilesρ (u)andρ (u),andis thus automat- A B ically additive overmacroscopic regions (even at critical points). Additivity is also true for the LDF of a system in full local thermal equilibrium (LTE), e.g. for the stationary nonequilibrium state of the zero range process. The microscopic origin of the non–locality of the LDF for the open SEP lies in the O(N−1) corrections to LTE whichextendoverdistances ofO(N); N is number of lattice sites, whichgoes to infinity in the hydrodynamicalscalinglimit [1,24]. So while the deviations from LTEvanishinthislimittheircontributionstotheLDF,whichinvolvessummations over regions of size N, does not The effect of these O(N−1) corrections to LTE is already present at the level of Gaussianfluctuationsaboutρ¯(u). ThesewerecomputedbySpohnin1983[24]who found that the contributions fromthe deviations fromLTE made a finite contribu- tiontothevarianceoftheseGaussianfluctuations,causingthemtodecrease,forthe SNS of the SEP from their LTE values. The reduction in the variance of Gaussian 2 fluctuations can be recovered from the LDF by setting ρ(u) = ρ¯(u)+N−1/2φ(u). Infactin[5,11]itisshownthatF(ρ)fortheSEPdominatesthe LDFcomingfrom the corresponding LTE state and therefore the fluctuations are suppressed. TheaboveobservationsabouttheSEPraisemanyquestionsaboutthenatureof the SNSofmorerealisticsystems. Do theirLDFandGaussianfluctuations behave similarly to those of the SEP? In particular, to what extent do the LDF for SNS play a “similar role” to free energies in equilibrium systems? In the absence of more solvedexamples it is difficult to answer these questions. It is therefore useful to find and investigate the SNS of other model systems for which the LDF can be found and compare them to that of the SEP. This is what we do in the present paper and then discuss the limited universality of the results. The SNS we consider here is a simple stochastic model of heat conduction in a crystal. Itiswellknown,seee.g.[20,23],thatharmonicchainsdonotobeyFourier’s law of heat conduction. On the other hand, Kipnis, Marchioro, and Presutti [18] introduced a model of mechanically uncoupled harmonic oscillators in which nearest neighbor oscillators redistribute randomly their energy. This system is then coupledto thermalreservoirsat differenttemperatures and, thanks to the stochastic dynamics, the validity of Fourier’s law is proven. In particular the stationary energy density θ¯(u) is a linear profile as in the SEP. We mention that a more sophisticated stochastic model of coupled harmonic oscillators has been recently investigated. The evolution is given by superimposing the Hamiltonian dynamics withastochasticoneinwhichtwonearestoscillatorsrandomlyexchangemomenta. Thismodelhastwoconservationslaws(energyandtotallength);thehydrodynamic limitisprovenin[2]fortheequilibriumcaseandin[3]fornonequilibrium,Gaussian fluctuations are analyzed in [16]. InthispaperweconsidertheKipnis–Marchioro–Presuttimodel,ourmainresult is the derivation of the corresponding LDF, that we denote by S(θ). It turns out that this function has both strong similarities and significant differences from that of the SEP.Like for the SEP the LDF is nonlocal and yields Gaussianfluctuations about θ¯(u). Unlike the SEP, however, it is obtained by minimization, rather than maximization, of a “proto LDF” and the variance is increased compared to that obtained from LTE. Also in contrast to the SEP the LDF, S(θ), is not convex. We discuss these similarities and differences in section 7, where we also give some generalization of our and previous results to a larger class of model systems. 2. The model and main result Following [18] we consider a chain of one–dimensional harmonic oscillators lo- cated at sites x ∈ [−N,N]∩Z =: Λ and described by the canonical coordinates N (q ,p ). TheoscillatorsaremechanicallyuncoupledsothattheHamiltonianofthe x x chain is H = (p2 +q2)/2. The harmonic oscillators are however coupled x∈ΛN x x by the following stochastic dynamics. Every pair of nearest neighbors sites waits P an exponential time of rate one and then the corresponding oscillators exchange energy. Moreprecisely,let(q ,p ),(q ,p )be thecanonicalcoordinatesatthe y y y+1 y+1 sites y, y+1; when the exponential clock between y and y+1 rings then the new values (q′,p′), (q′ ,p′ ) are distributed according to the uniform distribution y y y+1 y+1 on the surface of constant energy 1 1 1 1 (q′)2+(p′)2 + (q′ )2+(p′ )2 = q2+p2 + q2 +p2 2 y y 2 y+1 y+1 2 y y 2 y+1 y+1 Moreov(cid:2)erthebounda(cid:3)rysit(cid:2)e−N,respectively(cid:3)+N,(cid:2)waitsan(cid:3)expo(cid:2)nentialtimeo(cid:3)frate one and then the corresponding oscillator assume an energy distributed according to a Gibbs distribution with temperature τ , respectively τ . All the exponential − + clocks involved in the dynamics are independent. 3 Fromamathematicalpointofviewitissufficienttolookonlyatthelocalenergy givenbytherandomvariablesξ := p2+q2 /2,forwhichwegetaclosedevolution x x x described by the following Markov process. The state space is Σ := RΛN, an (cid:0) (cid:1) N + elementofΣ is denotedby ξ :={ξ , x∈Λ }. The infinitesimalgeneratorofthe N x N process is the sum of a bulk generator L plus two boundary generators L and 0 + L − L :=N2 L +L +L (2.1) N 0 − + in which we have speeded up the tim(cid:2)e by the factor(cid:3)N2, this corresponds to the diffusive scaling. The bulk dynamics L is defined as 0 N−1 L := L 0 x,x+1 x=−N X where 1 L f(ξ):= dp f(ξ(x,x+1),p)−f(ξ) (2.2) x,x+1 Z0 (cid:2) (cid:3) in which the configuration ξ(x,x+1),p is obtained from ξ by moving a fraction p of the total energy across the bond {x,x+1} to x and a fraction 1−p to x+1, i.e. ξ if y 6=x,x+1 y (ξ(x,x+1),p) := p(ξ +ξ ) if y =x y  x x+1 (1−p)(ξ +ξ ) if y =x+1  x x+1 The boundary generators L are defined by a heat bath dynamics with respect ± to thermostats at temperatures τ , i.e. ± ∞ 1 L f(ξ):= dr e−r/τ± f(ξ±N,r)−f(ξ) ± τ Z0 ± (cid:2) (cid:3) in which the configuration ξ±N,r is obtained from ξ by setting the energy at ±N equal to r, i.e. ξ if y 6=x (ξx,r) := y y r if y =x (cid:26) NotethatwehavesettheBoltzmannconstantequaltoone. Theprocessgenerated by (2.1), denoted by ξ(t), will be called the KMP process. We denote by u ∈ [−1,1] the macroscopic space coordinates and introduce the space of energy profiles as M:={θ ∈L ([−1,1],du) : θ(u)≥0}. We consider M 1 equipped with the weak topology namely, θ →θ iff for each continuous test func- n tion φ we have hθ ,φi →hθ,φi, where h·,·i is the inner product in L ([−1,1],du). n 2 Givenamicroscopicconfigurationξ ∈Σ ,weintroducetheempiricalenergyπ (ξ) N N by mapping ξ to the macroscopic profile N [π (ξ)](u):= ξ 1I (u) (2.3) N x x− 1 ,x+ 1 N 2N N 2N x=−N X (cid:2) (cid:3) note that π (ξ)∈M is a piecewise constant function. N In the case when τ = τ = τ it is easy to show that L is reversible with re- − + N specttotheproductofexponentialdistributionwithparameterτ,i.e.theinvariant measure is given by the equilibrium Gibbs measure at temperature τ, N dξ dµ (ξ)= x e−ξx/τ (2.4) N,τ τ x=−N Y 4 When ξ ∈Σ is distributed according to µ then the empirical energy π (ξ) N N,τ N concentrates, as N → ∞ on the constant profile τ according to the following law of large numbers. For each δ >0 and each continuous test function φ=φ(u) lim µ hπ (ξ),φi−hτ,φi >δ =0 (2.5) N,τ N N→∞ where τ ∈M is the constant fu(cid:16)n(cid:12)ction with that val(cid:12)ue. (cid:17) (cid:12) (cid:12) In this equilibrium case it is also easy to obtain the large deviation principle associated at the law of large numbers (2.5). More precisely, the probability that the empirical energy π (ξ) is close to some profile θ ∈ M different from τ is N exponentially small in N and given by a rate functional S 0 µ (π (ξ)∼θ)≍exp −NS (θ) (2.6) N,τ N 0 where πN(ξ)∼θ means closeness in the weak(cid:8)topology of(cid:9)M and ≍ denotes loga- rithmic equivalence as N →∞. The functional S is given by 0 1 θ(u) θ(u) 1 S (θ)= du −1−log = dus (θ(u),θ¯ ) (2.7) 0 0 0 τ τ Z−1 (cid:20) (cid:21) Z−1 where θ¯ = τ is the constant energy density profile for τ = τ = τ. The above 0 + − functional caninfact be obtainedas the Legendretransformof the pressureG (h) 0 S (θ)=sup hθ,hi−G (h) 0 0 h (cid:2) (cid:3) where G is defined as 0 1 1 G (h):= lim logE eNhh,πN(ξ)i =− du log[1−τh(u)] (2.8) 0 N→∞N µN,τ(cid:16) (cid:17) Z−1 in which E denotes the expectation with respect to µ . µN,τ N,τ If τ 6=τ the process generated by L is no longer reversible and its invariant − + N measure µ is not explicitly known. Theorem 4.2 in [18] implies however the N,τ± following law of large numbers. For each δ >0 and each continuous φ lim µ hπ (ξ),φi−hθ¯,φi >δ =0 (2.9) N→∞ N,τ± N where θ¯is the linear profile inte(cid:16)rp(cid:12)(cid:12)olating τ− and τ+(cid:12)(cid:12), i.e.(cid:17) 1−u 1+u θ¯(u)=τ + τ (2.10) − + 2 2 Itis naturalto lookfor the largedeviationsasymptotic forµ . Inthe caseof N,τ± the symmetric simple exclusion process (SEP) this program has been carried out in[5,6,10,11]. Themainresultofthispaperisanexpressionforthelargedeviation ratefunctionalforµ analogoustotheonefortheSEP.Thefunctionalweobtain N,τ± is nonlocal, as is the one for the SEP, but it turns out to be nonconvex while the one for SEP is convex. We mention that non convexity of the rate functional also occurs for the asymmetric exclusion process [12]. Withoutlossofgeneralityweassumeτ <τ andintroducethesetT :={τ ∈ − + τ± C1([−1,1]) : τ′(u)>0, τ(±1)=τ }, here τ′ is the derivative of τ. Given θ ∈M ± and τ ∈T we introduce the trial functional τ± 1 θ(u) θ(u) τ′(u) G(θ,τ):= du −1−log −log (2.11) τ(u) τ(u) [τ −τ ]/2 Z−1 h + − i In this paper we show that the empirical energy for µ satisfies the large devia- N,τ± tion principle with a nonlocal, nonconvex rate functional S(θ) given by S(θ)= inf G(θ,τ) (2.12) τ∈Tτ± 5 that is we have µ (π (ξ)∼θ)≍exp −NS(θ) (2.13) N,τ± N We note there is a very close similarity betwe(cid:8)en (2.12) a(cid:9)nd the analogous result for the SEP, we emphasize however that in (2.12) we minimize over the auxiliary profile τ, while in SEP one needs to maximize. This is, of course, related to the non convexity of our S versus the convexity of the rate functional for SEP. It would be very interesting to understand this basic difference also in terms of the combinatorial methods in [10–12] besides the dynamical approach presented here. Givenθ ∈M,weshowthattheminimizerin(2.12)isuniquelyattainedforsome profile τ(u) = τ[θ](u); therefore S(θ) = G(θ,τ[θ]). Moreover τ[θ](u) is the unique strictly increasing solution of the boundary value problem τ′′ τ2 +θ−τ =0  (τ′)2 (2.14)  τ(±1)=τ± which is the Euler–Lagrangeequation δG/δτ =0 when θ is kept fixed. We note that for θ = θ¯ the solution of (2.14) is given by τ[θ¯] = θ¯ therefore S(θ¯) = G(θ¯,θ¯) = 0. On the other hand, by the convexity of the real functions R ∋ x 7→ x−1−logx and R ∋ x 7→ −logx, for each θ ∈ M and τ ∈ T we + + τ± have G(θ,τ)≥0 hence S(θ)≥0. By the same argument we also get that S(θ)=0 if and only if θ = θ¯. This shows that the large deviation principle (2.13) implies the law of large numbers (2.9) and gives an exponential estimate as N → ∞. We finally remark that the reversible case (2.7) is recovered from (2.11)–(2.13) in the limit τ −τ →0 which impose τ(u) constant. + − Outline of the following sections. Our derivation of the rate functional S follows the dynamical/variational approach introduced in [4,5]. We look first, in Section 3, at the dynamical behavior inthe diffusive scalinglimitin abounded time interval[0,T]. Inparticular,weob- tain a dynamical large deviation principle which gives the exponential asymptotic for the event in which the empirical energy follows a prescribed space–time path. In Section 4 we introduce the quasi potential, it is defined by the minimal cost, as measured by the dynamical rate functional, to produce an energy fluctuation θ starting from the typical profile θ¯. By the arguments in [4,5], the quasi potential equals the rate functional S(θ) of the invariant measure µ . A mathematical N,τ± rigorous proof of this statement for the SEP is given in [7]. As discussed in [4,5], thequasipotentialistheappropriatesolutionofaHamilton–Jacobiequationwhich involves the transport coefficients of the macroscopic dynamics. The derivation of thefunctionalSisthencompletedbyshowingthat(2.12)istheappropriatesolution of this Hamilton–Jacobi equation. As in the case of the SEP we are also able, by followingthisdynamical/variationalapproach,tocharacterizetheminimizerforthe variationalproblemdefiningthequasipotential;thispathistheonefollowedbythe process,withprobabilitygoingtooneasN →∞,inthespontaneouscreationofthe fluctuationθ. InSection4wealsoshowthatthe functionalS is notconvex,obtain its expression for constant profiles θ, and derive an additivity principle analogous to the one for simple exclusion processes obtained in [11,12]. In the remaining part of the paper we discuss some extensions of the previous results. In particular, in Section 5 we discuss the KMP process in higher space dimension, d ≥ 1, and obtain an upper bound for the quasi potential in terms of the local equilibrium one. We note that for the SEP it is possible to prove [5,6] an analogous lower bound. We also discuss the Gaussian fluctuations around the 6 stationary profile θ¯; as for the SEP [5,9–11,24] the correction due to nonequilibrium is given by the Green function of the Dirichlet Laplacian. In particular, this correction is non local; as in the case of the SEP, this is due to the long range correlations [18]. However, for the KMP process, the nonequilibrium enhances the Gaussianfluctuations while inthe SEPit decreasesthem. As the covarianceofthe Gaussian fluctuations equals the inverse of the second derivative of S(θ) at θ¯, the enhancement of Gaussian fluctuations corresponds to the upper bound of S(θ) in terms of the local equilibrium functional. In the analysis in [18] a crucial role is playedby aprocess,induality withrespectto the KMPprocess,inwhichthelocal variable atthe site x takes integralvalues. In Section6 we discuss briefly the large deviations properties of this dual model and obtain the expression for the large deviation functional. Finally in Section 7 we discuss the derivation of the large deviationfunctionalforgenericone–dimensionalnonequilibriumsymmetricmodels with a single conservation law. We obtain a simple condition, which is satisfied by thezerorangeprocess,theGinzburg–Landaudynamics,theSEP,theKMPprocess and its dual, that allows the derivation of the large deviation function by means of asuitabletrialfunctional. Evenwhenthisconditionfailstohold,ityieldsasimple criterion to predict the enhancement/supressionof the Gaussian fluctuation in the stationary nonequilibrium state with respect to the full local thermal equilibrium. The discussion in this paper will be kept at the physicists level of mathematical rigor. However, for the more mathematically inclined reader, we shall point out the main differences and technical difficulties with respect to the case of the SEP, which has been analyzed in full mathematical rigor [6]. 3. Macroscopic dynamical behavior InthisSectionweconsidertheKMPprocessinaboundedtimeinterval[0,T]un- der the diffusive scaling limit. We discuss the law oflargenumbers (hydrodynamic limit) and the associated dynamical large deviations principle for the empirical energy (2.3). Givena continuous strictly positive energy profileθ ∈C([−1,1];R ), we denote + by νN the probability on Σ corresponding to a local equilibrium distribution θ N (LTE) with an energy profile given by θ. It is defined as N dνN(ξ):= dνN (ξ ) θ θ,x x x=−N Y where dξ ξ dνN := x exp − x θ,x θ(x/N) θ(x/N) n o Giventwoprobabilitymeasuresν,µonΣ wedenotebyh(ν|µ)therelativeentropy N of ν with respect to µ, it is defined as dν(ξ) dν(ξ) h(ν|µ):= dµ(ξ) log dµ(ξ) dµ(ξ) Z WeshallconsidertheKMPprocesswithinitialconditiondistributedaccordingto theproductmeasureνN forsomeenergyprofileθ . Astraightforwardcomputation θ0 0 then shows there exists a constant C (depending on θ ) such that for any N we 0 have the relative entropy bound h(νN|νN)≤CN (3.1) θ0 θ¯ where θ¯is the stationary energy profile (2.10). By the weak law of large numbers for independent variables we also have that νN is associated to the energy profile θ0 7 θ in the following sense. For each δ >0 and each continuous φ 0 lim νN hπ (ξ),φi−hθ ,φi >δ =0 (3.2) N→∞ θ0 N 0 We remark that for the SE(cid:16)P(cid:12) it is possible (and(cid:12)conv(cid:17)enient, see [6]) to consider (cid:12) (cid:12) deterministic initial conditions. For the KMP process, as the “single spin space” R is not discrete, such initial conditions do not satisfy the entropy bound (3.1), + which is required in the standard derivation, see e.g. [17,25], of the hydrodynamic limit. For this reasonwe have chosen the initial condition distributed according to the product measure νN. On the other hand, by the method developed in [21], it θ0 should be also possible to consider deterministic initial configurations. WedenotebyP thedistributionoftheKMPprocesswhentheinitialcondition νN θ0 is distributed according to νN. The measure P is a probability on the space θ0 νθN0 D([0,T];Σ ) of right continuous with left limit paths from [0,T] to Σ . The N N expectation with respect to P is denoted by E . νN νN θ0 θ0 3.1. Hydrodynamic limit. Equation (3.2) is the law of large number for the empiricalenergyattime t=0; the hydrodynamiclimit states that for eachmacro- scopic time t∈[0,T] there exists an energy profile θ(t) such we have the same law of large numbers lim P hπ (ξ(t)),φi−hθ(t),φi >δ =0 (3.3) νN N N→∞ θ0 Furthermore, we can obtain(cid:16)t(cid:12)he energy profile θ(t) b(cid:12)y sol(cid:17)ving the hydrodynamic (cid:12) (cid:12) equation. For the KMP process (as for the SEP) this is simply the the linear heat equation with boundary conditions τ , i.e. θ(t)=θ(t,u) solves ± 1 ∂ θ(t) = ∆θ(t) t 2  (3.4)  θ(t,±1) = τ± θ(0,u) = θ (u) 0 where ∆is the Laplacian. Note that the stationaryprofileθ¯in(2.10) is the unique stationary solution of (3.4). We give below a brief heuristic derivation, which is particularly simple for the KMPprocess,ofthehydrodynamiclimit. Wereferto[14,15]forarigorousproofin thecaseofthesocalledgradientnonequilibriummodelswithfinitesinglespinstate space; the extension to the KPM process should not present additional problems. Letφbeasmoothfunctionwhosesupportisasubsetof(−1,1);fromthegeneral theory of Markov processes, we have that d E hπ (ξ(t)),φi =E L hπ (ξ(t)),φi (3.5) dt νθN0 N νθN0 N N Since the support of φ(cid:0)is a strict su(cid:1)bset of(cid:0)[−1,1], only N2(cid:1)L contributes to 0 L hπ (ξ(t)),φi. A simple computation shows that, when y 6=±N, N N 1 L ξ = ξ +ξ −2ξ (3.6) 0 y y−1 y+1 y 2 we thus get (cid:2) (cid:3) N2 x/N+1/(2N) L hπ (ξ(t)),φi = ξ (t)+ξ (t)−2ξ (t) duφ(u) N N x−1 x+1 x 2 xX∈ΛN(cid:2) (cid:3)Zx/N−1/(2N) 1 1 ≈ ∆ φ(x/N)ξ (t)≈ hπ (ξ(t)),∆φi N x N 2N 2 xX∈ΛN 8 here ∆ φ(x/N) := N2 φ((x−1)/N)+φ((x+1)/N)−2φ(x/N) is the discrete N Laplacian. The first step above comes from (3.6) and (2.3), the second step from (cid:2) (cid:3) discrete integration by parts and last step from the regularity of φ. We have thus obtained the weak formulation of (3.4); it remains to show that also the boundary condition θ(t,±1)=τ is satisfied. For this we need to use the ± boundary generators N2L . These are Glauber like dynamics accelerated by N2 ± so that the energy has well thermalized to its equilibrium value. We get E (ξ (t))≈τ (3.7) νN ±N ± θ0 A standard martingale computation shows that, with a negligible error as N → ∞, π (ξ(t)) becomes non random. We can then remove the expectation value in N the previous equations and get (3.3). 3.2. Dynamic large deviations. We want next to obtain the large deviation principle associated to the law of large number (3.3); more precisely we want to estimate the probability that the empirical energy π (ξ(t)) does not follow the N solution of (3.4) but remains close to some prescribed path π = π(t,u). This probability will be exponentially small in N and we look for the exponential rate. Wefollowtheclassicprocedureinlargedeviationtheory: weperturbthedynamics in such a way that the path π becomes typical and compute the cost of such a perturbation. LetH =H(t,u)beasmoothfunctionvanishingattheboundary,i.e.H(t,±1)= 0. We then consider the following time dependent perturbations of the generators L in (2.2) x,x+1 1 LH f(ξ):= dpe[H(t,x/N)−H(t,(x+1)/N)][pξx+1−(1−p)ξx] f(ξ(x,x+1),p)−f(ξ) x,x+1 Z0 (cid:2) (cid:3) NotethatwehaveessentiallyjustaddedasmalldriftN−1∇H(t,x/N)intheenergy exchange across the bond {x,x+1}. We denote by PH the distribution on the νN θ0 path space D([0,T];Σ ) of this perturbed KMP process. As before EH is the N νN θ0 expectation with respect to PH . νN θ0 The first step to obtain the dynamic large deviations is to derive the hydrodynamic equation for the perturbed KMP process. We claim that for each t∈[0,T], each continuous φ, and each δ >0 we have lim PH hπ (ξ(t)),φi−hθ(t),φi >δ =0 (3.8) νN N N→∞ θ0 (cid:16)(cid:12) (cid:12) (cid:17) (cid:12) (cid:12) where θ(t)=θ(t,u) solves 1 ∂ θ(t) = ∆θ(t)−∇ θ(t)2∇H(t) t 2  θ(t,±1) = τ± (cid:0) (cid:1) (3.9) θ(0,u) = θ (u) 0  9 Theargumenttojustify(3.9)issimilartothepreviousone. Includingtheeffectof theperturbation,thecomputationfollowing(3.6)nowbecomes(asbeforey 6=±N) 1 LHξ = dpe[H(t,(y−1)/N)−H(t,y/N)][pξy−(1−p)ξy−1] (1−p)(ξ +ξ )−ξ 0 y y y−1 y Z0 1 (cid:2) (cid:3) + dpe[H(t,y/N)−H(t,(y+1)/N)][pξy+1−(1−p)ξy] p(ξ +ξ )−ξ y+1 y y Z0 ≈ ξy−1+ξy+1−2ξy + H(t,(y−1)/N)−H(t,(cid:2)y/N) ξyξy−1−ξy2−(cid:3) ξy2−1 2 3 (cid:2) −ξ ξ +ξ2+ξ2(cid:3) y y+1 y y+1 + H(t,y/N)−H(t,(y+1)/N) 3 (cid:2) (cid:3) As before, we consider a smooth function φ whose support is a strict subset of (−1,1); then only N2LH contributes to LHhπ (ξ(t)),φi and we get 0 N N 1 ξ (t)+ξ (t)−2ξ (t) LHhπ (ξ(t)),φi≈ N2φ(x/N) x−1 x+1 x N N N 2 x (cid:26) X ξ (t)ξ (t)−ξ (t)2−ξ (t)2 x x−1 x x−1 + H(t,(x−1)/N)−H(t,x/N) 3 (cid:2) (cid:3)−ξx(t)ξx+1(t)+ξx(t)2+ξx+1(t)2 + H(t,x/N)−H(t,(x+1)/N) 3 1(cid:2) (cid:3) (cid:27) ≈ ξ (t)∆ φ(x/N) x N N x 1X −ξ (t)ξ (t)+ξ (t)2+ξ (t)2 x x+1 x x+1 + ∇ H(t,x/N)∇ φ(x/N) N N N 3 x X where ∇ f(x/N) := N[f((x+1)/N)−f(x/N)] is the discrete gradient. In the N above computations we just used Taylor expansions and discrete integrations by parts. With respectto the verysimple casediscussedbefore,we facenow the main probleminestablishingthehydrodynamiclimit: theaboveequationisnotclosedin π (ξ(t)), i.e.itsrighthandsideisnotafunctionofπ (ξ(t)). Inordertoderivethe N N hydrodynamicequation(3.9),weneedtoexpress−ξ ξ +ξ2+ξ2 intermsofthe x x+1 x x+1 empiricalenergyπ (ξ). Thiswillbe done byassuminga“localequilibrium”state, N wereferto[6,14,15,17,25]forarigorousjustificationinthecontextofconservative interacting particle systems. Letusconsideramicroscopicsitexwhichisfarfromtheboundaryandintroduce a volume V, centered at x, which is very large in microscopic units, but still infini- tesimal at the macroscopic level. The time evolution in V is essentially given only by the bulk dynamics N2LH; since the total amount of energy in V changes only 0 via boundary effects and we are looking at what happens after O(N2) microscopic timeunits,weexpectthatthesysteminV hasrelaxedtothemicro–canonicalstate corresponding to the local empirical energy π (ξ(t))(x/N). To compute this state N let us construct first the canonicalmeasure in V with constant temperature τ >0, namely the product measure dνV,τ(ξ) := x∈V τ−1dξx e−ξx/τ. Let now mV,θ be the associated micro–canonicalmeasure with energy density θ, i.e. Q m (dξ):=ν dξ ξ =θ|V| V,θ V,τ x (cid:16) (cid:12)xX∈V (cid:17) (cid:12) (cid:12) We introduce the function σ(θ) defined by σ(θ):= lim E −ξ ξ +ξ2+ξ2 (3.10) V↑Z mV,θ x x+1 x x+1 (cid:0) (cid:1)