Thermal conductivity for coupled charged harmonic oscillators with noise in a magnetic field
TTHERMAL CONDUCTIVITY FOR COUPLED CHARGED HARMONICOSCILLATORS WITH NOISE IN A MAGNETIC FIELD
KEIJI SAITO AND MAKIKO SASADAA
BSTRACT . We introduce a d -dimensional system of charged harmonic oscillators in a magneticfield perturbed by a stochastic dynamics which conserves energy but not momentum. We studythe thermal conductivity via the Green–Kubo formula, focusing on the asymptotic behavior of theGreen–Kubo integral up to time t (i.e., the integral of the correlation function of the total energycurrent). We employ the microcanonical measure to calculate the Green–Kubo formula in generaldimension d for uniformly charged oscillators. We also develop a method to calculate the Green–Kubo formula with the canonical measure for uniformly and alternately charged oscillators indimension 1. We prove that the thermal conductivity diverges in dimension 1 and 2 while itremains finite in dimension 3. The Green–Kubo integral calculated with the microcanonicalensemble diverges as t / for uniformly charged oscillators in dimension 1, while it is known todiverge as t / without magnetic field. This is the first rigorous example of the new exponent1 / t / and t / . This means that the exponent depends not only on a non-zero magnetic field butalso on the charge structure of oscillators. Date : May 21, 2018.
Key words and phrases.
Thermal conductivity, harmonic chain of oscillators, anomalous diffusion, magneticfield, Hamiltonian system with noise, microcanonical state, canonical state. a r X i v : . [ m a t h . P R ] M a y HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 2 C ONTENTS
1. Introduction 31.1. Background and summary 31.2. Model and main result 52. Uniform charge model under microcanonical measures 102.1. Model 112.2. Conserved quantities and microcanonical states 112.3. Instantaneous energy current correlation 122.4. Proof of Proposition 2.4 152.5. Proof of Proposition 2.7 172.6. Analysis of the inverse Laplace transform 203. Uniform and alternate charge models under canonical measures 233.1. Model 233.2. Conserved quantities and canonical measures 233.3. Instantaneous energy current correlation 233.4. Reduction from the coordinate ( q , v ) to ( r , v ) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 3
1. I
NTRODUCTION
Background and summary.
In low dimensions, the thermal conductivity generally di-verges in the thermodynamic limit. This phenomenon is called anomalous heat transport orsuperdiffusion of energy, a research area of which has now become an interdisciplinary fieldfrom physics to mathematics [2, 6, 10, 11, 16].Diverging thermal conductivity and superdiffusion of energy have been studied for prototypi-cal interacting systems with several conservation laws [3–5, 13, 17]. Among them, the system ofoscillators is one of the most studied models. Using these models, the mechanism of divergingthermal conductivity has been studied mainly through the Green–Kubo formula. Consideringthe 1-dimensional periodic chain of N oscillators, we define the local energy E x of the oscillatorlabeled by x ∈ T N , where T N = Z / N Z is the discrete torus. From the conservation of total en-ergy, the local energy current is defined through the continuity equation with respect to energyas ∂ t E x ( t ) = J x − , x ( t ) − J x , x + ( t ) , where J x , x + is the energy current between the sites x and x + κ is written as κ = lim t → ∞ κ GK ( t ) , κ GK ( t ) = lim N → ∞ ∑ x ∈ T N (cid:90) t ds (cid:104) J x , x + ( s ) J , ( ) (cid:105) , where the thermal conductivity κ is obtained by taking the infinite time limit in the Green–Kubointegral κ GK ( t ) . The symbol (cid:104) ... (cid:105) implies the average over the microcanonical or canonicalensemble. Note that the Green–Kubo integral κ GK ( t ) contains the correlation function of theinstantaneous total energy current. In general low-dimensional systems, the correlation functionshows a power-law decay in the long-time regime, and this leads to the divergence of the Green–Kubo integral.So far, there have been many discussions on the divergence of the Green–Kubo formula interms of the asymptotic behavior of the Green–Kubo integral and the current correlation functionwith numerical and analytical approaches [10]. Recently, many theoretical advances have beenachieved. The key-ingredient of the theories is to focus on the conserved quantities. In any non-linear chain of oscillators connected with spring forces depending only on the distance betweenparticles, the momentum and energy are conserved. In addition, the so-called stretch is alsoconserved. Renormalization type arguments for the hydrodynamic equations of the conservedquantities are performed and they predict a t / asymptotic behavior of the Green–Kubo integralin 1-dimensional systems [12]. More recently, nonlinear fluctuating hydrodynamics argues fora remarkable connection between the dynamics of the sound modes in heat conduction and theKardar–Parisi–Zhang equation [16]. The asymptotic divergent behavior of the Green–Kubo in-tegral is classified into t / or t / depending on the symmetry of the potential function and thepressure [16]. The nonlinear fluctuating hydrodynamics also classifies the possible universalityclass of the dynamical exponent in general diffusive dynamics with several conservation laws,and a Fibonacci sequence of exponents, including the golden mean, is found [13].Another important approach to understand the diverging thermal conductivity relies on anexactly soluble models, among of them the momentum exchange model [1]. This model shows ahybrid dynamics consisting of a deterministic part and stochastic perturbation. In the momentumexchange model, the momenta of the nearest neighbor oscillators are stochastically exchanged so HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 4 that the conserved quantities defined for deterministic dynamics remain conserved. A rigorousanalysis shows the s − / power-law decay in the correlation of the total current if the chain isharmonic [1], and this leads to a t / divergence in the Green–Kubo integral. For this harmonicmomentum exchange model, it is also established that under a proper space–time scaling limit,the macroscopic energy diffusion is described by a fractional diffusion equation [7]. Recentlyvariants of the momentum exchange model have been proposed to simplify the analysis [4, 5].In this paper, we propose another variant of the harmonic momentum exchange model, wherethe oscillators are charged and a magnetic field is applied. We will show a new exponent inthe divergence of the Green–Kubo integral. The magnetic field induces cyclotron motion ofthe particle and hence the standard momentum conservation is broken. This model was partiallyanalyzed in the previous paper [15]. Herein we generalize the previous work and develop severalnovel mathematical techniques We consider a system where the charged harmonic oscillators arearranged on the d -dimensional lattice with periodic boundary conditions and each oscillator canmove in the d ∗ -dimensional space. We assume d ∗ ≥ d ( ≥ ) is arbitrary. The goal of our study is to derive the asymptotic behavior of the Green–Kubo integral for this model. To this end, we calculate the current correlation function withmicrocanonical and canonical measures in different lattice dimension d and charge structures ofoscillators.In Section 2, we study the thermal conductivity via the microcanonical Green–Kubo formulain general dimension d for uniformly charged oscillators, namely the case where the charges ofall oscillators are equal. Here, the microcanonical Green–Kubo formula implies that we employthe microcanonical ensemble for the correlation function of energy current in the Green–Kubointegral. We show that the thermal conductivity diverges in lattice dimensions 1 and 2, whileit is finite in d ≥
3. In dimension 1, the microcanonical Green–Kubo integral diverges as t / .This is a remarkable difference from the case with zero magnetic field having t / behavior asreported in [1]. This is the first rigorous example of the exponent 1 / s − / .In Section 3, we develop a technique to calculate the Green–Kubo integral with the canoni-cal measure in lattice dimension 1. To introduce the canonical measure, the coordinate of thesystem must be changed from the pair of position and velocity to the pair of deformation (thedifference in position between two neighboring oscillators) and velocity. We use this new co-ordinate for uniformly and alternately charged oscillators in dimension 1. Here, the alternatecharge means that the sign of charges is alternating, while the magnitude is the same for alloscillators. In addition, we apply this change of coordinates to a system of uncharged oscil-lators (which is equivalent to the original harmonic momentum exchange model), because the HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 5 canonical measures are not properly defined in [1]. For the canonical Green–Kubo integral, weagain find that the uniformly charged system asymptotically shows t / . However, unchargedand alternately charged systems show t / . This reveals that the exponent depends not only on anon-zero magnetic field, which breaks momentum conservation, but also on the charge structureof the oscillators. Heuristically the average effect from the magnetic field in the alternate chargemodel can be regarded as 0. However, we do not have any rigorous argument for the reason whyuncharged and alternately charged systems show the same exponent thus far.Concerning the proof, unlike the microcanonical case, we cannot solve the resolvent equa-tion directly because of the change of coordinates. Several nice symmetries exist for positioncoordinates, which are essential to solve the resolvent equation explicitly. Hence, we introducea new technique by using a solution in the original position coordinates. On the other hand,as the canonical measures are product measures in the new coordinates of deformations, theexpectation of observables are easy to compute and the equivalence of ensembles is not invoked.Once the divergence of the thermal conductivity is shown, it is natural to inquire about theexact nature of the superdiffusion, or more precisely about the macroscopic evolution equationof energy. We are currently working on this problem using the kinetic approach and expect acorresponding fractional diffusion equation.1.2. Model and main result.
We now introduce our model. We consider a d -dimensional chainof oscillators with periodic boundary conditions of length N moving in the d ∗ -dimensional space.The oscillators are labeled by x ∈ Z dN : = Z d / N Z d . We emphasize that d is not necessarily equalto d ∗ . In particular, as we aim to study the system in a magnetic field, we always assume d ∗ ≥ d ≥ x is denoted by v x ∈ R d ∗ and the displacementfrom its equilibrium position is denoted by q x ∈ R d ∗ .We denote by ( e , e , . . . , e n ) the canonical basis of R n and the coordinates of a vector u ∈ R n in this basis by ( u , . . . , u n ) . Its Euclidean norm | u | is defined by | u | = (cid:112) ( u ) + · · · + ( u n ) andthe scalar product of u and v is denoted by u · v . When it is clear from the context, we will notexplicitly mention whether a vector is in R d or R d ∗ . Instead, for better readability, we use thealphabets a , b for indexes in { , , . . . , d } and j , k , (cid:96), m for indexes in { , , . . . , d ∗ } . For example, x = ( x a ) a for x ∈ Z dN and v x = ( v j x ) j for v x ∈ R d ∗ .For two functions f ( t ) and g ( t ) defined on [ , ∞ ) , we denote by f ( t ) ∼ g ( t )( t → ∞ ) if thereexists a constant C > t , C g ( t ) ≤ f ( t ) ≤ Cg ( t ) .For F : Z dN → R or F : Z d → R , we introduce the discrete gradient of F in the direction e a defined by ( ∇ e a F )( x ) = F ( x + e a ) − F ( x ) and the discrete Laplacian of F defined by ( ∆ F )( x ) = ∑ | y − x | = ( F ( y ) − F ( x )) = d ∑ a = { F ( x + e a ) + F ( x − e a ) − F ( x ) } . The dynamics of the chain of oscillators in a magnetic field is given by the following differ-ential equations (cid:40) ddt q x = v x ddt v x = [ ∆ q ] x − B σ v x (1.1) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 6 for x ∈ Z dN where σ = ( σ j , k ) is d ∗ × d ∗ matrix with entries σ , = − , σ , = σ j , k = ( j , k ) (cid:54) = ( , ) , ( , ) and B ∈ R is the signed strength of the magnetic field. In this model, theoscillators are uniformly charged. For simplicity, we only consider the magnetic field orthogonalto the ( , ) plain herein, but we can replace the matrix B σ to an arbitrary real skew-symmetricmatrix (see Remark 2.1).The dynamics (1.1) is also written in the following way: (cid:40) ddt q j x = v j x ddt v j x = [ ∆ q j ] x + δ j , Bv x − δ j , Bv x (1.2)for j = , . . . , d ∗ and x ∈ Z dN . Remark 1.1.
If B = , the dynamics (1.1) is the Hamiltonian system with the following Hamil-tonian H N = ∑ x ∈ Z dN (cid:32) | v x | + ∑ | y − x | = | q y − q x | (cid:33) . For B (cid:54) = , the dynamics is also a Hamiltonian system with a canonical momentum p x : = v x + B σ q x and the Hamiltonian H BN = ∑ x ∈ Z dN (cid:32) | p x − B σ q x | + ∑ | y − x | = | q y − q x | (cid:33) . However, we do not use the canonical momentum in this paper.
It is obvious that the dynamics (1.1) conserves the total energy ∑ x ∈ Z dN E x , where we define theenergy of the oscillator x by E x = | v x | + ∑ | y − x | = | q y − q x | . Whereas the total velocity (or canonical momentum) ∑ x ∈ Z dN v x is also conserved if B =
0, this isnot the case if B (cid:54) =
0. Instead, the sum of the pseudomomentum ˜ p x ˜ p x : = v x + B σ q x is conserved, which is generally the case for Hamiltonian systems in a magnetic field [8]. Inparticular, if the dynamics starts from the initial value satisfying ∑ x ∈ Z dN q x = ∑ x ∈ Z dN v x = , thenfor any time t , ddt ∑ x ∈ Z dN q x = ∑ x ∈ Z dN v x = − B σ ∑ x ∈ Z dN q x , since ∑ x ∈ Z dN ˜ p x = . Hence, ∑ x ∈ Z dN q x = ∑ x ∈ Z dN v x = t . This observation showsthat the following microcanonical state space Ω N , E : = { ( q x , v x ) x ∈ Z dN ; ∑ x q x = ∑ x v x = , ∑ x E x = N d E } HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 7 is conserved by the dynamics for each fixed averaged energy E >
0. Note that the microcanonicalstate space does not depend on B . We present some discussions on the microcanonical state spacefor general initial configurations in Subsection 2.2.Now, we add a stochastic perturbation to this deterministic model. We introduce the velocityexchange noise, i.e., each pair of nearest neighbor oscillators exchanges the j -th component oftheir velocities at random exponential times with intensity γ > j ∈ { , , . . . , d ∗ } . γ represents the strength of the noise. This stochastic perturbation is given by the operator γ S acting on functions f : R d ∗ N → R as S f = ∑ x ∈ Z dN d ∑ a = d ∗ ∑ j = ( f ( q , v j , x , x + e a ) − f ( q , v )) . (1.3)Here, v j , x , y is obtained from v by exchanging the variables v j x and v j y . Remark 1.2.
This specific choice of noise is not important. Our proof is also applicable for thecontinuous noise used in [1], and yields the same divergence exponent of the thermal conduc-tivity.
The whole dynamics is generated by the operator L : = A + BG + γ S with the operators A and G acting on functions f ∈ C ( R d ∗ N d ) as A = ∑ x ∈ Z dN { v x · ∂ q x + [ ∆ q ] x · ∂ v x } = ∑ x ∈ Z dN d ∗ ∑ j = (cid:0) v j x ∂ q j x + [ ∆ q j ] x ∂ v j x (cid:1) , (1.4) G = ∑ x ∈ Z dN v x · σ ∂ v x = ∑ x ∈ Z dN (cid:0) v x ∂ v x − v x ∂ v x (cid:1) . Because ∑ x ∈ Z dN q x , ∑ x ∈ Z dN v x and ∑ x ∈ Z dN E x are conserved by the operator S , the microcanonicalstate space Ω N , E is also conserved by the whole dynamics. Moreover, the uniform probabilitymeasure on the space Ω N , E is stationary for the dynamics, and A and G are antisymmetric and S is symmetric with respect to this measure. We denote this microcanonical measure by µ N , E andthe expectation with respect to it by E N , E [ · ] .We aim to obtain the asymptotic behavior of the microcanonical Green–Kubo integral definedby taking the infinite size limit in the correlation of the integrated total current κ a , bN , E ( t ) ; κ a , bN , E ( t ) = N d E t E N , E [ (cid:0) ∑ x ∈ Z dN J x , x + e a ([ , t ]) (cid:1)(cid:0) ∑ x ∈ Z dN J x , x + e b ([ , t ]) (cid:1) ] (1.5)for a , b ∈ { , , . . . , d } where E N , E is the expectation for the dynamics starting from the micro-canonical measure µ N , E and J x , x + e a ([ , t ]) is the total energy current from x to x + e a up to time t . By symmetry, it is easy to see that κ a , bN , E ( t ) = δ a , b κ , N , E ( t ) (see Lemma 2.3).Our main result for the microcanonical Green–Kubo integral is the following: HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 8
Theorem 1.3.
For any d ≥ , d ∗ ≥ and t ≥ , κ , E ( t ) : = lim N → ∞ κ , N , E ( t ) exists. Moreover, ifB (cid:54) = , κ , E ( t ) ∼ t ( t → ∞ ) if d = and d ∗ = , κ , E ( t ) ∼ t ( t → ∞ ) if d = and d ∗ ≥ , κ , E ( t ) ∼ log t ( t → ∞ ) if d = and d ∗ ≥ , lim sup t → ∞ κ , E ( t ) < ∞ if d ≥ and d ∗ ≥ . (1.6) Remark 1.4.
Our proof also applies to the case B = and for that case the asymptotic behaviorof κ , E ( t ) is κ , E ( t ) ∼ t ( t → ∞ ) if d = , κ , E ( t ) ∼ log t ( t → ∞ ) if d = , lim sup t → ∞ κ , E ( t ) < ∞ if d ≥ for any d ∗ ≥ . Note that d ∗ = is also allowed at this remark because B = . This asymptoticbehavior has already been shown in Theorem 1 and 2 of [1] under the condition d = d ∗ . Remark 1.5.
The thermal conductivity via the microcanocial Green–Kubo integral in the direc-tion e is defined as the limit (when it exists) κ , E = lim t → ∞ lim N → ∞ E tN d E N , E [( ∑ x ∈ Z dN J x , x + e ([ , t ])) ] . Theorem 1.3 shows that the Green-Kubo integral κ , E ( t ) = lim N → ∞ E tN d E N , E [( ∑ x ∈ Z dN J x , x + e ([ , t ])) ] exists for any t, but κ , E = lim t → ∞ κ , E ( t ) = ∞ for d = , . Remark 1.6.
In this model, the dispersion relations are ˜ ω ± ( θ ) = (cid:113) ω θ + ( B ) ± B for d ∗ = ,and ˜ ω ± ( θ ) and ω θ for d ∗ ≥ where ω θ = (cid:113) ∑ da = sin ( πθ a ) is the dispersion relation of thedynamics with zero magnetic field, B = . In particular, ∂ θ ˜ ω ± ( θ ) = π sin ( πθ ) cos ( πθ ) √ ω θ +( B ) and thesound velocity of these modes vanish: lim θ → | ∂ ± θ ˜ ω ( θ ) | = . Next, we consider the canonical measure, which is defined in [1] by µ N , β ( dqdv ) = Z exp ( − β ∑ x ∈ Z dN E x ) Π x ∈ Z dN Π d ∗ j = dq j x dv j x . However, the partition function Z in the expression above is infinite since the total energy ∑ x ∈ Z dN E x is invariant under the translation ( q x ) x ∈ Z dN → ( q x + c ) x ∈ Z dN for any c ∈ R d ∗ . Hence,to handle the canonical measures, we redefine the dynamics with different coordinates and statespaces, which is physically identical to (1.1) in the bulk, but the boundary conditions are differ-ent. Since state spaces and canonical measures in the new coordinates become complicated for d ≥
2, we study only the case d =
1. Moreover, for notational simplicity, we restrict ourselves
HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 9 to the case d ∗ =
2. In this setting, we study three versions of the dynamics, namely ( ) zeromagnetic field, B =
0, (i) uniform charges, and (ii) alternate charges.We now introduce the dynamics in new coordinates. Suppose d = d ∗ =
2. The oscilla-tors are labeled by x ∈ Z N : = Z / N Z . We use x rather than x to emphasize that x is not a vector.The velocity of the oscillator x is denoted by v x ∈ R as before, and instead of q x , we introduce r x ∈ R which represents the difference in the displacement between the oscillators x and x + r x = q x + − q x but q x is no longer periodic. In other words, we do not assume ∑ x ∈ Z N r x = .The equations of motion for the three cases are ( ) (cid:40) ddt r jx = v jx + − v jx , ddt v jx = r jx − r jx − , ( i ) (cid:40) ddt r jx = v jx + − v jx , ddt v jx = r jx − r jx − + δ j , Bv x − δ j , Bv x , and ( ii ) (cid:40) ddt r jx = v jx + − v jx , ddt v jx = r jx − r jx − + ( − ) x ( δ j , Bv x − δ j , Bv x ) for j = , x ∈ Z N . For dynamics (ii), we assume that N is even. Note that dynamics (i) isobtained from the dynamics (1.2) by changing the coordinates formally as r x = q x + − q x .As before obvious by the dynamics (0), (i) and (ii) conserve the total energy ∑ x ∈ Z N E x , wherewe define the energy of the oscillator x by E x = | v x | + | r x | + | r x − | . We now consider the stochastic perturbation of these dynamics. The operator of the perturba-tion is given by the operator γ S r acting on functions f : R N → R as S r f = ∑ x ∈ Z N ∑ j = ( f ( r , v j , x , x + ) − f ( r , v )) (1.8)whose physical interpretation is the same as before. The generator of the whole dynamics is L r : = A r + BG ( ) r + γ S r with the operators A r and G ( ) r for = , i , ii acting on functions f ∈ C ( R N ) as A r = ∑ x ∈ Z N ∑ j = (cid:0) ( v jx + − v jx ) ∂ r jx + ( r jx − r jx − ) ∂ v jx (cid:1) , G ( ) r = , G ( i ) r = ∑ x ∈ Z N (cid:0) v x ∂ v x − v x ∂ v x (cid:1) , (1.9) G ( ii ) r = ∑ x ∈ Z N ( − ) x (cid:0) v x ∂ v x − v x ∂ v x (cid:1) . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 10
For each β >
0, the canonical measure µ N , β on R N is defined by µ N , β ( drd p ) = Z N β exp ( − β ∑ x E x ) Π x ∈ Z N Π j = dr jx dv jx = Z N β exp ( − β ∑ x ∑ j = (( v jx ) + ( r jx ) )) Π x ∈ Z N Π j = dr jx dv jx where Z β = (cid:113) πβ . Dynamics (0), (i) and (ii) are all stationary under µ N , β since A r and G ( ) r for = i , ii are antisymmetric and S r is symmetric with respect to this measure.We consider the canonical Green–Kubo integral defined by taking the infinite size limit in thecorrelation of the integrated total current κ ( ) N , β ( t ) ; κ ( ) N , β ( t ) = β Nt E ( ) N , β [ (cid:0) ∑ x ∈ Z N J x , x + ([ , t ]) (cid:1) ] where E ( ) N , β is the expectation for the dynamics generated by L ( ) starting from the canonicalmeasure µ N , β and J x , x + ([ , t ]) is the total energy current from x to x + t .Our main result for the canonical Green–Kubo integral is the following: Theorem 1.7.
For any t ≥ and = , i , ii, κ ( ) β ( t ) : = lim N → ∞ κ ( ) N , β ( t ) exists. Moreover, κ ( ) β ( t ) ∼ t ( t → ∞ ) , κ ( i ) β ( t ) ∼ t ( t → ∞ ) , κ ( ii ) β ( t ) ∼ t ( t → ∞ ) if γ ≤ . Remark 1.8.
We conjecture that the assumption γ ≤ is not essential and the asymptotic behav-ior of κ ( ii ) β ( t ) does not depend on the parameter γ . This assumption is needed only in Subsection3.6. We provide some discussions on it in Remark 3.8. The remainder of the article is organized as follows; In Section 2, we study the model withuniform charges under microcanonical measures in general dimensions, and establish Theorem1.3. Section 3 concerns the one-dimensional chain of uncharged oscillators and oscillators withuniform and alternate charges under canonical measures in the two-dimensional space, includingthe proof of Theorem 1.7.2. U
NIFORM CHARGE MODEL UNDER MICROCANONICAL MEASURES
This section provides a detailed study of the uniform charge model in general dimensionsin the coordinate ( q , v ) . We start by recalling the description of the model in Subsection 2.1.Subsection 2.2 is for the analysis of conserved quantities and microcanonical states. In Subsec-tion 2.3, we study the thermal conductivity in terms of the current-current correlation and give aproof of Theorem 1.3 by assuming one key proposition. The proof of this proposition is dividedinto several steps and given in the last three subsections. HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 11
Model.
We consider the Markov process ( q x ( t ) , v x ( t )) x ∈ Z dN on R d ∗ N d generated by theoperator L = A + BG + γ S where A , G and S are defined at (1.3) and (1.4). Note that B ∈ R and γ > Remark 2.1.
We can replace the matrix B σ by any real skew-symmetric matrix M with size d ∗ at(1.1) and consider the dynamics associated to this matrix, which conserves the total energy. Notethat any real skew-symmetric matrix M is given in the form M = U Σ U T where U is orthogonaland Σ is block diagonal with a form Σ = B J · · · · · · · · · · · · B J · · · · · · · · · ... · · · · · · · · · ... ... B m J · · · · · · ... ... ... · · · ... ... ... ... ... · · ·
00 0 0 0 0 · · · , J = (cid:18) − (cid:19) for some m ≤ d ∗ and real numbers B i (cid:54) = , i = , , . . . , m. In the above expression of Σ , each correspond to × size zero matrix, but only when d ∗ is odd, each in the last row is ascalar. Obviously, Σ = B σ if m = and B = B. By changing the coordinates ( q x , v x ) to ( ˜ q x = U q x , ˜ v x = U v x ) , the deterministic dynamics associated to the matrix M is rewritten as (cid:40) ddt ˜ q x = ˜ v x ddt ˜ v x = [ ∆ ˜ q ] x − Σ ˜ v x . Then, Theorem 1.3 is generalized to this dynamics with stochastic perturbation and the asymp-totic behavior of the microcanonical Green-Kubo integral is given as κ , E ( t ) ∼ t ( t → ∞ ) ( d = , d ∗ = m ) κ , E ( t ) ∼ t ( t → ∞ ) ( d = , d ∗ (cid:54) = m ) κ , E ( t ) ∼ log t ( t → ∞ ) ( d = , d ∗ ≥ ) lim sup t → ∞ κ , E ( t ) < ∞ ( d ≥ , d ∗ ≥ ) . In particular, if d ∗ is odd, then κ , E ( t ) ∼ t for d = . The generalization from B σ to Σ isstraightforward, so we omit the proof. Conserved quantities and microcanonical states.
As described in the Introduction, thedynamics (1.1) conserves the total energy ∑ x ∈ Z dN E x . In particular, G E x =
0. On the other hand,the dynamics does not conserved the total velocity ∑ x ∈ Z dN v x if B (cid:54) = G ∑ x ∈ Z dN v x (cid:54) = ∑ x q x = ∑ x v x =
0, then thecondition is conserved by the dynamics. We discuss what happens for general initial conditionshere. Let ¯ q = N d ∑ x q x and ¯ v = N d ∑ x v x . Then, it is easy to see that (cid:40) ddt ¯ q = ¯ v ddt ¯ v = − B σ ¯ v (2.1) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 12 and then | ¯ v | is conserved. Moreover, the centered configuration ( Q x : = q x − ¯ q , V x : = v x − ¯ v ) x ∈ Z dN also evolves according to the stochastic dynamics generated by L . In this way, the dy-namics of ( q x , v x ) is decomposed into the dynamics of ( Q x , V x ) and ( ¯ q , ¯ v ) . In particular, underany stationary measure, the distribution of ( Q x , V x ) is also stationary.In the next subsection, we will see that the total instantaneous energy current denoted by ∑ x j x , x + e is explicitly given as ∑ x j x , x + e = − d ∗ ∑ j = ∑ x v j x ( q j x + e − q j x − e ) and plays an essential role for the asymptotic behavior of the Green-Kubo integral. Since d ∗ ∑ j = ∑ x v j x ( q j x + e − q j x − e ) = d ∗ ∑ j = ∑ x V j x ( Q j x + e − Q j x − e ) , this total instantaneous energy current depends only on the centered configuration ( Q x , V x ) . Also,since ∑ x Q x = ∑ x V x = at time t =
0, it stays in Ω N , ˜ E for ˜ E = N d ∑ ˜ E x where ˜ E x is the energyof the oscillator x associated to the configuration ( Q x , V x ) . By definition, E : = N d ∑ x E x = N d ∑ x (cid:32) | v x | + ∑ | y − x | = | q y − q x | (cid:33) = N d ∑ x (cid:32) | V x | + V x · ¯ v + | ¯ v | + ∑ | y − x | = | Q y − Q x | (cid:33) = N d ∑ x ˜ E x + | ¯ v | = ˜ E + | ¯ v | . From the above observation, on a general microcanonical state with the averaged energy E andthe square of the norm of averaged velocity | ¯ v | , the distribution of the centered configuration ( Q x , V x ) is µ N , ˜ E where ˜ E = E − | ¯ v | , and the microcanonical Green-Kubo integral is given via κ a , bN , ˜ E .In this way, we can reduce the study of the Green-Kubo integral on a general microcanonicalstate to the one on the microcanonical state Ω N , E .2.3. Instantaneous energy current correlation.
In this subsection, we study the asymptoticbehavior of the correlation of the integrated energy current κ a , bN , E ( t ) . We follow the strategyof [1].The energy conservation law can be read locally as E x ( t ) − E x ( ) = d ∑ a = (cid:0) J x − e a , x ([ , t ]) − J x , x + e a ([ , t ]) (cid:1) where J x , x + e a ([ , t ]) is the total energy current between x and x + e a up to time t . This can bewritten as J x , x + e a ([ , t ]) = (cid:90) t j x , x + e a ( s ) ds + M x , x + e a ( t ) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 13 where M x , x + e a ( t ) is the martingale given by M x , x + e a ( t ) = − d ∗ ∑ j = (cid:90) t (cid:8) ( v j x + e a ) ( s − ) − ( v j x ) ( s − ) (cid:9) dM j , x , x + e a ( s ) with M j , x , x + e a ( t ) = N γ j , x , x + e a ( t ) − γ t and { N γ j , x , x + e a ( t ) } j = ,..., d ∗ , x ∈ Z dN , a = ,..., d are d ∗ dN d -independentPoisson processes with intensity γ .The instantaneous energy current j x , x + e a can be written as j x , x + e a = j a x , x + e a + j s x , x + e a with j a x , x + e a = − d ∗ ∑ j = ( q j x + e a − q j x )( v j x + e a + v j x ) , j s x , x + e a = − γ d ∗ ∑ j = (cid:8) ( v j x + e a ) − ( v j x ) (cid:9) where j a x , x + e a is the contribution of the deterministic dynamics to the instantaneous energy cur-rent and j s x , x + e a is the stochastic noise contribution to it. Remark 2.2.
Since G E x = , the instantaneous energy currents does not depend on B. Recall that the microcanocial Green-Kubo integral is defined by taking the infinite size limitin κ a , bN , E ( t ) = N d E t E N , E [ (cid:0) ∑ x ∈ Z dN J x , x + e a ([ , t ]) (cid:1)(cid:0) ∑ x ∈ Z dN J x , x + e b ([ , t ]) (cid:1) ] . We first give a simple lemma about a spatial symmetry of the system.
Lemma 2.3.
For any ≤ a , b ≤ d, κ a , bN , E ( t ) = δ a , b κ , N , E ( t ) . Proof.
Let τ : { , , . . . , d } → { , , . . . , d } be any permutation and ( τ x ) a = x τ ( a ) . From thesymmetry with respect to τ of the generator L or simply by the explicit form of (1.1) and (1.3),the distribution of the Markov process ( q τ x ( t ) , v τ x ( t )) with the initial measure µ N , E is the sameas that of ( q x ( t ) , v x ( t )) with the same initial measure. In particular, κ a , aN , E ( t ) = κ , N , E ( t ) for any1 ≤ a ≤ d . Next, let R a : Z dN → Z dN be the permutation ( R a x ) b = ( − ) δ a , b x b . By the same reason,the distribution of ( q R a x ( t ) , v R a x ( t )) is same as that of ( q x ( t ) , v x ( t )) with the initial measure µ N , E .For a (cid:54) = b , ∑ x ∈ Z dN J x , x + e a ([ , t ]) = − ∑ x ∈ Z dN J R a x , R a ( x + e a ) ([ , t ]) , ∑ x ∈ Z dN J x , x + e b ([ , t ]) = ∑ x ∈ Z dN J R a x , R a ( x + e b ) ([ , t ]) and so κ a , bN , E ( t ) = − κ a , bN , E ( t ) = (cid:3) We denote J e [ , t ] = ∑ x ∈ Z dN J x , x + e ([ , t ]) . Since ∑ x j s x , x + e =
0, we have1 N d E N , E [ J e [ , t ] ] = N d E N , E [( (cid:90) t ∑ x j a x , x + e ( s ) ds + M ( t )) ] where M is a martingale given by M ( t ) = − ∑ x ∈ Z dN d ∗ ∑ j = (cid:90) t (cid:8) ( v j x + e ) ( s − ) − ( v j x ) ( s − ) (cid:9) dM j , x , x + e ( s ) . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 14
Since ∑ x j a x , x + e = − ∑ d ∗ j = ∑ x v j x ( q j x + e − q j x − e ) is bounded under the condition ∑ x E x = EN d , A ( t ) : = (cid:82) t ∑ x j a x , x + e ( s ) ds is a bounded variation process. Then, by Ito’s formula for jumpprocesses (c.f. [14]), we have E N , E [ A ( t ) M ( t )] = E N , E [ (cid:90) t M ( s ) d A ( s )] = E N , E [ (cid:90) t M ( s ) ∑ x j a x , x + e ( s ) ds ]= (cid:90) t E N , E [ M ( s ) ∑ x j a x , x + e ( s )] ds . Then, by considering the time reversed process as in [1], we obtain E N , E [ M ( s ) ∑ x j a x , x + e ( s )] = s ≥
0, and hence E N , E [ A ( t ) M ( t )] =
0. Here, the generator of the time reversedprocess is − A − BG + γ S , but it does not make any difficulty to apply the argument in [1], sincethe current does not depend on B .Also, 1 N d E N , E [( M ( t )) ] = γ N d ∑ x ∈ Z dN d ∗ ∑ j = (cid:90) t E N , E [ (cid:8) ( v j x + e ) ( s − ) − ( v j x ) ( s − ) (cid:9) ] ds = γ t d ∗ ∑ j = E N , E [ (cid:8) ( v j e ) − ( v j ) (cid:9) ] = d ∗ γ t E N , E [ (cid:8) ( v e ) − ( v ) (cid:9) ] . Thanks to the equivalence of ensembles given in Lemma C.1 in Appendix C, the last quantityis equal to E γ td ∗ + o ( N ) where lim N → ∞ o ( N ) =
0. So far, we have shown that12 E tN d E N , E [( ∑ x ∈ Z dN J x , x + e ([ , t ])) ]= E tN d (cid:90) t (cid:90) t E N , E [ (cid:0) ∑ x j a x , x + e ( s ) (cid:1)(cid:0) ∑ x j a x , x + e ( u ) (cid:1) ] dsdu + γ d ∗ + o ( N ) . By the spatial translation symmetry and a simple computation, the last term is rewritten as12 E t (cid:90) t (cid:90) t E N , E [ ∑ x j a x , x + e ( s ) j a , e ( u )] dsdu + γ d ∗ + o ( N )= E (cid:90) t ( − st ) C N ( s ) ds + γ d ∗ + o ( N ) where C N ( s ) = E N , E [ ∑ x j a x , x + e ( s ) j a , e ( )] . Then, Theorem 1.3 follows immediately from thenext proposition. Proposition 2.4.
The sequence of functions { C N : [ , ∞ ) → R } converges compactly to a functionC ∞ : [ , ∞ ) → R as N → ∞ , namely the convergence is uniform on each compact subset of [ , ∞ ) .Moreover, C ∞ ( s ) = C ( s ) + C ( s ) + C ( s ) + ( d ∗ − ) C ( s ) where lim sup t → ∞ (cid:90) t ( − st ) C ( s ) < ∞ , C ( s ) ∼ s − d / − , C ( s ) ∼ s − d / − / and C ( s ) ∼ s − d / ass → ∞ . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 15
Remark 2.5.
As we will see in Subsection 2.6, C ( s ) is an oscillating term and in particular | C ( s ) | ∼ s − d / . In d = , this term is bigger than the term C ( s ) , which is the main term forthe Green-Kubo integral. This big oscillation of the current correlation function is also seennumerically in [15]. Proof of Proposition 2.4.
The proof of Proposition 2.4 is divided into several steps. Wefirst show the relative compactness of the set of functions { C N } N . Lemma 2.6.
The set of functions { C N : [ , ∞ ) → R } N is uniformly bounded and equicontinuous.Proof. By the translation invariance of the measure µ N , E and the Cauchy-Schwarz inequality, | C N ( s ) | = | N d E N , E [ ∑ x j a x , x + e ( s ) ∑ y j a y , y + e ( )] |≤ N d (cid:114) E N , E [( ∑ x j a x , x + e ( s )) ] E N , E [( ∑ y j a y , y + e ( )) ] . Using the time stationarity of the dynamics, we havesup s ∈ [ , ∞ ) | C N ( s ) | ≤ N d E N , E [( ∑ x j a x , x + e ) ] . Recall that ∑ x j a x , x + e = − ∑ d ∗ j = ∑ x ( q j x + e − q j x − e ) v j x . By the symmetry of the measure µ N , E under the change of variables v j → − v j for any fixed j , we have E N , E [( ∑ x j a x , x + e ) ] = E N , E [( d ∗ ∑ j = ∑ x ( q j x + e − q j x − e ) v j x ) ]= d ∗ ∑ j = ∑ x , y E N , E [( q j x + e − q j x − e ) v j x ( q j y + e − q j y − e ) v j y ] . Since the measure µ N , E is translation invariant and symmetric with respect to the components j = , , . . . , d ∗ , the last expression is equal to d ∗ N d ∑ x E N , E [( q x + e − q x − e ) v x ( q e − q − e ) v ] . By the symmetry of the measure µ N , E with respect to any permutation on the index set Z dN for v , E N , E [( q x + e − q x − e ) v x ( q e − q − e ) v ] = E N , E [( q x + e − q x − e ) v y ( q e − q − e ) v ] for any x , y (cid:54) = . Noting ∑ x v x = µ N , E , we have E N , E [( q x + e − q x − e ) v x ( q e − q − e ) v ] = − N d − E N , E [( q x + e − q x − e )( q e − q − e )( v ) ] for any x (cid:54) = . Therefore,4 d ∗ N d E N , E [( ∑ x j a x , x + e ) ]= E N , E [( q e − q − e ) ( v ) ] − N d − ∑ x (cid:54) = E N , E [( q x + e − q x − e )( q e − q − e )( v ) ] HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 16 = N d N d − E N , E [( q e − q − e ) ( v ) ] . For the last equality, we use the fact that ∑ x (cid:54) = q j x ± e = − q j ± e under µ N , E . Hence, the equivalenceof ensemble (Lemma C.1 (iv)) shows E N , E [( ∑ x j a x , x + e ) ] is of order N d and the set of functions { C N : [ , ∞ ) → R } N is uniformly bounded.Next, we show the equicontinuity. Note that L ( ∑ x j a x , x + e ) = − L ( d ∗ ∑ j = ∑ x ( q j x + e − q j x − e ) v j x )= − d ∗ ∑ j = ∑ x { ( v j x + e − v j x − e ) v j x + ( q j x + e − q j x − e )( ∆ q j x + δ j , Bv x − δ j , Bv x )+ γ ( q j x + e − q j x − e ) ∆ v j x } = − d ∗ ∑ j = ∑ x ( q j x + e − q j x − e )( δ j , Bv x − δ j , Bv x + γ ∆ v j x ) . We denote by W the last term, namely W = − d ∗ ∑ j = ∑ x ( q j x + e − q j x − e )( δ j , Bv x − δ j , Bv x + γ ∆ v j x ) . Then, ∑ x j a x , x + e ( t ) − ∑ x j a x , x + e ( ) = (cid:90) t W ( s ) ds + m t where m t is a martingale. Therefore, | C N ( t ) − C N ( s ) | = | N d E N , E [ ∑ x j a x , x + e ( t ) ∑ y j a y , y + e ( )] − N d E N , E [ ∑ x j a x , x + e ( s ) ∑ y j a y , y + e ( )] |≤ N d (cid:90) ts E N , E [ | W ( r ) ∑ y j a y , y + e ( ) | ] dr . By the Cauchy-Schwarz inequality and the time stationarity of the dynamics, E N , E [ | W ( r ) ∑ y j a y , y + e ( ) | ] ≤ (cid:114) E N , E [ | W ( r ) | ] E N , E [ | ∑ y j a y , y + e ( ) | ]= (cid:114) E N , E [ W ] E N , E [( ∑ y j a y , y + e ) ] . By the same argument for the term E N , E [( ∑ x j a x , x + e ) ] , we can show that E N , E [ W ] is of order N d . Hence, the term N d (cid:113) E N , E [ W ] E N , E [( ∑ y j a y , y + e ) ] is uniformly bounded in N and so the setof functions { C N : [ , ∞ ) → R } N is equicontinuous. (cid:3) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 17
Next, we show that the Laplace transform of C N , which we denote by ˜ C N converges pointwiseto a function ˜ C ∞ on ( , ∞ ) . For each λ >
0, let ˜ C N ( λ ) = (cid:82) ∞ C N ( s ) exp ( − λ s ) ds and ˜ C ∞ : ( , ∞ ) → R be a function given by˜ C ∞ ( λ ) = E d ∗ (cid:90) [ , ] d sin ( πθ ) ω θ ( λ + γω θ )( λ + λ γω θ + ω θ )( λ + γω θ ) ( λ + λ γω θ + ω θ ) + B λ ( λ + γω θ ) d θ + ( d ∗ − ) E d ∗ (cid:90) [ , ] d sin ( πθ ) ω θ λ + γω θ d θ (2.2)where ω θ = ∑ da = sin ( πθ a ) . Proposition 2.7.
For any λ > , lim N → ∞ ˜ C N ( λ ) = ˜ C ∞ ( λ ) . We give a proof of this proposition in the next subsection.From a general observation of the convergence of functions, Lemma 2.6 and Proposition 2.7imply the existence of the limiting function C ∞ on [ , ∞ ) whose Laplace transform is ˜ C ∞ (wegive a rigorous argument for this in Proposition A.1 in Appendix A for completeness).Finally, we study the asymptotic behavior of the inverse Laplace transform of ˜ C ∞ in Subsec-tion 2.6 which completes the proof of Proposition 2.4. Remark 2.8.
By taking B = in (2.2), we obtain (cid:90) ∞ C ∞ ( s ) exp ( − λ s ) ds = E d ∗ (cid:90) [ , ] d sin ( πθ ) ω θ λ + γω θ d θ which recovers the result (48) of [1] (where the stochastic noise is different, hence the constantdoes not coincide). Proof of Proposition 2.7.
We follow the strategy of [1] again, for which an explicit ex-pression of a resolvent equation is crucial.For each λ >
0, let u λ , N be the solution of the resolvent equation ( λ − L ) u λ , N = ∑ x j a x , x + e . Since L is the generator of the process, we have ˜ C N ( λ ) = E N , E [ u λ , N j a , e ] .From Lemmas B.1 and B.2 in Appendix B, u λ , N is explicitly given by u λ , N = ∑ x , y ∈ Z dN ( g λ , N ( x − y ) q x q y + g λ , N ( x − y )( q x v y + q x v y )+ g λ , N ( x − y )( q x v y − q x v y ) + g λ , N ( x − y ) v x v y )+ d ∗ ∑ j = ∑ x , y ∈ Z dN g λ , N ( x − y ) q j x v j y with the solutions of (B.1) and (B.2). By the symmetry of the measure µ N , E under the change ofvariables q j → − q j or v j → − v j for any fixed j , we have˜ C N ( λ ) = E N [ u λ , N j a , e ] = ∑ x , y ∈ Z dN g λ , N ( x − y ) E N [( q x v y + q x v y ) j a , e ] HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 18 + d ∗ ∑ j = ∑ x , y ∈ Z dN g λ , N ( x − y ) E N [ q j x v j y j a , e ] . Moreover, by the symmetry with respect to the components j = , , . . . , d ∗ , we have E N , E [ q j x v j y j a , e ] = − E N , E [ q x v y ( q e − q )( v e + v )] for any j . Therefore,˜ C N ( λ ) = − ∑ x , y ∈ Z dN ( g λ , N (cid:0) x − y ) + ( d ∗ − ) g λ , N ( x − y ) (cid:1) E N , E [ q x v y ( q e − q )( v e + v )]= − ∑ x , y ∈ Z dN ( g λ , N (cid:0) x − y ) + ( d ∗ − ) g λ , N ( x − y ) (cid:1) E N , E [ q x v y ( q e − q e − ) v ] . (2.3)For the second equality, we use the fact that the sum is taken over all x , y ∈ Z dN .By the symmetry of the measure µ N , E with respect to any permutation on the index set Z dN for v , E N , E [ q x v y ( q e − q − e ) v ] = E N , E [ q x v y (cid:48) ( q e − q − e ) v ] for any y (cid:54) = and y (cid:48) (cid:54) = . Noting ∑ x v x =
0, we have E N , E [ q x v y ( q e − q − e ) v ] = − N d − E N , E [ q x ( q e − q − e )( v ) ] for any y (cid:54) = , and so the last term of (2.3) is rewritten as − ∑ x ∈ Z dN (cid:0) g λ , N ( x ) + ( d ∗ − ) g λ , N ( x ) (cid:1) E N , E [ q x ( q e − q − e )( v ) ]+ ( N d − ) ∑ x , y ∈ Z dN , y (cid:54) = (cid:0) g λ , N ( x − y ) + ( d ∗ − ) g λ , N ( x − y ) (cid:1) E N , E [ q x ( q e − q − e )( v ) ] . Since g λ , N ( x ) = − g λ , N ( − x ) , g λ , N ( x ) = − g λ , N ( − x ) and in particular ∑ x g λ , N ( x ) = ∑ x g λ , N ( x ) =
0, we finally obtain that˜ C N ( λ ) = − N d N d − ∑ x ∈ Z dN (cid:0) g λ , N ( x ) + ( d ∗ − ) g λ , N ( x ) (cid:1) E N , E [ q x ( q e − q − e )( v ) ]= − N d N d − ∑ x ∈ Z dN (cid:0) g λ , N ( x ) + ( d ∗ − ) g λ , N ( x ) (cid:1) E N , E [( q x − q − x )( q e − q − e )( v ) ] . Let us define the Fourier transform ˆ f ( ξ ) , ξ ∈ Z dN of a function f on Z dN byˆ f ( ξ ) = ∑ z ∈ Z dN f ( z ) e − π i ξ N · z . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 19
The inverse Fourier transform is given by f ( z ) = N d ∑ ξ ∈ Z dN ˆ f ( ξ ) e π i ξ N · z . From Lemma C.1 (iv), we have E N , E [( q x − q − x )( q e − q − e )( v ) ]= N d (cid:18) E d ∗ + O ( N − d ) (cid:19) ∑ ξ ∈ Z dN , ξ (cid:54) = sin ( π ξ N · x ) sin ( π ξ N · e ) ∑ da = sin ( πξ a N ) . Since g λ , N ( x ) − g λ , N ( − x ) = iN d ∑ ξ ∈ Z dN sin ( π ξ N · x ) ˆ g λ , N ( ξ ) , we have ∑ x ∈ Z dN g λ , N ( x ) E N , E [( q x − q − x )( q e − q − e )( v ) ]= ∑ x ∈ Z dN ( g λ , N ( x ) − g λ , N ( − x )) E N , E [( q x − q − x )( q e − q − e )( v ) ]= ∑ x ∈ Z dN iN d ∑ ξ ∈ Z dN sin ( π ξ N · x ) ˆ g λ , N ( ξ ) N d (cid:18) E d ∗ + O ( N − d ) (cid:19) ∑ ζ ∈ Z dN , ζ (cid:54) = sin ( π ζ N · x ) sin ( π ζ N · e ) ∑ da = sin ( πζ a N ) = iN d (cid:18) E d ∗ + O ( N − d ) (cid:19) ∑ ξ ∈ Z dN , ξ (cid:54) = ˆ g λ , N ( ξ ) sin ( π ξ N · e ) ∑ da = sin ( πξ a N ) . At the last equality, we use the fact ∑ x ∈ Z dN sin ( π ξ N · x ) sin ( π ζ N · x ) = N d ( δ ξ , ζ − δ ξ , − ζ ) . On the other hand, by the direct computations ˆ g λ , N ( ξ ) = ˆ g λ ( ξ N ) where ˆ g λ ( θ ) = i sin ( πθ ) λ + γω θ . Inthis way, we have ∑ x ∈ Z dN g λ , N ( x ) E N , E [( q x − q − x )( q e − q − e )( v ) ] → − E d ∗ (cid:90) [ , ] d sin ( πθ ) ω θ λ + γω θ d θ as N → ∞ since the function sin ( πθ ) ω θ λ + γω θ is continuous except at the boundary and uniformlybounded on [ , ] d .In the same manner, we will have the following convergence ∑ x ∈ Z dN g λ , N ( x ) E N , E [( q x − q − x )( q e − q − e )( v ) ] → E id ∗ (cid:90) [ , ] d ˆ g λ ( θ ) sin ( πθ ) ω θ d θ if ˆ g λ , N ( ξ ) = ˆ g λ ( ξ N ) for some function ˆ g λ and ˆ g λ ( θ ) sin ( πθ ) ω θ behaves well at the boundary. Weprove that this is the case in the rest of this subsection. HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 20
From the explicit expression (B.2), it is easy to see that ˆ g λ , N ( ξ ) = ˆ g λ ( ξ N ) with ( ˆ g λ , ˆ g λ , ˆ g λ , ˆ g λ ) given as the solution of the following linear equation: λ ω θ λ + γω θ B − − B λ + γω θ ω θ − λ + γω θ ˆ g λ ˆ g λ ˆ g λ ˆ g λ = i sin ( πθ ) . Its explicit solution is ( ˆ g λ , ˆ g λ , ˆ g λ , ˆ g λ ) = i sin ( πθ ) Q ( λ ) (cid:0) − B ω θ ( λ + γω θ ) , P ( λ ) , B λ ( λ + γω θ ) , B λ (cid:1) with P ( λ ) = ( λ + γω θ )( λ + λ γω θ + ω θ ) and Q ( λ ) = ( λ + γω θ ) ( λ + λ γω θ + ω θ ) + B λ ( λ + γω θ ) . In particular,ˆ g λ ( θ ) sin ( πθ ) ω θ = i sin ( πθ ) ω θ P ( λ ) Q ( λ ) is continuous except at the boundary and uniformly bounded on [ , ] d . Therefore, we have ∑ x ∈ Z dN g λ , N ( x ) E N , E [( q x − q − x )( q e − q − e )( v ) ] → − E d ∗ (cid:90) [ , ] d sin ( πθ ) ω θ P ( λ ) Q ( λ ) d θ as N → ∞ .2.6. Analysis of the inverse Laplace transform.
In this section, we study the inverse Laplacetransform of (2.2) in detail and give the asymptotic behavior of C ( s ) .First, we study the term C ( t ) : = L − (cid:20) (cid:90) [ , ] d sin ( πθ ) ω θ λ + γω θ d θ (cid:21) ( t ) = (cid:90) [ , ] d sin ( πθ ) ω θ exp ( − t γω θ ) d θ where L − represents the inverse Laplace transform operator. The asymptotic behavior of thisterm is already studied in [1] as C ( t ) ∼ t − d / .To analyze the remaining term, we first study the inverse Laplace transform of P ( λ ) Q ( λ ) . Let ¯ λ = λ + γω θ . Then, P ( λ ) = ¯ λ ( ¯ λ − γ ω θ + ω θ ) and Q ( λ ) = ¯ λ + ( B − γ ω θ + ω θ ) ¯ λ − γ B ω θ .Define α ( θ ) ≥ , α ( θ ) ≥ (cid:40) α ( θ ) − α ( θ ) = B − γ ω θ + ω θ α ( θ ) α ( θ ) = γ B ω θ . We emphasize that any pair of the form ( ± α ( θ ) , ± α ( θ )) satisfies the equations (2.4), but wechoose nonnegative solutions and denote them by α ( θ ) and α ( θ ) which are unique. Then Q ( λ ) is factorized as Q ( λ ) = ( ¯ λ + α ( θ ) )( ¯ λ − α ( θ ) ) and P ( λ ) Q ( λ ) = ¯ λ ¯ λ − γ ω θ + ω θ ( ¯ λ + α ( θ ) )( ¯ λ − α ( θ ) ) = ¯ λ (cid:18) β ( θ ) ¯ λ + α ( θ ) + β ( θ ) ¯ λ − α ( θ ) (cid:19) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 21 where β ( θ ) = α ( θ ) + B α ( θ ) + α ( θ ) and β ( θ ) = α ( θ ) − B ( α ( θ ) + α ( θ ) ) . Therefore, P ( λ ) Q ( λ ) = β ( θ ) ¯ λ ¯ λ + α ( θ ) + β ( θ ) λ + α ( θ ) + β ( θ ) λ − α ( θ ) and we can calculate the inverse Laplace transform explicitly as L − (cid:20) P ( λ ) Q ( λ ) (cid:21) ( t ) = β ( θ ) exp ( − γω θ t ) cos ( α ( θ ) t )+ β ( θ ) exp ( − γω θ t − α ( θ ) t ) + β ( θ ) exp ( − γω θ t + α ( θ ) t ) . Therefore, we have L − (cid:20) (cid:90) [ , ] d sin ( πθ ) ω θ P ( λ ) Q ( λ ) d θ (cid:21) ( t )= (cid:90) [ , ] d sin ( πθ ) ω θ (cid:0) β ( θ ) exp ( − γω θ t ) cos ( α ( θ ) t )+ β ( θ ) exp ( − γω θ t − α ( θ ) t ) + β ( θ ) exp ( − γω θ t + α ( θ ) t ) (cid:1) d θ . (2.5)By definition, α ( θ ) = ( B − γ ω θ + ω θ ) + (cid:113) ( B − γ ω θ + ω θ ) + γ B ω θ α ( θ ) = − ( B − γ ω θ + ω θ ) + (cid:113) ( B − γ ω θ + ω θ ) + γ B ω θ . Hence, α , α , β , β are continuous functions on [ , ] d . Therefore, standard analysis showsthe behavior of the term (2.5) as t goes to infinity is governed by the behavior of the functions α , α and β , β around the minimal value of γω θ , γω θ + α ( θ ) and γω θ − α ( θ ) . By theexplicit expression, γ ω θ − α ( θ ) = B + γ ω θ + ω θ (cid:0) − (cid:115) − γ ω θ ( B + γ ω θ + ω θ ) (cid:1) ≥ . (2.6)Therefore, γω θ , γω θ + α ( θ ) and γω θ − α ( θ ) are 0 if and only if ω θ =
0. By symmetry, wetreat only the case θ = ( , , . . . , ) .We study the asymptotic behavior of the following three terms separately: C ( t ) : = (cid:90) [ , ] d sin ( πθ ) ω θ β ( θ ) exp ( − γω θ t ) cos ( α ( θ ) t ) d θ , C ( t ) : = (cid:90) [ , ] d sin ( πθ ) ω θ β ( θ ) exp ( − γω θ t − α ( θ ) t ) d θ , C ( t ) : = (cid:90) [ , ] d sin ( πθ ) ω θ β ( θ ) exp ( − γω θ t + α ( θ ) t ) d θ . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 22
Note that α ( θ ) − B = ω θ − γ ω θ + ( (cid:113) ( B − γ ω θ + ω θ ) + γ B ω θ − B )= ω θ + B ( (cid:115) + ω θ B + o ( | θ | ) − ) + o ( | θ | ) = ω θ + o ( | θ | ) as | θ | →
0. It implies β ( θ ) = B | θ | + o ( | θ | ) as | θ | → α ( θ ) = γ | θ | + o ( | θ | ) as | θ | →
0, we have C ( t ) ∼ (cid:90) [ , ] d | θ | | θ | | θ | exp ( − γ | θ | t ) d θ = t − d / − (cid:90) [ , √ t ] d u exp ( − γ | u | ) du ∼ t − d / − . Also, from (2.6), we have γ ω θ − α ( θ ) = γ ω θ B + o ( | θ | ) , γω θ + α ( θ ) = γω θ + o ( | θ | ) , and so γω θ − α ( θ ) = γω θ B + o ( | θ | ) . Then, C ( t ) ∼ (cid:90) [ , ] d | θ | | θ | | θ | exp ( − γ | θ | B t ) d θ = t − d / − / (cid:90) [ , t / ] d u exp ( − γ | u | B ) du ∼ t − d / − / . Finally, we study the oscillation term C ( t ) . We estimate the time integral of this term, namely, (cid:90) T ( − tT ) (cid:90) [ , ] d sin ( πθ ) ω θ β ( θ ) exp ( − γω θ t ) cos ( α ( θ ) t ) d θ dt = (cid:90) [ , ] d sin ( πθ ) ω θ β ( θ ) (cid:90) T ( − tT ) exp ( − γω θ t ) cos ( α ( θ ) t ) dtd θ . By a direct computation, (cid:90) T exp ( − γω θ t ) cos ( α ( θ ) t ) dt = exp ( − γω θ T )( − γω θ cos ( α ( θ ) T ) + α ( θ ) sin ( α ( θ ) T )) + γω θ γ ω θ + α ( θ ) and 1 T (cid:90) T t exp ( − γω θ t ) cos ( α ( θ ) t ) dt = exp ( − γω θ T )( − γω θ cos ( α ( θ ) T ) + α ( θ ) sin ( α ( θ ) T )) γ ω θ + α ( θ ) − exp ( − γω θ T )(( γ ω θ − α ( θ ) ) cos ( α ( θ ) T ) − γω θ α ( θ ) sin ( α ( θ ) T )) T ( γ ω θ + α ( θ ) ) + γ ω θ − α ( θ ) T ( γ ω θ + α ( θ ) ) . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 23
Then, we havelim sup T → ∞ | (cid:90) [ , ] d sin ( πθ ) ω θ β ( θ ) (cid:90) T ( − tT ) exp ( − γω θ t ) cos ( α ( θ ) t ) dtd θ |≤ (cid:90) [ , ] d | β ( θ ) | γω θ γ ω θ + α ( θ ) d θ < ∞ .
3. U
NIFORM AND ALTERNATE CHARGE MODELS UNDER CANONICAL MEASURES
This section provides a detailed study of the one-dimensional chain of oscillators with uni-form and alternate charges in two-dimensional space under canonical measures. In the sameway as the last section, we start by recalling the description of the model in Subsection 3.1,and characterize conserved quantities and introduce canonical measures in Subsection 3.2. Thestrategy of the proof of Theorem 1.7 given in Subsection 3.3 is essentially the same as that forTheorem 1.3, but we need one new and important step, which is given in Subsection 3.4. Themain result in this subsection allows us to reduce the problem to solve the resolvent equation in ( r , ( v ) coordinate to that in ( q , v ) coordinate. In the last two subsections, we give proofs of otheringredients of the proof of Theorem 1.7, mainly for the alternate charge model.3.1. Model.
As already discussed in the Introduction, we only consider the case d = d ∗ = ( r x ( t ) , v x ( t )) x ∈ Z N be a a Markov process on R N generated by L ( ) r = A r + BG ( ) r + γ S r where the operators A r , G ( ) r and S are given in (1.8) and (1.9) for = , i , ii .We call these dynamics as the dynamics (0), (i) and (ii) respectively.3.2. Conserved quantities and canonical measures.
As observed in the Introduction, the totalenergy ∑ x ∈ Z N E x is conserved under any of the dynamics (0), (i) and (ii). The total deformation ∑ x ∈ Z N r jx for j = , ∑ x ∈ Z N v jx for j = , p since it is given in terms of q . However,for the dynamics (ii), ∑ x ∈ Z eN ( v x + v x + + Br x ) and ∑ x ∈ Z eN ( v x + v x + − Br x ) are also conservedwhere Z eN : = { x ∈ Z N ; x ≡ } .By direct computations, one sees immediately that the dynamics (0), (i) and (ii) are all sta-tionary under µ N , β . More generally, for each β > τ = ( τ , τ ) ∈ R , µ N , β , τ ( drd p ) = Z N β , τ exp ( − β ( ∑ x E x + ∑ j = τ j ∑ x r jx )) Π x ∈ Z N Π j = dr jx dv jx are stationary measures for the dynamics (0), (i) and (ii). We mainly study the case τ =
0, butgive some discussion on the asymptotic behavior of the Green-Kubo integral in the case τ (cid:54) = Instantaneous energy current correlation.
Recall that J x , x + ([ , t ]) is the total energycurrent between x and x + t . Let j x , x + be the instantaneous energy current given by HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 24 j x , x + = j ax , x + + j sx , x + with j ax , x + = − ∑ j = r jx ( v jx + + v jx ) , j sx , x + = − γ ∑ j = (cid:32) ( v jx + ) − ( v jx ) (cid:33) . Note that these currents are common among the dynamics (0), (i) and (ii).The total energy current is written as J x , x + ([ , t ]) = (cid:90) t j x , x + ( s ) ds + ∑ j = (cid:90) t (cid:32) ( v jx + ( s − )) − ( v jx ( s − )) (cid:33) dM j , x , x + ( s ) where M j , x , x + ( t ) = N γ j , x , x + ( t ) − γ t and { N γ j , x , x + ( t ) } j = , , x ∈ Z N are 2 N -independent poisson pro-cesses with intensity γ .We can apply the same argument in Subsection 2.3 to obtain β tN E ( ) N , β [ (cid:0) ∑ x ∈ Z N J x , x + ([ , t ]) (cid:1) ] = β (cid:90) t ( − st ) D ( ) N ( s ) ds + γ D ( ) N ( s ) = E ( ) N , β [ ∑ x j ax , x + ( s ) j a , ( )] for = , i , ii . Here, E ( ) N , β is the expectation for thedynamics ( ) starting from the canonical measure µ N , β . Since the canonical measures is product,we do not need o ( N ) correction term.Now, the next proposition is suffice to prove Theorem 1.7. Proposition 3.1.
For each = , i , ii, the sequence of functions { D ( ) N : [ , ∞ ) → R } convergescompactly to a function D ( ) ∞ : [ , ∞ ) → R as N → ∞ , namely the convergence is uniform on eachcompact subset of [ , ∞ ) . Moreover, D ( ) ∞ ( s ) ∼ s − / as s → ∞ , D ( i ) ∞ ( s ) = D ( s ) + D ( s ) + D ( s ) where lim sup t → ∞ (cid:90) t ( − st ) D ( s ) < ∞ , D ( s ) ∼ s − / , D ( s ) ∼ s − / as s → ∞ . Also, if γ ≤ ,D ( ii ) ∞ ( s ) ∼ s − / . The main strategy of the proof of Proposition 3.1 is the same as that for Proposition 2.4.For each λ > = , i , ii , let ˜ D N ( λ ) = (cid:82) ∞ D ( ) N ( s ) exp ( − λ s ) ds be the Laplace transformof D ( ) N and ˜ D ( ) ∞ : ( , ∞ ) → R be functions defined by˜ D ( ) ∞ ( λ ) = β (cid:90) [ , ] cos ( πθ ) λ + γω θ d θ , ˜ D ( i ) ∞ ( λ ) = β (cid:90) [ , ] cos ( πθ ) ( λ + γω θ )( λ + λ γω θ + ω θ )( λ + γω θ ) ( λ + λ γω θ + ω θ ) + B λ ( λ + γω θ ) d θ , ˜ D ( ii ) ∞ ( λ ) = β (cid:90) [ , ] cos ( πθ ) R ( λ ) S ( λ ) d θ where ω θ = ( πθ ) . R ( λ ) and S ( λ ) are polynomial functions of order 5 and 6 respectively,and their explicit expressions are given at (3.3) where ¯ λ = λ + γ . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 25
We can follow the proof of Lemma 2.6 to prove the next lemma, so we omit the proof.
Lemma 3.2.
For each = , i , ii, the set of functions { D ( ) N : [ , ∞ ) → R } N is uniformly boundedand equicontinuous. We prove the next theorem in Subsection 3.5.
Proposition 3.3.
For any λ > and = , i , ii, lim N → ∞ ˜ D ( ) N ( λ ) = ˜ D ( ) ∞ ( λ ) . The detailed analysis of the inverse Laplace transform of ˜ D ( ) ∞ for = , i , ii completes theproof of Proposition 3.1. Since the case = i are studied in the last section, we onlyanalyze the inverse Laplace transform of ˜ D ( ii ) ∞ in Subsection 3.6.3.4. Reduction from the coordinate ( q , v ) to ( r , v ) . In the proof of Proposition 3.3, an ex-plicit expression of the resolvent equation (3.1) plays an essential role. However, it is not asimple problem to solve the equation in the coordinate ( r , v ) . In this subsection, we introducea technique to obtain this solution from the solution of the associated resolvent equation in thecoordinate ( q , v ) .We consider the change of variable Φ : ( r , v ) → ( q , v ) defined by q jx = − N ∑ y = x ( r jy − ¯ r j ) , ¯ r j = N N ∑ x = r jx for x = , , . . . , N and define q jx = q jx + N for any x ∈ Z N . Then, we have q jx + − q jx = r jx − ¯ r j .For each = , i , ii , let G ( ) be an operator which is formally given as G ( ) r but acting on func-tions f ( q , v ) rather than f ( r , v ) . Define the operator L ( ) = A + BG ( ) + γ S acting on functions f ( q , v ) ∈ C ( R N ) with the operators A and G defined at (1.4). Note that L ( i ) = L where L is thegenerator of the dynamics studied in Section 2. Proposition 3.4.
Suppose a pair of functions F ( q , v ) and H ( q , v ) satisfies ( λ − L ( ) ) F = Hand ∑ Nx = ∂ q jx F = for j = , . Then F r ( r , v ) and H r ( r , v ) satisfies ( λ − L ( ) r ) F r = H r whereF r ( r , v ) = F ( Φ ( r , v )) and H r ( r , v ) = H ( Φ ( r , v )) .Proof. Since G ( ) F r ( r , v ) = ( G ( ) F )( Φ ( r , v )) and S r F r ( r , v ) = ( SF )( Φ ( r , v )) , we only need toshow that A r F r ( r , v ) = ( AF )( Φ ( r , v )) . By definition, we have q jx + − q jx + q jx − = − N ∑ z = x + ( r jz − ¯ r j ) + N ∑ z = x ( r jz − ¯ r j ) − N ∑ z = x − ( r jz − ¯ r j ) = r jx − r jx − . Also, ∂ r jx F r ( r , v ) = N ∑ y = ∂ q jy ∂ r jx ( ∂ q jy F )( Φ ( r , v )) = N ∑ y = ( − x ≥ y + N )( ∂ q jy F )( Φ ( r , v )) and hence ( ∂ r jx − ∂ r jx − ) F r ( r , v ) = − ( ∂ q jx F )( Φ ( r , v )) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 26 for x = , , . . . , N . On the other hand, by assumption, ( ∂ r j − ∂ r jN ) F r ( r , v ) = N ∑ y = ( ∂ q jy F )( Φ ( r , v )) = − ( ∂ q j F )( Φ ( r , v )) . Therefore, A r F r ( r , v ) = ( AF )( Φ ( r , v )) . (cid:3) Proof of Proposition 3.3.
For λ >
0, let v ( ) λ , N be the solution of the resolvent equation(3.1) ( λ − L ( ) r ) v ( ) λ , N = ∑ x j ax , x + for = , i , ii . Then, ˜ D ( ) N ( λ ) = E N , β [ v ( ) λ , N j a , ] .Unlike the microcanonical case, it is not easy to solve the equation (3.1) directly. So, weconsider the associated resolvent equation for the coordinate ( q , v ) ∈ R N and use its solution toobtain the solution of (3.1). Let u ( ) λ , N be the solution of the resolvent equation(3.2) ( λ − L ( ) ) u ( ) λ , N = − ∑ x ∑ j = ( q jx + − q jx )( v jx + + v jx ) . For = , i , ii , the explicit form of u ( ) λ , N is given in Lemmas B.1,B.2 and B.3 respectively. FromProposition 3.4 and Lemma B.4, ( λ − L ( ) r ) v ( ) λ , N , ∗ = − ∑ x ∑ j = ( r jx − ¯ r j )( v jx + + v jx ) = ∑ x j ax , x + + N ∑ j = ¯ r j ¯ v j where v ( ) λ , N , ∗ ( r , v ) = u ( ) λ , N ( Φ ( r , v )) , ¯ r j = N ∑ x ∈ Z N r jx and ¯ v j = N ∑ x ∈ Z N v jx .Let v ( ) λ , N , ∗∗ be the solution of the resolvent equation ( λ − L ( ) r ) v ( ) λ , N , ∗∗ = N ∑ j = ¯ r j ¯ v j . Then, we have v ( ) λ , N = v ( ) λ , N , ∗ − v ( ) λ , N , ∗∗ .By direct computations, we have v ( ) λ , N , ∗∗ = (cid:40) N λ ∑ j = ¯ r j ¯ v j for = , − N ( ¯ v λ ¯ r − B ¯ r λ + B + ¯ v B ¯ r + λ ¯ r λ + B ) for = i , and v ( ) λ , N , ∗∗ = − N λ ( λ + γλ + + B ) (cid:0) ( λ + γλ + ) ¯ r ¯ v − B λ ¯ r ˇ v + B λ ¯ r ˇ v + ( λ + γλ + ) ¯ r ¯ v + B ( ¯ r ˇ r − ¯ r ˇ r ) (cid:1) for = ii where ˇ r j = N ∑ x ∈ Z N ( − ) x r jx and ˇ v j = N ∑ x ∈ Z N ( − ) x v jx . For any j , k = ,
2, we have E N , β [ ¯ r j ¯ v k j a , ] = − δ j , k E N , β [ ¯ r j r j ] E N , β [ ¯ v j v j ] = − δ j , k N β . Hence,lim N → ∞ E N , β [ v ( ) λ , N , ∗∗ j a , ] = HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 27 for = , i . Similarly, we can also check thatlim N → ∞ E N , β [ v ( ii ) λ , N , ∗∗ j a , ] = . Therefore, E N , β [ v ( ) λ , N , ∗ j a , ] is the only term which contributes to the limit of ˜ D ( ) N ( λ ) . Remark 3.5.
Under general canonical measures µ N , β , τ , lim N → ∞ E N , β , τ [ v ( ) λ , N , ∗∗ j a , ] = − λ τ + τ β , lim N → ∞ E N , β , τ [ v ( i ) λ , N , ∗∗ j a , ] = − λλ + B τ + τ β , lim N → ∞ E N , β , τ [ v ( ) λ , N , ∗∗ j a , ] = − λ + γλ + λ ( λ + γλ + + B ) τ + τ β and so the term E N , β , τ [ v ( ) λ , N , ∗∗ j a , ] does not vanish in the limit. On the other hand, the behaviorof the term E N , β , τ [ v ( ) λ , N , ∗ j a , ] does not depend on τ , which we can see from the argument below.Therefore, by the inverse Laplace transform, we can see that the current correlation D ( ) ∞ ( t ) does not decay under the dynamics (0) and (ii). On the other hand, under the dynamics (i) theterm E N , β , τ [ v ( i ) λ , N , ∗∗ j a , ] is an oscillation term and so the asymptotic behavior of the Green-Kubointegral does not depend on τ . We study the term E N , β [ v ( ) λ , N , ∗ j a , ] by the same strategy in the last section. First, let usconsider the uniform case. From Lemma B.2 in Appendix B, we have u ( i ) λ , N = ∑ x , y ∈ Z N ( g λ , N ( x − y ) q x q y + g λ , N ( x − y )( q x v y + q x v y )+ g λ , N ( x − y )( q x v y − q x v y ) + g λ , N ( x − y ) v x v y ) where ( g λ , N , g λ , N , g λ , N , g λ , N ) is the solution of (B.2). As already discussed, v ( i ) λ , N , ∗ is obtainedfrom u ( i ) λ , N by replacing q jx by − ∑ Ny = x ( r jy − ¯ r j ) . Since the measure µ N , β is product, it is easy tosee that E N , β [ v ( i ) λ , N , ∗ j a , ] = ∑ x , y ∈ Z N g λ , N ( x − y ) E N , β [(( − N ∑ z = x ( r z − ¯ r )) v y + ( − N ∑ z = x ( r z − ¯ r )) v y ) j a , ]= ∑ x , y ∈ Z N g λ , N ( x − y ) E N , β [ N ∑ z = x ( r z − ¯ r ) r ] E N , β [ v y ( v + v )]= β ∑ x ∈ Z N ( g λ , N ( x ) + g λ , N ( x − )) E N , β [ N ∑ z = x ( r z − ¯ r ) r ]= β ∑ x ∈ Z N g λ , N ( x ) E N , β [ (cid:0) N ∑ z = x ( r z − ¯ r ) + N ∑ z = x + ( r z − ¯ r ) (cid:1) r ] . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 28
Let L ( x ) : = E N , β [ (cid:0) ∑ Nz = x ( r z − ¯ r ) + ∑ Nz = x + ( r z − ¯ r ) (cid:1) r ] for x ∈ Z N . By simple computations, L ( x + ) − L ( x ) + L ( x − ) = E N , β [( − r x + + r x − ) r ] = β ( − δ x , − + δ x , ) . Since ∑ x ∈ Z N g λ , N ( x ) = E N , β [ v ( i ) λ , N , ∗ j a , ] = β ∑ x ∈ Z N g λ , N ( x ) L ( x ) = β ∑ x ∈ Z N g λ , N ( x )( L ( x ) − ¯ L ) where ¯ L = N ∑ x L ( x ) and by Parseval’s identity, E N , β [ v ( i ) λ , N , ∗ j a , ] = − β N ∑ ξ ∈ Z N , ξ (cid:54) = ˆ g λ , N ( ξ ) i sin ( πξ N ) ( πξ N ) . In this way, for ˆ g λ ( ξ N ) = ˆ g λ , N ( ξ ) , we have˜ D ( i ) N ( λ ) → − i β (cid:90) [ , ] ˆ g λ ( θ ) sin ( πθ ) sin ( πθ ) d θ = − i β (cid:90) [ , ] ˆ g λ ( θ ) sin ( πθ ) ω θ d θ . The inverse Laplace transform of this limiting function has been already studied in the lastsection, so we conclude Proposition 3.3 for the uniform case, namely the case = i . The case = u ( ii ) λ , N = ∑ x ≡ y mod 2 (cid:0) h λ , N ( x − y )( − ) y q x q y + h λ , N ( x − y )( − ) y v x v y ) (cid:1) + ∑ x , y (cid:0) h λ , N ( x − y )( q x v y + q x v y ) + h λ , N ( x − y )( − ) y ( q x v y − q x v y ) (cid:1) where ( h λ , N , h λ , N , h λ , N , h λ , N ) is the solution of (B.3). Let ˆ h λ , N be the discrete Fourier transformof h λ , N . We can apply the same argument as above to show that˜ D ( ii ) N ( λ ) → − i β (cid:90) [ , ] ˆ h λ ( θ ) sin ( πθ ) sin ( πθ ) d θ = − i β (cid:90) [ , ] ˆ h λ ( θ ) sin ( πθ ) ω θ d θ if ˆ h λ , N ( ξ ) = ˆ h λ ( ξ N ) for some function ˆ h λ and ˆ h λ ( θ ) sin ( πθ ) ω θ behaves well at the boundary. Weprove this is the case in the rest of this subsection.We define the discrete Fourier transform ˆ h i λ , N , e and ˆ h i λ , N , o for i = , , , h i λ , N , e ( ξ ) = ∑ x :even h i λ , N ( x ) e − π i ξ N x , ˆ h i λ , N , o ( ξ ) = ∑ x :odd h i λ , N ( x ) e − π i ξ N x where the first sum is taken over all x ≡ x ≡ N to be even for the alternate case. From the explicit expression(B.3), we haveˆ h λ , N , e ( ξ ) = λ ( − ˆ h λ , N , e ( ξ ) − cos ( πξ N ) ˆ h λ , N , o ( ξ )) , ˆ h λ , N , e ( ξ ) = λ + γ ˆ h λ , N , e ( ξ ) . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 29
With these expressions, it is easy to see that ˆ h i λ , N , e ( ξ ) = ˆ h λ , e ( ξ N ) and ˆ h i λ , N , o ( ξ ) = ˆ h λ , o ( ξ N ) for i = , ( ˆ h λ , o , ˆ h λ , e , ˆ h λ , o , ˆ h λ , e ) is the solution of the following linear equation: ¯ λ − γ cos ( πθ ) B − γ cos ( πθ ) ¯ λ B − B λ ( πθ )( γ − λ + γ ) − B ( πθ )( γ + λ − γ ) ¯ λ ( + λ − γ ) ˆ h λ , o ˆ h λ , e ˆ h λ , o ˆ h λ , e = i sin ( πθ ) where ¯ λ = λ + γ . Then, by a direct computation, we haveˆ h λ ( θ ) − i sin ( πθ ) ω θ = cos ( πθ ) R ( λ ) S ( λ ) where R ( λ ) = ¯ λ (( B + ¯ λ )( − γ + ¯ λ ) − B )+ ( B ( λ + γ ( − γ + ¯ λ )) + γ ¯ λ ( − γ + ¯ λ )) cos ( πθ )+ ( + γ − γ ( + ¯ λ ))( λ cos ( πθ ) + γ cos ( πθ )) , (3.3) S ( λ ) = ( B + ¯ λ )(( B + ¯ λ )( − γ + ¯ λ ) − B )+ ( − B γ ( − γ + ¯ λ ) + ¯ λ ( + γ − γ ( + ¯ λ ))) cos ( πθ ) − γ ( + γ − γ ( + ¯ λ )) cos ( πθ ) . To simplify the notation, we introduce Y = X − γ . Then, we have S ( λ ) = ( B + Y + γ )(( B + Y + γ )( + Y ) − B )+ ( − B γ ( + Y ) + ( Y + γ )( − γ − γ Y ))( − sin ( πθ )) − γ ( − γ − γ Y )( − sin ( πθ )) = Y + ( + B + γ sin ( πθ )) Y + ( B + ( + γ sin ( πθ )) B +
16 cos ( πθ ) + γ sin ( πθ ) + γ sin ( πθ )) Y + B γ sin ( πθ ) + γ sin ( πθ )( − sin ( πθ )) + γ sin ( πθ ) . We call the last expression ˜ S ( Y ) . Since Y = λ + λ γ > θ ∈ [ , ) S ( λ ) = inf θ ∈ [ , ) ˜ S ( Y ) ≥ Y > . Therefore, cos ( πθ ) R ( λ ) S ( λ ) is continuous and bounded on [ , ] and so we complete the proof ofProposition 3.3. HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 30
Analysis of the inverse Laplace transform.
In this subsection, we study the inverseLaplace transform of (cid:82) cos ( πθ ) S ( λ ) R ( λ ) d θ and its asymptotic behavior. First note that, (cid:90) cos ( πθ ) R ( λ ) S ( λ ) d θ = (cid:90) cos ( πθ ) R ( λ ) S ( λ ) d θ since R ( λ ) and S ( λ ) are symmetric in θ . Also, since sin ( πθ ) = sin ( π ( − θ )) , cos ( πθ ) = − cos ( π ( − θ )) and cos ( πθ ) = sin ( π ( − θ )) , (cid:90) cos ( πθ ) R ( λ ) S ( λ ) d θ = (cid:90) ¯ λ R ( λ ) S ( λ ) d θ + (cid:90) cos ( πθ ) R ( λ ) S ( λ ) d θ where R ( λ ) = ( B + ¯ λ )( − γ + ¯ λ ) − B + ( + γ − γ ( + ¯ λ )) cos ( πθ ) , R ( λ ) = ( B ( λ + γ ( − γ + ¯ λ )) + γ ¯ λ ( − γ + ¯ λ )) cos ( πθ )+ γ ( + γ − γ ( + ¯ λ )) cos ( πθ ) . Let ¯ R ( λ ) = ¯ λ R ( λ ) + R ( λ ) cos ( πθ ) . Then, from the above observations, (cid:90) cos ( πθ ) R ( λ ) S ( λ ) d θ = (cid:90) ¯ R ( λ ) S ( λ ) d θ . To obtain the inverse Laplace transform of ¯ R ( λ ) S ( λ ) , we compute the partial fraction decomposi-tion of ¯ R ( λ ) S ( λ ) . For this, we first study the zero points of S ( λ ) , namely the solution of S ( λ ) = B = B and T ( Y ) = ˜ S ( Y ) . Then, by a simple computation, we have T ( Y ) = Y + ( + ˜ B + γ θ ) Y + ( ˜ B + ( + γ θ ) ˜ B + cos ( πθ ) + γ θ + γ θ ) Y + B γ θ + γ θ cos ( πθ ) + γ θ where γ θ = γ sin ( πθ ) . In particular, for θ = T ( Y ) = Y ( Y + ˜ B + ) . So, T ( Y ) = , − ˜ B − Lemma 3.6.
Assume γ ≤ . Then, for each fixed θ ∈ ( , ] , the third order polynomial T ( Y ) = has three distinct solutions ˜ α ( θ ) , ˜ α ( θ ) , ˜ α ( θ ) ∈ R such that > ˜ α ( θ ) > − γ > ˜ α ( θ ) > − ˜ B − > ˜ α ( θ ) .Proof. For any θ ∈ ( , ] and Y ≥
0, obviously T ( Y ) >
0. Also, T ( − γ ) = − γ (cid:0) ˜ B + − γ + sin ( πθ )( γ − ) (cid:1) < , T ( − ˜ B − ) = ( ˜ B + − γ ) sin ( πθ ) − γ ( ˜ B γ + − γ ) sin ( πθ ) ≥ ( ˜ B + − γ ) sin ( πθ ) − ( ˜ B + − γ ) sin ( πθ ) > . (cid:3) We define ˜ α ( ) = , ˜ α ( ) = ˜ α ( ) = − ˜ B −
1. Then, by the continuity of the coefficientsof polynomial T ( Y ) , ˜ α i ( θ ) are continuous on [ , ] . From this, we have S ( λ ) = ∏ i = ( Y − ˜ α i ( θ )) = ∏ i = ( Y − α i ( θ )) = ∏ i = ( ¯ λ − γ − α i ( θ )) . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 31
Next, we work on the numerator ¯ R ( λ ) . By a direct computation, we have¯ R ( λ ) = ¯ λ { ( B + Y + γ )( + Y ) − B + ( + B − γ − γ Y ) cos ( πθ ) } + ( B γ ( + Y ) + γ ( Y + γ )( Y + )) cos ( πθ ) + γ ( − γ − γ Y ) cos ( πθ ) where we use the notation Y = ¯ λ − γ . Let U ( Y ) = ( B + Y + γ )( + Y ) − B + ( + B − γ − γ Y ) cos ( πθ ) , U ( Y ) = ( B γ ( + Y ) + γ ( Y + γ )( Y + )) cos ( πθ ) + γ ( − γ − γ Y ) cos ( πθ ) so as ¯ R ( λ ) = ¯ λ U ( Y ) + U ( Y ) .Now, we give the partial fraction decomposition of ¯ R ( λ ) S ( λ ) . Let α ( θ ) = γ + α ( θ ) > α i ( θ ) = − γ − α i ( θ ) > i = , α i ( θ ) > i = , , θ ∈ [ , ] . Though wealso use the notation α ( θ ) , α ( θ ) in Subsection 2.6 which are clearly different from the oneswe have just defined, since there is no room for misunderstanding, we use the same notationhere. Also, define β i ( θ ) = U ( α i ( θ )) ∏ j (cid:54) = i ( ˜ α i ( θ ) − ˜ α j ( θ )) , β i + ( θ ) = U ( α i ( θ )) ∏ j (cid:54) = i ( ˜ α i ( θ ) − ˜ α j ( θ )) for θ ∈ ( , ] and i = , , θ ∈ ( , ] ,¯ λ U ( Y ) S ( λ ) = ¯ λ U ( Y ) ∏ i = ( Y − α i ( θ )) = ¯ λ ∑ i = β i ( θ ) Y − α i ( θ )= β ( θ ) λ − α ( θ ) + β ( θ ) λ + α ( θ ) + β ( θ ) ¯ λ ¯ λ + α ( θ ) + β ( θ ) ¯ λ ¯ λ + α ( θ ) and U ( Y ) S ( λ ) = U ( Y ) ∏ i = ( Y − α i ( θ )) = ∑ i = β i + ( θ ) Y − α i ( θ )= β ( θ ) α ( θ ) λ − α ( θ ) − β ( θ ) α ( θ ) λ + α ( θ ) + β ( θ ) λ + α ( θ ) + β ( θ ) λ + α ( θ ) . Therefore, we have L − (cid:34) (cid:90) ¯ λ U ( Y ) S ( λ ) d θ (cid:35) ( t )= (cid:90) L − (cid:20) β ( θ ) λ − α ( θ ) + β ( θ ) λ + α ( θ ) + β ( θ ) ¯ λ ¯ λ + α ( θ ) + β ( θ ) ¯ λ ¯ λ + α ( θ ) (cid:21) ( t ) d θ = exp ( − γ t ) (cid:90) (cid:18) β ( θ ) { exp ( α ( θ ) t ) + exp ( − α ( θ ) t ) } + β ( θ ) cos ( α ( θ ) t ) + β ( θ ) cos ( α ( θ ) t ) (cid:19) d θ and L − (cid:34) (cid:90) U ( Y ) S ( λ ) d θ (cid:35) ( t ) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 32 = (cid:90) L − (cid:20) β ( θ ) α ( θ ) λ − α ( θ ) − β ( θ ) α ( θ ) λ + α ( θ ) + β ( θ ) α ( θ ) α ( θ ) ¯ λ + α ( θ ) + β ( θ ) α ( θ ) α ( θ ) ¯ λ + α ( θ ) (cid:21) ( t ) d θ = exp ( − γ t ) (cid:90) (cid:18) β ( θ ) α ( θ ) { exp ( α ( θ ) t ) − exp ( − α ( θ ) t ) } + β ( θ ) α ( θ ) sin ( α ( θ ) t ) + β ( θ ) α ( θ ) sin ( α ( θ ) t ) (cid:19) d θ . Since β ( ) : = lim θ → β ( θ ) = ( B + )( B + ) < ∞ , β ( θ ) is continous on [ , ] and | exp ( − γ t ) (cid:90) β ( θ ) ( − α ( θ ) t ) d θ | ≤ C exp ( − γ t ) for some positive constant C .In the same way, | exp ( − γ t ) (cid:90) β ( θ ) α ( θ ) exp ( − α ( θ ) t ) d θ | ≤ C exp ( − γ t ) for some positive constant C .Next, we consider the term (cid:90) (cid:0) β ( θ ) cos ( α ( θ ) t ) + β ( θ ) cos ( α ( θ ) t ) (cid:1) d θ . Lemma 3.7.
There exists a constant C > such that sup θ ∈ [ , ] | β ( θ ) cos ( α ( θ ) t ) + β ( θ ) cos ( α ( θ ) t ) | ≤ Ct . Proof.
We only need to show that for some C > θ → | β ( θ ) cos ( α ( θ ) t ) + β ( θ ) cos ( α ( θ ) t ) | ≤ Ct . By definition, β ( θ ) cos ( α ( θ ) t ) + β ( θ ) cos ( α ( θ ) t )= α ( θ ) − α ( θ ) (cid:18) U ( − α ( θ ) − γ ) cos ( α ( θ ) t ) α ( θ ) + α ( θ ) − U ( − α ( θ ) − γ ) cos ( α ( θ ) t ) α ( θ ) + α ( θ ) (cid:19) = α ( θ ) − α ( θ ) (cid:18) V ( − α ( θ ) − γ ) cos ( α ( θ ) t ) α ( θ ) + α ( θ ) − V ( − α ( θ ) − γ ) cos ( α ( θ ) t ) α ( θ ) + α ( θ ) (cid:19) + cos ( πθ ) α ( θ ) − α ( θ ) (cid:18) V ( − α ( θ ) − γ ) cos ( α ( θ ) t ) α ( θ ) + α ( θ ) − V ( − α ( θ ) − γ ) cos ( α ( θ ) t ) α ( θ ) + α ( θ ) (cid:19) where V ( Y ) = ( B + Y + γ )( + Y ) − B and V ( Y ) = ( + B − γ − γ Y ) . Let f i , t ( x ) = V i ( − x − γ ) cos ( xt ) γ + x for i = ,
2. Then, since lim θ → α ( θ ) = γ lim θ → (cid:0) β ( θ ) cos ( α ( θ ) t ) + β ( θ ) cos ( α ( θ ) t ) (cid:1) = lim θ → (cid:0) − α ( θ ) + α ( θ ) f , t ( α ( θ )) − f , t ( α ( θ )) α ( θ ) − α ( θ ) + − cos ( πθ ) α ( θ ) + α ( θ ) f , t ( α ( θ )) − f , t ( α ( θ )) α ( θ ) − α ( θ ) (cid:1) . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 33
Applying lim θ → α ( θ ) = lim θ → α ( θ ) = (cid:112) B + − γ >
0, we havelim θ → (cid:0) β ( θ ) cos ( α ( θ ) t ) + β ( θ ) cos ( α ( θ ) t ) (cid:1) = − (cid:112) B + − γ ∑ i = f (cid:48) i , t ( (cid:112) B + − γ ) . From the explicit expression of f i , t for i = , | f (cid:48) i , t ( (cid:112) B + − γ ) | ≤ Ct for some C >
0, andhence we complete the proof. (cid:3)
From the above lemma, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp ( − γ t ) (cid:90) ( β ( θ ) cos ( α ( θ ) t ) + β ( θ ) cos ( α ( θ ) t )) d θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct exp ( − γ t ) for some positive constant C .In the same way, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp ( − γ t ) (cid:90) (cid:18) β ( θ ) α ( θ ) sin ( α ( θ ) t ) + β ( θ ) α ( θ ) sin ( α ( θ ) t ) (cid:19) d θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct exp ( − γ t ) for some positive constant C .Finally, we study the term (cid:90) (cid:18) β ( θ ) + β ( θ ) α ( θ ) (cid:19) exp (( − γ + α ( θ )) t ) d θ . Since α ( θ ) < γ for θ (cid:54) =
0, we only need to consider its behavior around θ =
0. For this,we study the behavior of ˜ α ( θ ) ≤ α ( ) = T ( ˜ α ( θ )) =
0. By expanding T ( c sin ( πθ )) around θ =
0, we have(3.4) lim θ → ˜ α ( θ ) sin ( πθ ) = − γ ( B + )( ˜ B + ) = − γ ( B + )( B + ) . Namely, − α ( θ ) = γ − α ( θ ) = O ( θ ) as θ →
0. Hence 2 γ − α ( θ ) ∼ O ( θ ) as θ → θ → (cid:18) β ( θ ) + β ( θ ) α ( θ ) (cid:19) = B + (cid:54) = . Therefore, (cid:90) (cid:18) β ( θ ) + β ( θ ) α ( θ ) (cid:19) exp (cid:0) ( − γ + α ( θ )) t (cid:1) d θ ∼ t − / . Remark 3.8. If γ > , T ( Y ) = may have a multiple root for some θ (cid:54) = . Even for such a case,we expect the asymptotic behavior of the thermal conductivity is same as the case for γ ≤ bythe following reason. First, for general γ > , ˜ α i ( θ ) , i = , , , the solutions of T ( Y ) = , areall continuous and if ˜ α i ( θ ) ∈ R , then ˜ α i ( θ ) < for θ ∈ ( , ] . Moreover, ˜ α ( θ ) ∈ R for smallenough θ and the asymptotic behavior of ˜ α ( θ ) as θ → , which is given at (3.4), also holds.Namely, among poles of ¯ R ( λ ) S ( λ ) , there is just one pole which can contribute to the divergence ofthe thermal conductivity. Since the exponent is determined by the asymptotic behavior (3.4) and(3.5) which both hold for general γ > , we should have D ( ii ) ( t ) ∼ t − in general. However, arigorous argument will be too complicated, so we do not pursue it here. HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 34 A CKNOWLEDGEMENT
The authors express their sincere thanks to Herbert Spohn and Stefano Olla for insightfuldiscussions, and to Shuji Tamaki for careful reading of the manuscript and helpful comments.The authors also thank anonymous referees for their constructive comments that significantlyimprove the manuscript. KS was supported by JSPS Grants-in-Aid for Scientific Research No.JP26400404 and No. JP16H02211. MS was supported by JSPS Grant-in-Aid for Young Scien-tists (B) No. JP25800068.A
PPENDIX
A. C
ONVERGENCE OF FUNCTIONS AND L APLACE TRANSFORM
Proposition A.1.
Suppose a set of functions f N : [ , ∞ ) → R indexed by N is uniformly boundedand equicontinuous. Moreover, assume that for any λ > , lim N → ∞ (cid:90) ∞ e − λ t f N ( t ) dtexists. Then, there exists a function f ∞ : [ , ∞ ) → R such that f N converges compactly to f ∞ ,namely the convergence is uniform on each compact subset of [ , ∞ ) , and also lim N → ∞ (cid:90) ∞ e − λ t f N ( t ) dt = (cid:90) ∞ e − λ t f ∞ ( t ) dt . Proof.
By Arzel´a-Ascoli theorem, if f N is restricted to each compact interval in [ , ∞ ) , thereexists a subsequence N k such that the function f N k converges in the uniform topology on thisinterval. Therefore, by the diagonal argument, we can construct a subsequence ˜ N k such that f ˜ N k converges compactly on [ , ∞ ) . Denote its limit by f ∞ . Then, by the Lebesgue’s convergencetheorem, lim k → ∞ (cid:90) ∞ e − λ t f ˜ N k ( t ) dt = (cid:90) ∞ e − λ t f ∞ ( t ) dt since { f N } is uniformly bounded. Then, by the injectivity of the Laplace transform, the originalsequence f N also should converge to f ∞ ( t ) . (cid:3) A PPENDIX
B. R
ESOLVENT EQUATION
In this section, we give an explicit solution of the resolvent equation ( λ − L ( ) ) u λ , N = ∑ x j a x , x + e = d ∗ ∑ j = ∑ x , y ∈ Z dN H ( x − y ) q j x v j y for each λ > H ( z ) = ( δ z + e − δ z − e ) and L ( ) = A + BG ( ) + γ S for = , i , ii . Notethat L ( i ) and G ( i ) are denoted by L and G in Subsection 2 respectively.First, we concern the components for which the magnetic field does not effect. Let g λ , N : Z dN → R be the solution of the equation(B.1) λ g λ , N ( z ) − γ ∆ g λ , N ( z ) = H ( z ) which satisfies g λ , N ( z ) = − g λ , N ( − z ) . The existence and the uniqueness of such a solution fol-lows straightforwardly by the Fourier transform. HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 35
Lemma B.1.
For any j = , , . . . , d ∗ , let u λ , N , j = ∑ x , y ∈ Z dN g λ , N ( x − y ) q j x v j y . Then, ( λ − ( A + γ S )) u λ , N , j = ∑ x , y ∈ Z dN H ( x − y ) q j x v j y . Proof.
By direct computations, Au λ , N , j = ∑ x , y ∈ Z dN g λ , N ( x − y ) v j x v j y + ∑ x , y ∈ Z dN g λ , N ( x − y ) q j x [ ∆ q j ] y = ∑ x , y ∈ Z dN g λ , N ( x − y ) v j x v j y + ∑ x , y ∈ Z dN [ ∆ g λ , N ]( x − y ) q j x q j y = g λ , N ( z ) = − g λ , N ( − z ) . Also, a simple computation shows that γ Su λ , N , j = γ ∑ x , y ∈ Z dN g λ , N ( x − y ) q j x [ ∆ v j ] y = γ ∑ x , y ∈ Z dN [ ∆ g λ , N ]( x − y ) q j x v j y which concludes the proof. (cid:3) From this lemma, we have ( λ − L ( ) ) d ∗ ∑ j = u λ , N , j = d ∗ ∑ j = ∑ x , y ∈ Z dN H ( x − y ) q j x v j y and ( λ − L ( ) ) d ∗ ∑ j = u λ , N , j = d ∗ ∑ j = ∑ x , y ∈ Z dN H ( x − y ) q j x v j y for = i , ii with u λ , N , j given in Lemma B.1.Next, we work on the components j = , g i λ , N : Z dN → R ( i = , , , ) be the solution of the simultaneous equations λ g λ , N ( z ) − ∆ g λ , N ( z ) = , ( λ − γ ∆ ) g λ , N ( z ) + Bg λ , N ( z ) = H ( z ) , − g λ , N ( z ) − Bg λ , N ( z ) + ( λ − γ ∆ ) g λ , N ( z ) − ∆ g λ , N ( z ) = , − g λ , N ( z ) + ( λ − γ ∆ ) g λ , N ( z ) = g i λ , N ( z ) = − g i λ , N ( − z ) for i = , , ,
4. The existence and the uniqueness followsby the Fourier transform again.
Lemma B.2.
Define u λ , N byu λ , N = ∑ x , y ∈ Z dN (cid:0) g λ , N ( x − y ) q x q y + g λ , N ( x − y )( q x v y + q x v y )+ g λ , N ( x − y )( q x v y − q x v y ) + g λ , N ( x − y ) v x v y (cid:1) . Then, ( λ − L ) u λ , N = ( λ − L ( i ) ) u λ , N = ∑ x , y ∈ Z dN ∑ j = , H ( x − y ) q j x v j y . HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 36
Proof.
It follows by the direct computations: L ( i ) (cid:0) ∑ x , y ∈ Z dN g λ , N ( x − y ) q x q y (cid:1) = ∑ x , y ∈ Z dN g λ , N ( x − y )( v x q y + q x v y )= ∑ x , y ∈ Z dN g λ , N ( x − y )( q x v y − q x v y ) , L ( i ) (cid:0) ∑ x , y ∈ Z dN g λ , N ( x − y )( q x v y + q x v y ) (cid:1) = ∑ x , y ∈ Z dN g λ , N ( x − y ) { ( v x v y + v x v y ) + ( q x [ ∆ q ] y + q x [ ∆ q ] y )+ B ( q x v y − q x v y ) + γ ( q x [ ∆ v ] y + q x [ ∆ v ] y )= ∑ x , y ∈ Z dN { g λ , N ( x − y ) B ( q x v y − q x v y ) + γ [ ∆ g λ , N ]( x − y )( q x v y + q x v y ) } , L ( i ) (cid:0) ∑ x , y ∈ Z dN g λ , N ( x − y )( q x v y − q x v y ) (cid:1) = ∑ x , y ∈ Z dN g λ , N ( x − y ) { ( v x v y − v x v y )+ ( q x [ ∆ q ] y − q x [ ∆ q ] y ) + B ( − q x v y − q x v y ) + γ ( q x [ ∆ v ] y − q x [ ∆ v ] y ) } = ∑ x , y ∈ Z dN { g λ , N ( x − y ) v x v y + [ ∆ g λ , N ]( x − y ) q x q y − B ( q x v y + q x v y ) + γ [ ∆ g λ , N ]( x − y )( q x v y − q x v y ) } , L ( i ) (cid:0) ∑ x , y ∈ Z dN g λ , N ( x − y ) v x v y (cid:1) = ∑ x , y ∈ Z dN g λ , N ( x − y )([ ∆ q ] x v y + v x [ ∆ q ] y + B ( v x v y − v x v y ) + γ ([ ∆ v ] x v y + v x [ ∆ v ] y ))= ∑ x , y ∈ Z dN { [ ∆ g λ , N ]( x − y )( q x v y − q x v y ) + γ [ ∆ g λ , N ]( x − y ) v x v y } . (cid:3) Finally, we study the case with alternate charges. For this case, we take d = d ∗ =
2. Weintroduce ( ¯ ∆ F )( z ) = ∑ | y − z | = ( F ( y ) + F ( z )) = F ( z + ) + F ( z ) + F ( z − ) for F : Z N → R . Let h i λ , N : Z N → R ( i = , , , , , ) be the solution of the simultaneousequations λ h λ , N ( z ) + ∆ h λ , N ( z ) = z ≡ , ( λ + γ ) h λ , N ( z ) − h λ , N ( z ) = z ≡ , ( λ − γ ∆ ) h λ , N ( z ) + Bh λ , N ( z ) = H ( z ) for all z , ( λ + γ ¯ ∆ ) h λ , N ( z ) − h λ , N ( z ) + h λ , N ( z ) − Bh λ , N ( z ) = z ≡ , ( λ + γ ¯ ∆ ) h λ , N ( z ) − h λ , N ( z − ) − h λ , N ( z + ) − Bh λ , N ( z ) = z ≡ , (B.3)and satisfies h i λ , N ( z ) = − h i λ , N ( − z ) for i = , , , , ,
6. The existence and the uniqueness fol-lows by the Fourier transform again.
HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 37
Lemma B.3.
Define u λ , N byu λ , N = ∑ x ≡ y mod 2 (cid:0) h λ , N ( x − y )( − ) y q x q y + h λ , N ( x − y )( − ) y v x v y ) (cid:1) + ∑ x , y (cid:0) h λ , N ( x − y )( q x v y + q x v y ) + h λ , N ( x − y )( − ) y ( q x v y − q x v y ) (cid:1) . Then, ( λ − L ( ii ) ) u λ , N = ∑ x , y ∈ Z N ∑ j = , H ( x − y ) q jx v jy . Proof.
It follows by the following direct computations. The sum over x ≡ y means the sum overall pairs ( x , y ) satisfying x − y ≡ x ≡ y + ( x , y ) satisfying x − y ≡ L ( ii ) (cid:0) ∑ x ≡ y h λ , N ( x − y )( − ) y q x q y (cid:1) = ( ∑ x ≡ y h λ , N ( x − y )( − ) y ( v x q y + q x v y )= ∑ x ≡ y h λ , N ( x − y )( − ) y ( q x v y − q x v y ) , L ( ii ) (cid:0) ∑ x ≡ y h λ , N ( x − y )( − ) y v x v y (cid:1) = ∑ x ≡ y h λ , N ( x − y ) (cid:0) ( − ) y ([ ∆ q ] x v y + v x [ ∆ q ] y )+ B (( − ) y ( − ) x v x v y − ( − ) y ( − ) y v x v y ) + ( − ) y γ ([ ∆ v ] x v y + v x [ ∆ v ] y ) (cid:1) = ∑ x ≡ y (cid:0) − h λ , N ( x − y )( − ) y ( q x v y − q x v y ) (cid:1) + ∑ x ≡ y + (cid:0) ( h λ , N ( x − y + ) + h λ , N ( x − y − ))( − ) y ( q x v y − q x v y ) (cid:1) − γ ∑ x ≡ y h λ , N ( x − y )( − ) y v x v y , L ( ii ) (cid:0) ∑ x , y ∈ Z N h λ , N ( x − y )( q x v y + q x v y ) (cid:1) = ∑ x , y ∈ Z N h λ , N ( x − y ) (cid:0) ( v x v y + v x v y ) + ( q x [ ∆ q ] y + q x [ ∆ q ] y )+ ( − ) y B ( q x v y − q x v y ) + γ ( q x [ ∆ v ] y + q x [ ∆ v ] y ) (cid:1) = ∑ x , y ∈ Z N (cid:0) h λ , N ( x − y )( − ) y B ( q x v y − q x v y ) + γ [ ∆ h λ , N ]( x − y )( q x v y + q x v y ) (cid:1) , L ( ii ) (cid:0) ∑ x , y ∈ Z dN h λ , N ( x − y )( − ) y ( q x v y − q x v y ) (cid:1) = ∑ x , y ∈ Z dN h λ , N ( x − y ) (cid:0) ( − ) y ( v x v y − v x v y )+ ( − ) y ( q x [ ∆ q ] y − q x [ ∆ q ] y ) + B ( − q x v y − q x v y ) + ( − ) y γ ( q x [ ∆ v ] y − q x [ ∆ v ] y ) (cid:1) = ∑ x , y ∈ Z N (cid:0) h λ , N ( x − y ) { ( − ) x + ( − ) y } v x v y − [ ¯ ∆ h λ , N ]( x − y ) { ( − ) x + ( − ) y } q x q y − B ( q x v y + q x v y ) + γ [ ¯ ∆ h λ , N ]( x − y )( − ) y ( q x v y − q x v y ) (cid:1) = ∑ x ≡ y (cid:0) h λ , N ( x − y )( − ) y v x v y − [ ¯ ∆ h λ , N ]( x − y )( − ) y q x q y (cid:1) + ∑ x , y (cid:0) − Bh λ , N ( x − y )( q x v y + q x v y ) − γ [ ¯ ∆ h λ , N ]( x − y )( − ) y ( q x v y − q x v y ) (cid:1) . (cid:3) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 38
Lemma B.4.
Let d = and d ∗ = . Then, for j = , , we have N ∑ x = ∂ q jx F = if F = u λ , N , or u λ , N , given in Lemma B.1, or F = u λ , N given in Lemma B.2 or B.3.Proof. If F = u λ , N , or u λ , N , given in Lemma B.1, then N ∑ x = ∂ q jx F = ∑ x , y ∈ Z N g λ , N ( x − y ) v jy = ∑ x g λ , N ( x ) = j = , F = u λ , N given in Lemma B.2, N ∑ x = ∂ q x F = ∑ x , y ∈ Z N (cid:0) g λ , N ( x − y ) q y + g λ , N ( x − y ) v y + g λ , N ( x − y ) v y (cid:1) = ∑ x g λ , N ( x ) = ∑ x g λ , N ( x ) = ∑ x g λ , N ( x ) =
0. For j =
2, we can apply the same argument.If F = u λ , N given in Lemma B.3, N ∑ x = ∂ q x F = ∑ x ≡ y mod 2 h λ , N ( x − y )( − ) y q y + ∑ x , y (cid:0) h λ , N ( x − y ) v y + h λ , N ( x − y )( − ) y v y (cid:1) = ∑ x : even h λ , N ( x ) = ∑ x h λ , N ( x ) = ∑ x h λ , N ( x ) =
0. For j =
2, we can apply the same argu-ment. (cid:3) A PPENDIX
C. E
QUIVALENCE OF ENSEMBLES
We list the consequence of the equivalence of ensembles used in Section 2.
Lemma C.1.
For j = , , . . . , d ∗ , the followings hold:i) E N , E [( v j ) ] = Ed ∗ for any N,ii) E N , E [( v j ) ] → E ( d ∗ ) as N → ∞ ,iii) For any x ∈ Z d \ { } , E N , E [( v j ) ( v j x ) ] → E ( d ∗ ) as N → ∞ ,iv) For any x ∈ Z dN ,E N , E [( q j x − q j − x )( q j e − q j − e )( v j ) ] = N d (cid:18) E d ∗ + O ( N − d ) (cid:19) ∑ ξ ∈ Z dN , ξ (cid:54) = sin ( π ξ N · x ) sin ( π ξ N · e ) ∑ dk = sin ( πξ k N ) . Moreover, the term O ( N − d ) does not depend on x . This is essentially done in Lemma 7 of [1], but for completeness, we give a proof.
Proof.
Define ˜ q ( ξ ) = ( − δ ( ξ )) ω N ( ξ ) ˆ q ( ξ ) , ˜ v ( ξ ) = N − d / ( − δ ( ξ )) ˆ v ( ξ ) HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 39 where ω N ( ξ ) = N − d / (cid:113) ∑ da = sin ( πξ a N ) and ˆ q , ˆ v are the Fourier transform of q and v . Thefactor 1 − δ in the definition is due to the condition ∑ x q x = ∑ x v x = assumed in the micro-canonical state. The total energy is written as ∑ x E x = ∑ ξ (cid:54) = {| ˜ q ( ξ ) | + | ˜ v ( ξ ) | } = ∑ ξ (cid:54) = { ℜ ( ˜ q ( ξ )) + ℑ ( ˜ q ( ξ )) + ℜ ( ˜ v ( ξ )) + ℑ ( ˜ v ( ξ )) } . On Z dN \ { } , we define an equivalence relation ξ ∼ ξ (cid:48) if and only if ξ = − ξ (cid:48) . Denote the classof representatives for ∼ by U dN . If ξ ∼ ξ (cid:48) , then we have ℜ ( ˜ q ( ξ )) = ℜ ( ˜ q ( ξ (cid:48) )) , ℜ ( ˜ v ( ξ )) = ℜ ( ˜ v ( ξ (cid:48) )) ℑ ( ˜ q ( ξ )) = − ℑ ( ˜ q ( ξ (cid:48) )) , ℑ ( ˜ v ( ξ )) = − ℑ ( ˜ v ( ξ (cid:48) )) . Therefore, if N is odd ∑ x E x = ∑ ξ ∈ U dN { ℜ ( ˜ q ( ξ )) + ℑ ( ˜ q ( ξ )) + ℜ ( ˜ v ( ξ )) + ℑ ( ˜ v ( ξ )) } and if N is even, ∑ x E x = ∑ ξ ∈ U dN , ξ (cid:54) = ξ N { ℜ ( ˜ q ( ξ ))+ ℑ ( ˜ q ( ξ ))+ ℜ ( ˜ v ( ξ ))+ ℑ ( ˜ v ( ξ )) } + { ℜ ( ˜ q ( ξ N ))+ ℜ ( ˜ v ( ξ N )) } where ξ N = ( N , N , . . . , N ) .Note that the cardinality of U dN is N d − if N is odd and N d if N is even. If N is odd, under themeasure µ N , E , the random variables ( ℜ ( ˜ q ( ξ )) , ℑ ( ˜ q ( ξ )) , ℜ ( ˜ v ( ξ )) , ℑ ( ˜ v ( ξ )) ξ ∈ U dN are distributed according to the uniform measure on 2 d ∗ ( N d − ) -dimensional sphere of radius √ N d E . If N is even, under the measure µ N , E , the random variables ( ℜ ( ˜ q ( ξ )) , ℑ ( ˜ q ( ξ )) , ℜ ( ˜ v ( ξ )) , ℑ ( ˜ v ( ξ )) ξ ∈ U dN , ξ (cid:54) = ξ N , (cid:0) ℜ ( ˜ q ( ξ N )) √ , ℜ ( ˜ v ( ξ N )) √ (cid:1) are distributed according to the uniform measure on 2 d ∗ ( N d − ) -dimensional sphere of radius √ N d E .Then, we can conclude (i) since if N is odd, E N , E [( v j ) ] = E N , E [( N d ∑ ξ ∈ Z dN ˆ v j ( ξ )) ] = N d E N , E [( ∑ ξ ∈ Z dN , ξ (cid:54) = ˜ v j ( ξ )) ]= N d ∑ ξ ∈ U dN E N , E [ ℜ ( ˜ v j ( ξ )) ] = N d N d − N d E d ∗ ( N d − ) = Ed ∗ . We can also do a similar computation and conclude the same result if N is even.For (ii), if N is odd, we have E N , E [( v j ) ] = E N , E [( N d ∑ ξ ∈ Z dN ˆ v j ( ξ )) ] = N d E N , E [( ∑ ξ ∈ Z dN , ξ (cid:54) = ˜ v j ( ξ )) ] HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 40 = N d ∑ ξ ∈ U dN E N , E [ ℜ ( ˜ v j ( ξ )) ] + N d ∑ ξ (cid:54) = ξ (cid:48) ∈ U dN E N , E [ ℜ ( ˜ v j ( ξ )) ℜ ( ˜ v j ( ξ (cid:48) )) ]= N d N d − E N , E [ ℜ ( ˜ v j ( e )) ] + N d ( N d − )( N d − ) E N , E [ ℜ ( ˜ v j ( e )) ℜ ( ˜ v j ( e )) ] . Then, we can apply the classical equivalence of ensembles under the uniform measure on thesphere to conclude (ii). Similar arguments also work for the case that N is even, and also for(iii).Finally, we prove (iv). We only deal with the case that N is odd, since the other case can beshown in the same way. By definition, q j x − q j − x = iN d ∑ ξ ∈ Z dN sin ( π ξ N · x ) ˆ q j ( ξ ) , v j = N d ∑ ξ ∈ Z dN ˆ v j ( ξ ) and so E N , E [( q j x − q j − x )( q j e − q j − e )( v j ) ]= − N d ∑ ξ , ξ (cid:48) , ζ , ζ (cid:48) ∈ Z dN sin ( π ξ N · x ) sin ( π ξ (cid:48) N · e ) E N , E [ ˆ q j ( ξ ) ˆ q j ( ξ (cid:48) ) ˆ v j ( ζ ) ˆ v j ( ζ (cid:48) )]= − N d ∑ ξ , ξ (cid:48) , ζ , ζ (cid:48) ∈ Z dN , ξ , ξ (cid:48) , ζ , ζ (cid:48) (cid:54) = sin ( π ξ N · x ) sin ( π ξ (cid:48) N · e ) E N , E [ ˆ q j ( ξ ) ˆ q j ( ξ (cid:48) ) ˆ v j ( ζ ) ˆ v j ( ζ (cid:48) )] . The last term is equal to − N d ∑ ξ , ξ (cid:48) , ζ , ζ (cid:48) ∈ Z dN , ξ , ξ (cid:48) , ζ , ζ (cid:48) (cid:54) = sin ( π ξ N · x ) sin ( π ξ (cid:48) N · e ) ω N ( ξ ) ω N ( ξ (cid:48) ) E N , E [ ˜ q j ( ξ ) ˜ q j ( ξ (cid:48) ) ˜ v j ( ζ ) ˜ v j ( ζ (cid:48) )] . From the explicit distribution of ( ˜ q , ˜ v ) studied above, if ξ (cid:28) ξ (cid:48) , then E N , E [ ˜ q j ( ξ ) ˜ q j ( ξ (cid:48) ) ˜ v j ( ζ ) ˜ v j ( ζ (cid:48) )] = − E N , E [ ˜ q j ( ξ ) ˜ q j ( ξ (cid:48) ) ˜ v j ( ζ ) ˜ v j ( ζ (cid:48) )] = . The same is true if ζ (cid:28) ζ (cid:48) . Also, if ξ = ξ (cid:48) , then E N , E [ ˜ q j ( ξ ) ˜ q j ( ξ (cid:48) ) ˜ v j ( ζ ) ˜ v j ( ζ (cid:48) )]= E N , E [ (cid:0) ℜ ( ˜ q j ( ξ )) + i ℑ ( ˜ q j ( ξ )) (cid:1) ˜ v j ( ζ ) ˜ v j ( ζ (cid:48) )]= E N , E [( ℜ ( ˜ q j ( ξ )) − ℑ ( ˜ q j ( ξ )) ) ˜ v j ( ζ ) ˜ v j ( ζ (cid:48) )] + iE N , E [ ℜ ( ˜ q j ( ξ )) ℑ ( ˜ q j ( ξ )) ˜ v j ( ζ ) ˜ v j ( ζ (cid:48) )] = . The same is true if ζ = ζ (cid:48) . Therefore, the term E N , E [ ˜ q j ( ξ ) ˜ q j ( ξ (cid:48) ) ˜ v j ( ζ ) ˜ v j ( ζ (cid:48) )] (cid:54) = ξ = − ξ (cid:48) and ζ = − ζ (cid:48) . Moreover, E N , E [ ˜ q j ( ξ ) ˜ q j ( − ξ ) ˜ v j ( ζ ) ˜ v j ( − ζ )] = E N , E [ (cid:0) ℜ ( ˜ q j ( ξ )) + ℑ ( ˜ q j ( ξ )) (cid:1)(cid:0) ℜ ( ˜ v j ( ζ )) + ℑ ( ˜ v j ( ζ )) (cid:1) ]= E N , E [ ℜ ( ˜ q j ( e )) ℜ ( ˜ v j ( e )) ] does not depend on ξ , ζ . From the explicit distribution4 E N , E [ ℜ ( ˜ q j ( e )) ℜ ( ˜ v j ( e )) ] = E d ∗ + O ( N − d ) . Therefore, E N , E [( q x − q − x )( q e − q − e )( v ) ] HERMAL CONDUCTIVITY IN A MAGNETIC FIELD 41 = N d (cid:18) E d ∗ + O ( N − d ) (cid:19) ∑ ξ , ζ ∈ Z dN , ξ , ζ (cid:54) = sin ( π ξ N · x ) sin ( π ξ N · e ) ω N ( ξ ) ω N ( ξ )= N d (cid:18) E d ∗ + O ( N − d ) (cid:19) ∑ ξ ∈ Z dN , ξ (cid:54) = sin ( π ξ N · x ) sin ( π ξ N · e ) ∑ da = sin ( πξ a N ) . (cid:3) R EFERENCES [1] G. B
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EPARTMENT OF P HYSICS , F
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