VOLUME 78, NUMBER 10 P HYS I CAL RE VI EWL E T T ERS 10 MARCH 1997Heat Conduction in Chains of Nonlinear OscillatorsStefano Lepri,1,3Roberto Livi,1,3and Antonio Politi21Dipartimento di Fisica, Universit di Bologna and Istituto Nazionale di Fisica Nucleare, I-40127 Bologna, Italy2Istituto Nazionale di Ottica and Istituto Nazionale di Fisica Nucleare, I-50125 Firenze, Italy3Istituto Nazionale di Fisica della Materia, I-50125 Firenze, Italy(Received 22 October 1996; revised manuscript received 30 January 1997)We numerically study heat conduction in chains of nonlinear oscillators with time-reversiblethermostats. Anontrivial temperature prole is found to set in, which obeys a simple scalingrelationfor increasingthenumber Nofparticles. Thethermal conductivitydivergesapproximatelyasN12, indicatingthatchaoticbehaviorisnot enoughtoensuretheFourierlaw. Finally, weshowthat themicroscopicdynamicsensuresfulllment ofamacroscopicbalanceequationfortheentropyproduction. [S0031-9007(97)02611-2]PACS numbers: 44.10.+i, 05.45.+b, 05.60.+w, 05.70.LnTheapproachtononequilibriumstatistical mechanicsthrough the introduction of microscopically time-reversiblemodelshasbeenshowntoberatherpowerfulin the context of many-particle dynamics [1]. If thereversibility propertyis supplementedbythe so-calledchaotic hypothesis, the tools developed for strictlyhyperbolic systems allowed making general statisticalpredictions that have been successfully tested [2]. Amongthe achievements of this approach, we recall the derivationof theOnsagerreciprocityrelations[3] andtheexpres-sionof entropyproductionintermsof aself-generateddissipation rate [4].So far, however, most of the numerical efforts inthisareahavebeenrestrictedmainlytothedescriptionof gases anduids, where the thermostats, introducedto keep the energy constant, affect each particle [1](witha fewexceptions, suchas Refs. [3,5,6]). Inthepresent Letter we investigate the possibility of extendingthe above approach to a chain of coupled nonlinearoscillators, with specic reference to heat-conductionpropertiesofinsulatingsolids. Inthiscontext, themostnatural choice is to put only the chain extrema in contactwith two thermal baths at different temperatures.Afurthermotivationforthepresent workisthelackof convincingresults about thevalidityof theFourierconductionlawin1Dsystems. Let us brieyreviewthe current state of the arts. In the simplest case ofcoupled harmonic oscillators, it was rigorously shown[7] that, if theextremaof thechainareput incontactwith stochastic heat reservoirs operating at differenttemperatures, a nonequilibriumstationary state sets inwithnotemperaturegradient inthebulk. Asaresult,thermal conductivity kturns out tobeproportional tothenumberofoscillatorsN. Suchadivergencesimplyfollows from the existence of extended waves (phonons)freely traveling and carrying energy along the latticewithout attenuation.Afterwards,theroleofimpuritieshasbeentakenintoaccount, since it was expected that phonon waves shouldbe damped by the scattering processes due to defects, thuspossibly removing the divergence of k. Unfortunately, itwasfoundthatalthoughisotopicdisorderinaharmonicchainyieldsanonzerotemperaturegradient inthebulk[8,9], it still implies a diverging conductivity k N12[10,11]. A nite k has been obtained only by placing allthe oscillators in contact with independentthermal baths[12]. Asaresult, onecanconcludethat nophysicallysounddescriptionof Fourier lawcanbeobtainedwithharmonic chains.Morethandisorder, anharmonicityhasbeeninvokedasthekeyfeatureofreal solidsresponsiblefor normalheat conduction[13]: Nonlinearitiesmakephononsin-teract among themselves, thus impeding free propagation.In the spirit of the general theory of dynamicalsystems,nonintegrability, rather thananharmonicity, istheprop-ertythat shouldberesponsibleforaniteconductivity.In fact, nonlinear normal modes (solitons) freely transportenergyalongthechain. Ournumericalsimulationsper-formedwithaTodachain[seeEqs. (1)and(2)fortheprecisedenitionofthemodel]dorevealthesamesce-nario as for linear chains (see also [14]).Numerical experiments for chains with chaotic smoothpotentials [14,15] have been performed with too fewparticles to allow, even in the most detailed investigation[16], a conclusive study of the dependence of k onN. The same can be said for the case where bothanharmonicity and disorder have been simultaneouslyincluded [17,18].Finally, we must recall two somehow articial modelsthat lead to contradictory conclusions: The rst one is achain of harmonic oscillators with an innite barrier set ata given distance [19]; the second is the so-called ding-a-ling model, where harmonic oscillators alternate with freeparticles [20]. While in the former case the conductivityhas again been found to diverge, in the latter the authorsfound convincing evidence that it attains a nite value.Inthis Letter westudytheFermi-Pasta-Ulam(FPU)model, which represents the simplest anharmonic1896 0031-90079778(10)1896(4)$10.00 1997 The American Physical SocietyVOLUME 78, NUMBER 10 P HYS I CAL RE VI EWL E T T ERS 10 MARCH 1997approximationof amonoatomicsolid. Specically, weconsider a chain of N oscillators, indicating with qithe displacement of the ith particle fromits equilib-riumposition. Fixedboundaryconditions areassumedq0 qN11 0, while the dynamics of the centralN2 2oscillators isruledbythe equationsof motion qi fi2 fi11 , (1)where fi2V0qi2 qi21 and Vx x221bx44is theinteractionpotential ( bhas beenxedequal to0.1). Nos-Hoover thermostats [1,21] act on the rst andthe last particle, keeping them at temperature T1 and T2,respectively, q12z1 q11 f12 f2 ,z1 q21T12 1 , qN2z2 qN1 fN2 fN11 ,z2 q2NT22 1 . (2)The dynamical equations are left invariant under time re-versalcomposed with the involutionpi !2pi. Recentnumerical observations [22] showthat time-reversiblenonequilibriumdynamicsyieldsresultscompatiblewiththe predictions of Ref. [2], despite the fact that thesystemunder investigationis not strictlyAnosov. Weexpect thatthisshouldholdalsoforourmodelatsuf-ciently high temperatures. However, we shall not furtheraddress this point here; this will be the subject of a forth-comingpaper[23].We have performedextensive numerical simulationswith several values of N and T6, integrating the equationsof motionwithanimprovedfourth-order Runge-Kutta-Ghil algorithm. The rst clear result is the convergencetoawelldenedspatialproleofthelocaltemperatureTi p2i (?denotingtimeaverage). Theasymptoticstationary state satises the local equilibrium condition, asconrmed by the uctuations ofTithat are in agreementwith the canonical ones. The only exceptions are repre-sented by the particles close to the boundaries, where thetemperature prole seems to exhibit a singularity. Glob-ally, the proles satisfy a simple scaling relation, as clearlyshown in Fig. 1, where the values of Ti, corresponding todifferent chain lengths (and the same boundary tempera-tures), are plotted versus iN. The adoption of the abovescaled units is tantamount to considering the continu-um limit with the lattice spacing a equal to 1N. How-ever, this is to be taken only as a formal interpretation, asthemassdensityobviouslydivergeswhenN !`;con-versely, iftheequationsarerescaledinsuchawaythatboth energy and mass densities are kept constant, one ndsthat the nonlinearity coefcient b should diverge.The nonlinear shape of the proles could be interpretedas an indication of a temperature-dependent conductivity,but this is incorrect, since simulations done with such smalltemperature differences as T12 T2 4 still reveal cleardeviations from linearity. This is rather an indication ofFIG. 1. Scalingof the temperature proles for the FPUbmodel. The imposed temperatures are T1 152 and T2 24,and chain lengths areN 128,194, and256 (dashed, dotted,and solid lines, respectively). Averages are carried over a timeinterval 106, after a transient 104.the relevant role played by boundary conditions; indeed, aseemingly square-root-type singularity in the temperatureprole is always observed at the chain extrema.The next result concerns the local heat ux Jx, t,which is implicitly dened by the continuity equation,Hx, t1 divJx, t 0 , (3)where HPiHidx2 xi, Hi p2i 21 Vqi2qi21 and xi ia1 qi. By Fourier transforming (inspace) Eq. (3), and upon expanding in powers of thewave number k, one eventually nds that the heat ux atthe ith position is given by [23,24]Jit apifi11 , (4)where pifi11has the simple interpretation of the ow ofpotential energy from theith to the neighboring particle.Wehavecheckedthat J Jitisindependentofthelatticepositioni, asitshouldindeedbeforastationarynonequilibrium state.The only physically meaningful setting for the compari-son of heat uxes for different values of N is achieved byxinga 1, as itisthecasein realsystemswherethelatticespacingisdeterminedbythemutual interactions.The data reported in Fig. 2 shows that Jscales to zero asN2a, with a 0.556 0.05. The same scaling behaviorhas been obtained for different choices of the temperaturesT1and T2, provided that they are sufciently largeto ensure a chaotic behavior. This implies that theconductivity,kJdTdx , (5)diverges as N12a, since the temperature gradient vanishesas N21. Therefore, we are forced to conclude that1897VOLUME 78, NUMBER 10 P HYS I CAL RE VI EWL E T T ERS 10 MARCH 1997FIG. 2. Scaling of the heat ux J with the number ofoscillatorsNfortheFPUbmodel (sametemperaturesasinFig. 1). Theinset referstothecaseof animposedconstantgradient T12 T2N (see text) with the same boundarytemperatures: ScalingwithN21impliesthattheconductivityis constant.Fourier lawis not satised in the present frameworkand that chaoticity is not sufcient to ensure its validity.Surprisingly, the above behavior is similar tothe onefound in harmonic chains with randommasses [11],as if disorder andanharmonicityplayedthesamerole.However, we have no explanation for this fact.Two further remarks should be added as a comment tothe scaling behavior of J. First, from the very denitions[Eqs. (4)and(5)], onerealizesthattheassumptionaN214impliesanasymptoticallynitek, but thisisnomorethanjust aformal statement. Second, noticethatin the present philosophy, which is the standard oneadopted in the literature, T1and T2are kept xed whileNdiverges, sothat the temperature gradient (i.e., theexternal eld) goes to0. Accordingly, inthelimit oflargeN, thechaingetscloserandclosertoequilibriumso that, independently of T12 T2, a linear regime(intheGreen-Kubosense) iseventuallyattained. Thisis at variance with other physical settings, such aselectric charge transport, where the external eld is a freeparameter whosemagnitudecanbexedindependentlyof the system size. Accordingly, it is not obvious how tostudy nonlinear corrections in this framework, if they arerelevant at all.As a last comment onthermal conductivityinFPUchains, we want to stress that a truly nitekis observedwhen each particle is thermostated independently, accord-ing to a linear temperature prole. Obviously, in this case,Jitdependsonthelatticesite, but itsaveragevalueoverall sitesisfoundtoscaleasN21(seetheinset ofFig. 2). This is analogous to what was found in Ref. [12]withstochasticheat baths. Needlesstosay, thisresultsoundsa bitarticial, as the prole is imposed from theoutside.Returningtothe usualcase,theenergy balance atthechain extrema implies thatJ2z6p21,N 2z6T6 , (6)where the last equalityis obtainedfromthe conditiondz26dt 0. The above equation expresses the generalscenario arising in time-reversible models that a nonequi-libriumstationary state corresponds to a spontaneousemergence of dissipation [2,4]. The global volume con-traction rategin phase space is given by the average ofthe divergence of the velocity eld, i.e.,g z11z2.Inall simulations we checkedthat g. 0, as longasT1 T2, consistently with a theorem recently proved byRuelle[25]. Inanycase, z1isalwaysnegative(pro-vided that T1. T2), as indeed prescribed by energy bal-ance. In fact, the energy is pumped in from the hot reser-voir, ows through the chain, and is eventually absorbedinthecoldreservoir. Dynamically, itisatleast bizarrethat the hot thermostat is characterized by a local expan-sion of volumes: This iscompletely opposite to the ap-proachintermsofstochasticbaths, wheredissipationisalwaysassumed. To what extentthis peculiar feature isphysicallymeaningfulisunclear;nevertheless,theinter-pretationinterms of entropyproductionmakes perfectsense. In fact, Eq. (6) can be rewritten asz11 z2 J1T221T1, (7)with the convention that J. 0 is an incoming ux.Equation (7) canbephysicallyinterpretedasabalancerelation for the global entropy production. According tothe general principles of irreversible thermodynamics, thelocal rate of entropy production s in the bulk is given bysx Jddx1Tx. (8)UponintegratingEq. (8), theright-handsideof Eq. (7)isobtained, whichcanthusbeinterpretedastheglobalproduction rate of entropy in the bulk. On the other hand,according to general arguments on reversible thermostats[2], theleft-handsideof Eq. (7) is identiedwiththeentropy production from the heat baths. Equation (7) hasbeen numerically tested in a wide range of temperatures.Arelevant consequence of Eq. (6) is that z6areproportional toJ, sothat not onlytheuxes but alsothe dissipation g vanish in the thermodynamic limitN !`. This is indeed a remarkable difference withrespectto othermodelsof gases anduidsstudied, e.g.,in Refs. [2,4,5], where the dissipation is always extensive.In our opinion, this is due to the vanishing of dTdx, andnot to the fact that thermostats act only at the boundaries.Indeed, bygloballythermostatingthetwohalvesofthelattice at two different temperatures [26], we nd thatg,1898VOLUME 78, NUMBER 10 P HYS I CAL RE VI EWL E T T ERS 10 MARCH 1997which is now the sum of N contributions, still goes to zeroas N212.Although it is generally believed that nonlinearityyields a nite thermal transport coefcient, we haveshown that this is not true in FPUchains equippedwithtime-reversiblethermal baths. Thespecicchoiceof thethermostat does not affect thegeneralityof ourconclusions since the repetition of some simulationswith isokinetic (Gaussian) thermostats [1] lead to similarresults. Anotherpossibleexplanationofthedivergenceofk could be the dimensionality of the system: It mightbe that solitonlike propagation is generically favoredin 1Dsystems. At the present stage this is only aspeculation that needs further investigation. Another itemthat should be seriously taken into account concernsboundary conditions. If, as we suspect, the nonlinearshapeoftheprolestemsfromthepeculiarfunctioningof the thermostats, it is not to be excluded that a differentsetupcouldleadtomorephysicallymeaningful results.Some preliminary simulations performed by thermostatinganumber of ON particlesonboththeleft andrightsides suggest a possible slowconvergence to a nitekvalue. 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