Heat conduction in quantum harmonic chains with alternate masses and self-consistent thermal reservoirs

Heat conduction in quantum harmonic chains with alternate masses and self-consistent

thermal reservoirs

Antônio Francisco Neto*

Núcleo de Física, Campus Prof. Alberto Carvalho UFS, 49500-000 Itabaiana, SE, Brazil

Humberto C. F. Lemos†and Emmanuel Pereira‡

Departamento de Física-ICEx, UFMG, CP 702, 30.161-970 Belo Horizonte MG, Brazil 共Received 11 June 2007; published 12 September 2007兲

We consider the analytical investigation of the heat current in the steady state of the quantum harmonic chain of oscillators with alternate masses and self-consistent reservoirs. We analyze the thermal conductivity␬and obtain interesting properties: in the high temperature regime, where quantum and classical descriptions coin-cide, ␬ does not change with temperature, but it is quite sensitive to the difference between the alternate masses; and contrasting with this behavior, in the low temperature regime,␬becomes an explicit function of the temperature, but its dependence on the masses difference disappears. Our results reinforce the message that quantum effects cannot be neglected in the study of heat conduction in low temperatures.

DOI:10.1103/PhysRevE.76.031116 PACS number共s兲: 05.70.Ln, 05.40.⫺a, 05.45.⫺a, 44.10.⫹i

I. INTRODUCTION

Despite its fundamental importance for nonequilibrium statistical physics, the complete understanding of the heat conduction from first principles is still an open problem. For example, the derivation of the phenomenological Fourier’s law is still unknown, which states that the heat flow is proportional to the local temperature gradient, J共x兲

= −␬共T共x兲兲⵱T共x兲; ␬ is the thermal conductivity. After years of study共see e.g.,关1,2兴for reviews兲, the precise conditions in microscopic models of interacting particles that lead to the macroscopic Fourier’s law are still ignored. The detailed in-vestigation of the mechanisms behind the heat conduction 共including other questions besides the derivation of the Fou-rier’s law兲 involves considerable technical difficulties and even contradictions in numerical simulations, a scenario that makes the analysis of simple treatable analytical models a problem of interest 共more comments are presented, e.g., in 关3,4兴兲.

Recently, the model of the harmonic chain of oscillators with self-consistent stochastic thermal reservoirs at each site has been revisited and rigorously analyzed 关5兴 共this model was previously proposed in关6,7兴兲. The authors prove that the Fourier’s law holds in such a model. They still prove that there is a local thermal equilibrium and the heat conductivity ␬ is a constant 共it does not depend on the temperature兲. In this simple model, the reservoirs at each site describe the anharmonic degrees of freedom present in a more realistic 共and much more intricate兲interaction. The “self-consistent” condition means that there is no heat flow between an inner reservoir and its site, i.e., the inner reservoirs do not inject energy into the system. The quantum version of this model 共proposed in关8兴兲has been revisited quite recently in关9兴. The study of the quantum version is important to understand the

behavior of the thermal conductivity in the low temperature regions, where a quantum description may introduce signifi-cant changes. In particular, it is shown 共for this quantum model关8兴兲that the thermal conductivity becomes dependent on the temperature.

In the present paper, also with the aim of understanding the properties of the thermal conductivity of simple Hamil-tonian models, we consider the quantum chain of harmonic oscillators with reservoirs at each site and study, in detail, the heat flow for the case of particles共oscillators兲with different masses 共precisely, we take a chain with alternate masses— details ahead兲. The physical interest of models with unequal masses is well known. We recall, first, a few examples of analytical studies. The one-dimensional共1D兲harmonic chain with baths at the boundaries has been rigorously studied a long time ago: first the version with equal masses 关10兴, where it is proved that the heat current Jis independent of the system size N 共the Fourier’s law holds if J⬃N−1_{兲. In}

sequel, versions with different masses have been treated in 关11,12兴. In particular, for a random mass distribution, it is proved thatJ⬃N−1/2关13兴. For the harmonic Fibonacci chain, a model with the sequence of the particle masses given by 兵mi兩i= 1 , . . . ,N;mi=m␣ orm␤其 according to the Fibonacci

sequence, it is shown that J⬃共lnN兲−1 _{关14兴. Unfortunately,}

these models do not obey Fourier’s law and the investigation, in such case, is mainly related to the dependence of the heat flow on the system size. Recently, we have analyzed a sys-tem with unequal masses and normal conductivity关15兴: the harmonic chain with stochastic self-consistent reservoirs and unequal masses. We show that the thermal conductivity does not depend on temperature, but it is quite sensitive to the difference between the masses of the particles. Turning to the models treated by numerical simulations, there are many other problems where the presence of particle with unequal or alternate masses lead to interesting physical results: e.g., a nontrivial steady state is obtained for the two-mass problem in 关16兴; the relevance of the total momentum conservation for the validity of Fourier’s law is investigated in a model with alternate masses in关17兴; a strange behavior for Fermi-*afneto@fisica.ufmg.br

†

hcfl@fisica.ufmg.br

‡_{emmanuel@fisica.ufmg.br}

Pasta-Ulam chains with alternate masses in the steady state is numerically shown in关18兴.

Now, in the present work, we turn to the quantum version of this harmonic chain with self-consistent reservoirs and alternate masses. Using the techniques presented in 关9,19兴, we derive a formula for the thermal conductivity which pre-sents interesting new properties and shows that the quantum effects are quite significant. Besides the explicit dependence on temperature共that does not happen in the classical version, but appears in the quantum model with equal masses兲, we show that, for very small temperatures, the difference be-tween the masses, which is significant for the conductivity behavior in high temperature, is wiped out. In short, our results emphasize the significance of quantum effects in the low temperature region, which indicates that, in some recent approaches proposed to understand the heat mechanism and involving classical models and the region ofT→₀共_{see, e.g.,}

关20兴兲, the inclusion of quantum effects may give important corrections.

The rest of the paper is organized as follows. In Sec. II we introduce the model and the expressions for the heat cur-rents. In Sec. III we analyze the steady heat current in the self-consistent condition and derive the expression for the thermal conductivity, which is studied in the low and high temperature regimes in Sec. IV. Section V is devoted to the final comments, and the Appendix to a technical point, namely, the inversion of a tridiagonal matrix.

II. PRELIMINARIES: THE MODEL AND FORMALISM FOR THE STEADY HEAT CURRENT

Now we introduce the model and, briefly, schematize the derivation of the formula for the steady heat current as pre-sented in关9兴; there, the authors obtain a formalism to analyze quantum harmonic lattices following a Ford-Kac-Mazur pro-gram关21兴. In a few words, the approach considers harmonic lattices connected to baths modeled also as mechanical har-monic systems with initial coordinates and momenta distrib-uted according to some statistical distribution. The quantum dynamical equations are solved and the steady properties are obtained by taking stochastic distributions for the initial co-ordinates of the reservoirs and the limitt→⬁. For clearness, we will use, essentially, the same notation of关9兴.

We consider a harmonic system which consists of a chain 共W兲 with each site coupled to a bath共B兲 described also by harmonic interactions. The Hamiltonian of the chain and baths is given by

H₌1₂X˙TMX˙+12X

T_⌽X

=H_W+

兺

i=1

N

H_B i+

兺

i=1

N

V_B i, H_W=1

2X˙W T

MWX˙W+

1

2XW

T

⌽WXW,

H_B i=

1 2X˙B_i

T_M B_iX˙B_i+

1 2XB_i

T_⌽ B_iXB_i,

V_B i=XW

T

VB_iXB_i, 共1兲

M,M_W,M_B

iare diagonal matrices representing masses of the

particles in the entire system, chain, and baths, respectively;

Nis the number of sites in the chain. The symmetric matrices

⌽, ⌽W, ⌽B_i give the potential quadratic energies, and VB_i gives the interaction between theith site of the chain and its bath.X=关X1,X2, . . . ,XN_s兴T, whereXiis the position operator of theith particle,N_sis the number of elements in the entire system. We still haveX˙=M−1P, where Pris the momentum operator of therth particle satisfying the commutation rela-tions 关Xr,Pl兴=iប␦rl. In fact, we will be more specific: we will consider a one-dimensional harmonic chain with

H_W=

兺

l=1

N

冉

ml

2 X

˙

l

2

+m0 2 ␻0

2

X_l2

冊

+

兺

l=1

N+1

m0

2 ␻c

2_共

Xl−Xl−1兲2, 共2兲

wherem0␻0 2

is the on-site potential strength constant; m0␻c

2

is the interparticle potential strength; m0 is a constant with

unit of mass. We still considerm_l=m1iflis odd,ml=m2ifl

is even; and we take Dirichlet boundary conditions, i.e.,X0

=XN+1= 0.

The Heisenberg equations共giving the dynamic兲 for sys-tem and baths are

MWX¨W= −⌽WXW−

兺

i

VB_iXB_i,

MB_iX¨B_i= −⌽B_iXB_i−VB_i T

XW. 共3兲

The procedure is to treat the equations of the baths as linear inhomogeneous equations. Then, the solutions are plugged back into the equation for the chain, and the initial conditions of the baths are assumed to be distributed according to equi-librium phonon distribution functions 共with temperatures properly chosen—details ahead兲. All the reservoirs are taken to be Ohmic, and the coupling to the reservoirs is given by a dissipation constant␥.

The stationary properties are obtained turning to the equa-tions of motion, takingt→_⬁andt0→−⬁, and manipulating

them by using Fourier transforms. As said, details are pre-sented in 关9兴 and references therein. For the heat current from thelth reservoir into the chain we obtain

Jl=

兺

m=1

N ␥2

冕

−⬁

⬁

d␻␻2_兩关G

W

+

共␻兲兴l,m兩2

ប␻

␲ 关f共␻,Tl兲−f共␻,Tm兲兴, 共4兲

where

G_W+共␻兲=

冉

−␻2_M

W+⌽W−

兺

l

⌺l

+_共␻兲

冊

−1

,

M_W=

冢

m1

m2

⌽W=m0␻c

2

冢

2 +␯2

− 1 − 1 2 +␯2

_{− 1}

− 1 2 +␯2

冣

, 共5兲

with␯2

=␻₀2

/␻_c2

. The self-energy matrix has only one nonva-nishing element, 共⌺l

+_兲

ll=i␥␻. The variable ␻ appears with the Fourier transform

X ˜

W共␻兲= 1 2␲

冕

−⬁

⬁

dtXW共t兲ei␻t,

etc. And f共␻,Tl兲is the phonon distribution function

f共␻,Tl兲=

1

exp共ប␻/k_BT_l兲− 1. 共6兲 To proceed we consider the linear response regime with the temperature difference⌬T=兩T₁−T_N兩≪T=共T₁+T_N兲/ 2. In this situation, we may simplify the expression forJ_labove by expanding the phonon distribution functions 共6兲 about the mean temperatureT to get

Jl=␥2

冕

−⬁

⬁

d␻ប␻

3

␲ ⳵f共␻,T兲⳵Tm

兺

=1

N

兩关G_W+共␻兲兴l,m兩2共Tl−Tm兲.

共7兲

For the heat current inside the chain, from the sitel tol

+ 1, in the linear response limit, we have

Jl,l+1= −

m0␻c

2_␥

␲

冕

−⬁ ⬁

d␻␻

冉

ប␻ 2kBT

冊

2

cosech2

冉

ប␻ 2kBT

冊

⫻

兺

m=1

N

k_BT_mIm兵关G_W+共␻兲兴l,m关GW

+_共␻兲兴

l+1,m

* _其

, 共8兲

where * denotes the complex conjugate.

III. ANALYSIS OF THE STEADY HEAT CURRENT

To perform a detailed investigation of the steady heat cur-rents of our specific model, according to the preceding sec-tion, we need to know the matrixG_W+ 共5兲and the temperature profile, which is determined by the self-consistent condition

Jl= 0 for all inner sites, i.e. there is no heat flow between an inner site and the reservoir connected to it.

To determine关G_W+共␻兲兴l,m, we write关see Eq.共5兲兴

G_W+共␻兲= Z

−1

m0␻c

2, 共9兲

whereZ is a tridiagonal共Jacobi兲matrix

Z=

冢

z₁ − 1 − 1 z2 − 1

− 1 z3 − 1

− 1

_{zN−1 − 1} − 1 z_N

冣

, 共10兲

with

zj共␻兲=

冉

2 +␯2−

mj

m0

␻2

␻c

2

冊

−

i␥␻

m₀␻c

2, 共11兲

with mj=m1 for j odd, and mj=m2 for j even. There is a

general procedure to calculate the inverse of a tridiagonal matrix关22兴—we present the details for our specific case共10兲

in the Appendix. We have

共Z−1兲lm=clm共Z−1兲lm,

c_lm=

冦

冑

z2

z₁ iflandmare odd,

冑

z1

z2

iflandmare even,

1 otherwise,

共12兲

whereZis the tridiagonal matrix

Z₌

冢

z − 1

− 1 z

_{− 1}

− 1 z

冣

, z共␻兲=

冑

z1共␻兲z2共␻兲,

共13兲

with inverse

共Z−1_兲

lm=

冦

sinh关共N−m+ 1兲␣兴sinh共l␣兲

sinh共␣兲sinh关共N+ 1兲␣兴 ifm艌l, sinh关共N−l+ 1兲␣兴sinh共m␣兲

sinh共␣兲sinh关共N+ 1兲␣兴 ifm⬍l, 共14兲

where␣ is given by

e␣= z 2±

冑

冉

z

2

冊

2

− 1, e−␣= z 2⫿

冑

冉

z

2

冊

2

− 1, 共15兲

i.e.,z= 2 cosh␣. Any of the two roots can be taken in共15兲. As said, the detailed computation ofZ−1 _{is presented in the}

Appendix.

For points in the bulk of the chain, i.e., far from the boundaries共N→_{⬁兲, we have with a simple algebra}

共GW

+_兲

lm= 共Z−1兲lm

m0␻c

2 ⬇

clme−␣兩l−m兩 2m0␻c

2

To carry out the analysis of the heat currents, we still need to know the temperature profile: it must be taken such that

Jl= 0 for the inner pointsl, i.e., to assure the self-consistent condition. For the classical harmonic chain it is rigourously proved关5兴that there is a unique profile 共the linear one兲that leads to the self-consistent condition. As the quantum and classical models coincide at high temperatures, this linear profile is assumed for the quantum version with equal masses in关9兴: it is shown that the linear profile leads toJl= 0 for the inner points 共and it is also reobtained there by numerical simulations兲. Here we follow the same strategy共a direct deri-vation of the temperature profile for the quantum models is very intricate兲: if we turn to the classical model with unequal masses as treated in 关15兴 共i.e., using the techniques devel-oped by us—some of the authors and collaborators关23,24兴兲, we may easily find that the temperature profile for this clas-sical version is the linear one—see the general derivation given by Eqs.共22兲–共25兲presented in关23兴, and compare with Eqs. 共16兲–共18兲 in关15兴. We remark that, in a finite system, close to the boundaries, where the real thermal reservoirs are linked, the temperature profile shall deviate from the linear form共see关9兴for more discussions兲. We also recall that this model involving the self-consistent condition does not allow the inner baths to inject energy into the system, and so, the inner reservoirs shall be understood as a mechanism of pho-non scattering共i.e., in some sense they act as the anharmonic degrees of freedom not present in the potential兲 and not as real thermal reservoirs. It is also worth to recall that all of the situations, and the heat flow in the system, shall change if we leave the self-consistent condition and let the inner reservoirs inject energy into the system—see关25兴, for example, of sys-tems with strange heat flows. In a recent work关26兴, a quan-tum chain system with baths at the boundaries is analyzed and significant boundary effects are presented共see also关27兴

for other interesting results in quantum chains兲. But there, the system has a small length and differs very much from our problem: we are in the bulk of the system, far from the boundaries共the length of our chain goes to infinite兲.

Now, we take the linear temperature profile and show, for our quantum version, that it leads to the self-consistent con-ditionJl= 0, for l in the bulk of the chain. Indeed, forl far from the boundaries 共and N→_⬁兲_{, from} 共₇兲 _{we essentially}

have

Jl⬀

兺

m=−⬁

⬁

共l−m兲兩关G_W+共␻兲兴lm兩2

=

兺

m=−⬁ ⬁

兩clm兩2

兩e−␣兩l−m兩_兩2

4m₀2␻c

4

sinh2␣共l−m兲.

We write m=l±k, with k= 1 , 2 , . . . 共for k= 0, we have l−m

= 0兲. Then, in the sum we get

S₁₌

兺

m=−⬁ ⬁

兩clm兩2兩e−␣兩l−m兩兩2共l−m兲

=

兺

k=1 ⬁

兵兩cl,l−k兩2e−2␣兩l−共l−k兲兩关l−共l−k兲兴 +兩c_l,l+k兩2e−2␣兩l−共l+k兲兩_{关l−共l+k兲兴其}

=

兺

k=1 ⬁

k共兩cl,l−k兩2−兩cl,l+k兩2兲e−2␣k= 0,

since cl,l−k=cl,l+k: see 共12兲 and note that cl,r=cl,r±2=cl,r±2m,

∀l,r,m_苸Z.

Now we turn to the heat current inside the chain Jl,l+1.

From共8兲and共16兲we have

Jl,l+1=

−␥ 8m0␻c

2_␲

i

冕

−⬁ ⬁

d␻ ␻ 兩sinh␣兩2

冉

ប␻ 2kBT

冊

2

cosech2

冉

ប␻ 2kBT

冊

⫻

兺

m

kBTm关cl,mcl+1,m

*

e−␣兩l−m兩e−␣*兩l+1−m兩− c.c.兴, 共17兲 where “c.c.” means the complex conjugate. We still intro-duce the linear temperature profile

Tm=TL+

m− 1

N− 1共TR−TL兲⇒Tm=Tl+

m−l

N− 1共TR−TL兲. We writeTR andTL for the temperatures at the boundaries, instead of TN and T1, respectively. With the expression for

T_m, we get in the sum共17兲terms such as

兺

m=−⬁ ⬁

F共l,m兲 and

兺

m=−⬁

⬁

共m−l兲F共l,m兲,

with F共l,m兲=f共l,m兲−f*共l,m兲, where f共l,m兲

=c_l,mc_l*_+1,me−␣兩l−m兩_e−␣*兩l+1−m兩_{. Again, we consider l far from}

boundaries andN→⬁. To exploit the symmetry in the expo-nential, we write

S₂⬅

兺

m=−⬁ ⬁

F共l,m兲=

兺

k=0 ⬁

关F共l,l−k兲+F共l,l+ 1 +k兲兴.

To carry out the summation above, we note that we have for

kevenc_l,l−k=cl,l,cl+1,l+1+k=cl+1,l+1andcl,l+1+k= 1 =cl+1,l−k. On the other hand, for k odd we have cl,l+1+k=cl,l, cl+1,l−k =cl+1,l+1, andcl,l−k= 1 =cl+1,l+1+k.

It is convenient to split the sum into the terms withkeven and withkodd. We have

S₂=S₂odd+S₂even

=关共c_l,l−c_l+1,l+1兲共e␣+e−␣兲− c.c.兴 e

␣+␣*

e2共␣+␣*兲− 1. But, if we assumel odd共similar manipulation follows forl

even兲

共cl,l−cl+1,l+1兲共e␣+e−␣兲=共c1,1−c2,2兲z

=

冉

冑

z2

z1

−

冑

z1

z2

冊

冑

z₁z₂=z₂−z₁. And, as we can see from共11兲,z2−z1 is real,

z2−z1=

m1−m2

m0

␻2

␻c

2.

That is,

Let us analyze S₃₌兺_m⬁_=−⬁共m−l兲F共l,m兲. As before, we writem=l−kandm=l+ 1 +k, withk_苸兵0 , 1 , 2 , . . .其, and split the expression into terms withkeven andkodd,

S₃=

兺

k=0 ⬁

关共−k兲F共l,l−k兲+共k+ 1兲F共l,l+ 1 +k兲兴

=

兺

k=0 ⬁

F共l,l+ 1 +k兲+

兺

k=0 ⬁

k关F共l,l+ 1 +k兲−F共l,l−k兲兴.

After a considerable algebrism, we obtain

S_{3= − 2}关共cl,l+cl+1,l+1兲共e␣−e−␣兲− c.c.兴

e␣+␣* 共e2共␣+␣*兲− 1兲2 +关2cl,le−␣−cl+1,l+1共e␣−e−␣兲− c.c.兴

e␣+␣* e2共␣+␣*兲− 1. To make explicit the symmetry l↔l+ 1, we write for the second term above

关2cl,le−␣−cl+1,l+1共e␣−e−␣兲− c.c.兴

=关共c_l,l+cl+1,l+1兲e−␣+cl,le

−␣_−c

l+1,l+1e␣兴− c.c. 共19兲

Using that 共cl,l−cl+1,l+1兲共e␣+e−␣兲− c.c.= 0, from 共18兲, we

note that

共cl,le␣+cl,le−␣−cl+1,l+1e␣−cl+1,l+1e−␣兲− c.c. = 0

⇒共cl,le−␣−cl+1,l+1e␣− c.c.兲=共cl+1,l+1e−␣−cl,le␣− c.c.兲

⇒共cl,le−␣−cl+1,l+1e␣− c.c.兲

=1₂共cl,le−␣−cl+1,l+1e␣+cl+1,l+1e−␣−cl,le␣− c.c.兲. We rewriteS_3as

S₃₌共cl,l+cl+1,l+1兲

再

− sinh共␣兲e−共␣+␣*兲

sinh2共␣+␣*_兲 +

关e−␣− sinh␣兴 2 sinh共␣+␣*_兲

冎

− c.c. 共20兲

We remark that, in the particular case of massesm1=m2,

we havec_l,l=c_l+1,l+1= 1, and so, after some algebraic manipu-lations we have

S₃共m1=m2兲=

sinh␣*_{− sinh␣}

4 sinh2_␣

R

, 共21兲

where␣R= Re共␣兲=共␣+␣*兲/ 2. It leads to the same expression of关9兴 关see Eq.共5.8兲there in兴, where the case of equal masses was treated.

Finally, turning to the expression共17兲 for Jl,l+1, now we

have

Jl,l+1=

冋

−

␥kB 8m₀␻c

2_␲i

冕

−⬁

⬁

d␻ ␻ 兩sinh␣兩2

冉

ប␻ 2kBT

冊

2

⫻cosech2

冉

ប␻ 2kBT

冊

S₃共␻兲

册

TR−TL

N− 1 , 共22兲 withS₃共␻兲 given by共20兲. In short, the Fourier’s law holds with␬ given by the expression that multiplies 共TR−TL兲/共N − 1兲 in共22兲.

In the next section we study the behavior of ␬ as we change the temperature.

IV. THE EFFECTS OF TEMPERATURE ON THE THERMAL CONDUCTIVITY

We are mainly interested in the region of small tempera-tures, where the difference between the quantum and the classical description of the model may be significant.

As we have described in the preceding section, the differ-ence in␬ for the quantum systems with equal and unequal masses comes essentially from the factor 共c_l,l+c_l+1,l+1兲 that appears in the expression of S₃共␻兲, see 共20兲 and共21兲. For small temperatures, the main contribution to ␬ comes from small␻—see Eq.共22兲 共we give details below兲. Hence, let us investigate the factor共cl,l+cl+1,l+1兲for small␻. Recall that

cl,l+cl+1,l+1=

冑

z1

z2

+

冑

z2

z1

=z1+z2

z , 共23兲

wherez_jis given by共11兲. Thus, simple manipulations give us

z⬅

冑

z1z2= 2 +␯2−

1 2

m1+m2

m0

␻2

␻c

2−

i␥␻

m0␻c

2+O共␻

3_兲

,

i.e., up toO共␻2_{兲, the expression ofzbecomes that ofz}

jwith

m_j=共m₁+m₂兲/ 2, i.e.,z=共z₁+z₂兲/ 2. Hence,

cl,l+cl+1,l+1=

z1+z2

z = 2,

that is, our ␬ becomes the same of a system with equal masses, where the particle mass is 共m₁+m₂兲/ 2. In 关9兴, nu-merical computations for smallT in the quantum chain with equal masses are carried out, and graphics␬⫻T are plotted 共see Figs. 1 and 2 in关9兴兲. We present a brief analytical study which coincides with their results asT→_{0. From Eqs.共22兲}

and共20兲we get ␬= ␥kB

16m0␻c

2

␲i

冕

−⬁ ⬁

d␻

冉

ប␻ 2k_BT

冊

2

cosech2

冉

ប␻

2k_BT

冊

⫻4m0␻c 2

i

␥

sin2␣Icosh␣R sinh␣R关cosh2␣R− cos2␣I兴

,

where ␣I= Im共␣兲=共␣−␣*兲/共2i兲. As T→0, cosech2共ប␻/ 2kBT兲⬇4e−ប兩␻兩/kBT, and so

␬=C1

冕

0 ⬁

d␻

冉

ប␻ 2kBT

冊

2

e−ប␻/kBT

⫻ sin

2_␣

Icosh␣R sinh␣R共cosh2␣R− cos2␣I兲

=C1ប2␤2

冕

0 ⬁

d␻e−ប␻␤␻2_{f共␻兲, 共24兲}

whereC1is a constant depending onm0,␻c

2

follow the procedure given by the Laplace method 关28兴. It establishes that for a function such as

F共␶兲=

冕

0 ⬁

g共t兲e−␶h共t兲dt,

withh共t兲⬎h共0兲,h

_⬘

共0兲= 0, andh

_⬙

共0兲⬎0 共i.e.,h with a mini-mum int= 0兲, the asymptotic behavior ofF共␶兲 for ␶→⬁ is determined by the minimum of h. In a general case, with

h共t兲=h共0兲+1₂t2_h

⬙

共0兲+¯_{, g}共_t兲_=g共₀兲_+tg

_⬘

共₀兲₊¯_{, and} simple manipulations, we have

F共␶兲 ⬃g共0兲

冉

␲ 2␶h

_⬙

共0兲

冊

1/2

e−␶h共0兲_+e−␶h共0兲_O

冉

1

␶

冊

. Ifg共0兲= 0, to determine the asymptotic behavior ofF共␶兲, we take the next term in the expression above, etc. From共24兲, if we write␻=u2_{, we obtain}

␬=C␤2

冕

0 ⬁

due−បu2␤_u5_f共u2_{兲. 共25兲}

To examinef共u2_{兲, we need to split the analysis into the cases}

␯⫽_{0 and␯= 0. From}共₁₁兲_andz= 2 cosh␣, we have 关 recall-ing that our analysis is aroundu2=␻⬇0, the minimum ofh

in the exponential term in共25兲兴

2 cosh␣Rcos␣I⬇2 +␯2−

m¯

m0

␻2

␻c

2⬇2 +␯

2

,

2 sinh␣Rsin␣I⬇− ␥

m0␻c

2␻,

where m¯=共m₁+m₂兲/ 2. Therefore, f共␻兲⬇Csin2_␣

I⬇C␻2 =Cu4, whereC is some constant. Hence,

␬⬇C␤2

冕

0 ⬁

due−u2ប␤_u9

=C/␤3

.

That is, for␯⫽_{0, we have ␬}⬃_CT3_.

For␯= 0, again from 共11兲andz= 2 cosh␣, we have 2 cosh␣Rcos␣I⬇2 −

m¯

m0

␻2

␻c

2⬇2,

2 sinh␣Rsin␣I⬇− ␥

m0␻c

2␻,⇒sinh␣R⬇sin␣I⬇␻1/2.

Recalling the expression for f共␻兲,

f共␻兲= sin

2_␣

Icosh␣R sinh␣R关cosh

2_␣

R− cos

2_␣

I兴

⬇cosh␣R sinh␣R

⬇␻−1/2_.

Therefore,

␬⬇C␤2

冕

0 ⬁

due−u2ប␤u5f共u2兲

⬇C␤2

冕

0 ⬁

due−u2ប␤_u4_=C␤2

冉

1

␤

冊

5/2

.

That is,␬⬃CT1/2_{, if ␯= 0. Note that theC in␬ involves␥,}

m0,共m1+m2兲/ 2, but not 兩m1−m2兩, which appears in the

ex-pression of␬ for the classical version关15兴. As the quantum and classical versions shall coincide in the high temperature regime, our previous computations show that considerable changes appear with the temperature in the quantum model. We give more comments ahead.

Let us investigate the high temperature region. From共22兲, we note that the contribution of large␻ for ␬ goes to zero exponentially fast as␻increases. For bounded␻, we have

lim T→⬁

冉

ប␻ 2kBT

冊

2

cosech2

冉

ប␻ 2kBT

冊

→1,

and so, asTincreases,␬ becomes a constant function onT. In a previous work on the classical harmonic chain with self-consistent stochastic reservoirs and alternate masses 关15兴, using the approach developed in关23,24兴and a pertur-bative analysis, we show that, for weak coupling between the neighbor sites, i.e., for ␻_c2

small, and an interaction with nonzero on-site potential, we have

␬=共2m0␻c

2

兲2_␥m 1

−1_m

2

−1

冋

_关␻

c

2

共2 +␯2_兲兴2

冉

1

m1

− 1

m2

冊

2

冊

冋

+ 2␥2_␻

c

2_共

2 +␯2_兲

冉

1

m1

+ 1

m2

冊

册

−1

. 共26兲

It is interesting to note that, for a huge difference between the massesm₁andm₂, the behavior of␬is determined by the inverse of the square of the difference of masses. But ifm1

=m2, this behavior becomes related to the inverse of the

mass, and not to the inverse of the square of the mass. In other words, ␬ is more sensitive to changes in the particle masses for the system with alternate masses. As already men-tioned, it is also interesting to note that such a behavior is wiped out for the quantum system in a small temperature region, where␬ depends onm1+m2 only. In short, large T,

␬⬃1 /关const+共m1−m2兲2兴; small T, ␬ does not depend on

共m₁−m₂兲.

A detailed expression for␬in the high temperature region for the quantum chain with alternate masses requires a huge algebrism and we will not present it here. But it is easy to see that ␬ depends on 共m1−m2兲2: we turn to the factor cl,l +cl+1,l+1=共z1+z2兲共z1z2兲−1/2and write

z1+z2= 2¯z, ¯z⬅

冉

2 +␯2−

m1+m2

2m₀

␻2

␻c

2−i

␥␻

m0␻c

2

冊

,

冑

z1z2=

冋

冉

¯z+

m₂−m₁

2m₀

␻2

␻c

2

冊冉

¯z−

m₂−m₁

2m₀

␻2

␻c

2

冊

册

1/2

=

再

冋

¯z2−

冉

m2−m1 2m0

␻2

␻_c2

冊

2

册

冎

1/2

,

z1+z2

冑

z1z2

= 2

冋

1 −

冉

m2−m1 2m0

␻2

␻_c2

冊

2

共¯z2_兲−1

册

−1/2

.

mod-els in the high temperature region are the same, as shown in 关9兴 by using the same approach for the heat current of that adopted here.

V. FINAL COMMENTS

In this paper, we consider the investigation of the heat current in the steady state of the quantum harmonic chain of oscillators with alternate masses and self-consistent reservoirs—the anharmonic interactions which lead to a nor-mal thernor-mal conductivity 共i.e., to Fourier’s law兲 are repre-sented, in this model, by the reservoirs connected to each site. We use a recently proposed an approach for the study of the heat current 关9兴, and investigate, in detail, the thermal conductivity, presenting analytical results.

Models of unequal masses, as recalled in the introduction, have been used in several works related to the analysis of heat conduction, with important physical information. Here, for this quantum chain of particle with alternate masses, we show interesting properties of the thermal conductivity␬: in the high temperature regime, where the quantum and classi-cal descriptions coincide, ␬ is constant, i.e., it does not change with temperature, but it is quite sensitive to the dif-ference between the alternate masses共m1−m2兲; however, in

the low temperature regime, ␬ becomes a function of the temperatureT共␬⬃CT3_{for an interaction with on-site}

poten-tial; otherwise, ␬⬃CT1/2_{兲, and more, the dependence on}

共m₁−m₂兲is wiped out. Note that the difference between sys-tems with and without on-site potential is physically ex-pected: the on-site potential inhibits the heat transport, and so, decreases the thermal conductivity.

In a few words, our results reinforce the message that quantum effects cannot be neglected in the study of heat conduction in low temperatures.

ACKNOWLEDGMENT

The authors acknowledge financial support from CNPq 共Brazil兲.

APPENDIX: INVERSE OF THE TRIDIAGONAL MATRIX

We start from the formula for the inverse of a general tridiagonal matrix

A=

冢

b1 c1 0 0

a2 b2 c2

0 a3 b3 c3

a4

_{bn−1 cn−1}

0 an bn

冣

. 共A1兲

In关22兴it is proved, for␾ij=共A−1兲ij, that ␾j j=

冉

bj−ajcj−1

zj−2

zj−1

−aj+1cj

yj+2

yj+1

冊

−1

, 共A2兲

for j= 1 , 2 , . . . ,n, and

␾ij=

冦

−ci

zi−1

z_i ␾i+1,j ifi⬍j,

−ai

yi+1

yi

␾i−1,j ifi⬎j,

共A3兲

wherezjandyjare defined according the recurrence relations

zi=bizi−1−aici−1zi−2, z0= 1, z1=b1,

y_j=b_jy_j+1−aj+1cjyj+2, yn+1= 1, yn=bn, withi= 1 , 2 , . . . ,n, and j=n− 1 ,n− 2 , . . . , 1.

In our specific problem we havec_i=a_j= −1,∀i,j;b_j=b1if

jis odd, andbj=b2if j is even.

Let us consider first the simpler casebj=b,∀j, related to the problem of equal masses, since we will need it. In this case, we will denote the 共A1兲 matrix by A共b兲. Now, the recurrence relations above become

zi=bzi−1−zi−2, z0= 1, z1=b,

y_j=by_j+1−yj+2, yn+1= 1, yn=b.

Defining xi⬅yn+1−i, we have x0=yn+1= 1, x1=yn=b, and xi =bx_i−1−xi−2. Thus, the recurrence relation forxiis the same one ofzi.

Denoting byD_j共b兲the determinant of theA共b兲matrix of size j⫻j, it is easy to see 共e.g., by induction兲 that D_j共b兲

follow the same recurrence relation of zj, namely, D_j共b兲

=bD_j−1共b兲−D_j−2共b兲 关define first D_0共b兲⬅1兴. It follows that, ifb艌2, thenD_j共b兲⬎0, and soA is invertible. We have the following:

Lemma 1. For then⫻n matrixA共b兲, we have

D_n共b兲=sinh关共n+ 1兲␣兴 sinh␣ , where

e␣=b±

冑

b

2

− 4

2 . 共A4兲

Proof.The recurrence relationD_n共b兲=bD_n−1共b兲−D_n−2共b兲

reminds us that a second-order differential equation with constant coefficients f=bf

_⬘

−f

_⬙

, whose solutions aree±␣x_,␣ to be determined. If we try to writeD_n共b兲 as e␣n _ande−␣n_, from the recurrence relation, we obtain the restriction fore␣

given by the formula 共A4兲 above. Moreover, if we write

D_n共b兲=c1e␣n+c2e−␣n, with 共say, the boundaries condition兲

D_1共b兲=bandD_2共b兲=b2_{− 1, we findc}

1andc2which leads to

the formula of the lemma. 䊏

Theorem 1.Let A共b兲 be aN⫻N matrix, with b_j=b艌2. ThenA共b兲is invertible and

共A−1_兲

l,m=

冦

D_l−1D_N−m

D_N ifm⬎l, D_m−1D_N−l

D_N ifm艋l.

Proof.It follows from the general formulas for␾ij, 共A2兲 and共A3兲, restricted to this special case, from the observation that zj and Dj共b兲 have the same recurrence relation, and

some algebrism. 䊏

Now we turn to the case of our interest, the matrix with alternate masses, i.e., ai=cj= −1 and bj=b1 if j is odd and

bj=b2 if j is even. We will denote this matrix by ˜A共b1,b2兲.

For simplicity, we will be restricted to the cases of A˜ in a

N⫻N matrix, withNodd 共then,bN=b1兲. Now we have

z

˜i=bi˜zi−1−˜zi−2, ˜z0= 1, ˜z1=b1,

y

˜_j=bj˜yj+1−˜yj+2, ˜yn+1= 1, ˜yn=b1,

again with i= 1 , 2 , . . . ,N, and j=N− 1 ,N− 2 , . . . , 1. For odd

N, we have˜x_j⬅˜y_n+1−j=˜zj. It follows then thatD˜j共b1,b2兲 关the

determinant of aj⫻j A˜共b1,b2兲matrix兴and˜zjhave the same recurrence relation, namely,D˜k=bkD˜k−1−D˜k−2共again, we

de-fine D˜0= 1兲. We can obtain a relation between the

determi-nant of the matrix with alternate masses D˜k共b1,b2兲 and the

determinant of the specific matrix with equal masses

D_k共

冑

b1b2兲. We have the following lemma.

Lemma 2.We have

D˜k共b1,b2兲=

冦

冑

b1

b2

D_k共

冑

b1b2兲 ifkis odd,

D_k共

冑

b1b2兲 ifkis even.

Proof. It follows by induction and simple

manipula-tions. 䊏

Finally, let us calculateA˜−1.

From the formulas 共A2兲 and 共A3兲, and from the recur-rence relations, it follows that

共A˜−1兲ij=

冦

z ˜_i−1˜y_j+1

z

˜_j˜y_j+1−˜z_j−1˜y_j+2 ifi⬍j, z

˜_j−1˜y_i+1

z

˜_j−1˜y_j−˜z_j−2˜y_j+1 ifi艌j,

共A6兲

and we have the same relations for 共A−1_{兲ij, but, of course,}

with its ownzjandyjterms. We have seen that˜zj=D˜jforA˜,

z_j=D_jforA, and lemma 2 gives us the relation betweenD˜_j

andD_{j. Then, using} 共_A5兲 _{it is easy to prove the following}

result.

Theorem 2.We have

共A˜−1兲ij=

冦

冑

b2

b1

共A−1兲_ij ifi andjare odd,

冑

b₁

b2

共A−1兲_ij ifi andjare even, 共A−1_兲

ij otherwise.

共A7兲

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