Anomalous heat transport in classical many-body systems: overview and perspectives
AAnomalous heat transport in classical many-body systems:overview and perspectives
G. Benenti,
1, 2, 3, ∗ S. Lepri,
4, 5, † and R. Livi
6, 5, 4, ‡ Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Universit`a degli Studi dell’Insubria,via Valleggio 11, 22100 Como, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy NEST, Istituto Nanoscienze-CNR, I-56126 Pisa, Italy Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10 I-50019 Sesto Fiorentino,Italy Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1 I-50019, Sesto Fiorentino, Italy Dipartimento di Fisica e Astronomia and CSDC, Universit`a di Firenze, via G. Sansone 1 I-50019, Sesto Fiorentino,Italy
In this review paper we aim at illustrating recent achievements in anomalous heat diffusion, whilehighlighting open problems and research perspectives. We briefly recall the main features of thephenomenon for low-dimensional classical anharmonic chains and outline some recent develop-ments on perturbed integrable systems, and on the effect of long-range forces and magnetic fields.Some selected applications to heat transfer in material science at the nanoscale are described. Inthe second part, we discuss of the role of anomalous conduction on coupled transport and describehow systems with anomalous (thermal) diffusion allow a much better power-efficiency trade-offfor the conversion of thermal to particle current.
PACS numbers: 63.10.+a 05.60.-k 44.10.+i
I. INTRODUCTION
Anomalous diffusion is a well-established concept in statistical physics and has been invoked to describe manydiverse kinetic phenomena. A very detailed insight has been achieved by generalizations of the motion of Brownianparticles as done for the continuous-time random walk, L`evy flights and walks. A formidable body of literature onthe topic exists, and we refer for example to [1] as well as the present issue for an overview.The above particle models are based on a single-particle description, whereby the single walker performs a non-standard diffusive motion. How do these features emerge when dealing with a many-body problem? What are theconditions for a statistical system composed of many interacting particles to yield effectively anomalous diffusion ofparticles or quasi-particles? A further question regards how such anomalies in diffusion can be related to transportand whether they can be somehow exploited to achieve some design principle, like efficiency of energy conversion.In fact, although thermoelectric phenomena are known since centuries, it is only recently that a novel point of viewon the problem has been undertaken [2]. Generally speaking, this renewed research activity is motivated also by thepossibility of applications of the thermodynamics and statistical mechanics to nano and micro-sized systems withapplications to molecular biology, micro-mechanics, nano-phononics etc. This requires treating systems far fromthermodynamic limit, where fluctuations and interaction with the environment are essentially relevant and require tobe treated in detail.In the present contribution, we first review how anomalous energy diffusion arises in lattices of classical oscillatoras a joint effect of nonlinear forces and reduced dimensionality (and in this respect we will mostly discuss one-dimensional chains). In words, this amounts to say that the anomalous dynamics of energy carriers is an emergentfeature stemming from correlations of the full many-body dynamics. As a consequence, Fourier’s law breaks down :motion of energy carriers is so correlated that they are able to propagate faster than diffusively. In the second part ofthe paper, we discuss how this feature influences coupled transport and how it can be used to enhance the efficiencyof thermodiffusive processes.We conclude this review about the multifaceted problem of heat transport in classical systems with a short overviewsummarizing possible extensions to the quantum domain, with reference to related open problems, which certainly ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: roberto.livi@unifi.it a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p will deserve attention in the near future. II. ANOMALOUS HEAT TRANSPORT IN CLASSICAL ANHARMONIC LATTICES
The presence of a diverging heat conductivity with the systems size in a chain of coupled nonlinear oscillators wasfirst pointed out in [3, 4]. This was the beginning of a research field that, over more than two decades, has beendevoted to the understanding of the mechanisms yielding anomalous transport in low-dimensional systems. Far frombeing a purely academic problem, it has unveiled the possibility of observing such peculiar effects in nanomaterials,like nanotubes, nanowires or graphene [5, 6]. Extended review articles on this problem exist since many years [7, 8],while a collection of contributions about more recent achievements is contained in [9] and in a review article of thepresent issue [10]. Here it is useful to provide first a short summary about the state of the art in this field, while themain effort will be devoted to some recent achievements that point to promising and challenging directions for futureinvestigations.In any model where anomalous transport has been observed it emerges as a hydrodynamic effect due to theconspiracy of reduced space-dimensionality and conservation laws, yielding non standard relaxation properties evenin a linear response regime. As a reference we may consider the basic class of models, represented by a Hamiltonianof the following form: H = L (cid:88) n =1 (cid:20) p n m + V ( q n +1 − q n ) (cid:21) . (1)Typical choices for the interaction is the famous Fermi-Pasta-Ulam-Tsingou (FPUT) potential where V ( x ) = V F P UT ( x ) ≡ x + α x + β x and rotor (or Hamiltonian XY) model V ( x ) = V XY ( x ) ≡ − cos x . For whatconcerns conservation laws, in d = 1 anomalous transport has been generically observed in Hamiltonian models oftype (1), where energy and momentum and the ”stretch” variable (cid:80) n ( q n +1 − q n ) are conserved. It is worth recallingthat any approach aiming at describing out-of-equilibrium conditions, like stationary transport processes, has to bebased on the hydrodynamic equations associated with such locally conserved quantities. One-dimensional oscillatormodels with only one (e.g. the Frenkel-Kontorova or φ models [11]) or two conserved quantities like the rotor model[12, 13] or the discrete nonlinear Schr¨odinger lattice [14, 15] show instead standard diffusive transport. Intuitively,this is due to the presence of scattering sources for acoustic waves propagating through the lattice imposed by thevery presence of a local nonlinear potential, which breaks translation invariance (i.e. momentum conservation). Thisargument does not apply to the rotor model, where stretch only is not conserved, due to the angular nature of the q n variables: anyway standard diffusion is allowed because of the boundedness of the cosine potential. For what concernsdimensionality, in d = 3 normal diffusion regimes are expected to characterize heat transport in nonlinear lattices.Only in d = 2 one can find evidence of a diverging heat conductivity according to a logarithmic dependence with thesystem size L [16–18].The main distinctive feature of anomalous heat transport in one-dimensional Hamiltonian models of anharmoniclattices is that the finite-size heat conductivity κ ( L ) diverges in the limit of a large system size L → ∞ [3] as κ ( L ) ∝ L γ , with 0 < γ ≤
1, (the case γ = 1 corresponding to integrable models, like the Toda lattice discussed in Sec. II.B).This implies that this transport coefficient is,in the thermodynamic limit, not well-defined. In the linear responseregime, this is equivalent to find that the equilibrium correlator of the energy current J ( t ) displays for long times t anonintegrable power-law decay, (cid:104) J ( t ) J (0) (cid:105) ∝ t − (1 − δ ) , (2)with 0 ≤ δ <
1. Accordingly, the Green-Kubo formula yields an infinite value of the heat conductivity and allows toestablish the equivalence of the exponents, i.e. γ = δ , provided sound velocity is finite [4]. In Fig.1 we illustrate twotypical simulations of the FPUT model demonstrating the results above.The most basic issue of anomalous feature is related to anomalous dynamical scaling of equilibrium correlation ofthe hydrodynamic modes. A simple way to state this is to say that fluctuations of the conserved quantities withsmall wavenumber k evolve on time scales of order τ ( k ) ∼ | k | − z . For standard diffusion one has z = 2. Withinthe nonlinear fluctuating hydrodynamics approach it has been shown [19, 20] that models like (1) belong genericallyto the universality class of the famous Kardar-Parisi-Zhang (KPZ) equation, originally formulated in the context ofgrowing interfaces. It is well known that this equation in d = 1 is characterized by the dynamical exponent z = 3 / L10 κ (L) 10 -4 -2 ω -2 S (ω) -5 -4 -3 -2 -1 log ω -2-10 δ eff L γ ω −δ (a) (b) FIG. 1 Anomalous thermal conductivity for the FPUT model with cubic and quartic potential term ( α = 0 . β = 1); (a)finite-size conductivity measured in the nonequilibrium steady state and (b) power spectrum of heat current fluctuations S ( ω )(i.e. the Fourier transform of (cid:104) J ( t ) J (0) (cid:105) ); the long-time tail (2) in (2) corresponds to a divergence ω − δ at small frequencies.The inset reports the logarithmic derivative δ eff = d log S/d log ω . Data are compatible with δ = γ = 1 / . modes. This leads to the prediction γ = (2 − z ) /z = 1 / V ( x ) = V ( − x ) should belong to the a different universality class having a different exponent γ . Actually,the precise value γ is still somehow controversial: the theoretical prediction from mode-coupling approximation of thehydrodynamic theory yields γ = 1 / γ = 2 / γ = 1 / γ = 1 / σ ∝ t η with η >
1. These empiricalobservation have a theoretical justification within the nonlinear fluctuating hydrodynamics. Indeed the theory predictsa hydrodynamic ”heat mode” that has the characteristic shape given by a L´evy-stable distribution, see [20, 36] fordetails. Further support comes from mathematical results: superdiffusive behavior has been proven for one dimensionalinfinite chain of harmonic oscillators undergoing stochastic collisions conserving energy and momentum [37, 38]. Inthe same spirit, the more difficult case of nonlinear oscillators with conservative noise has been discussed [39]. Forexponential interactions (the Kac-van Moerbecke model), superdiffusion of energy is again demonstrated, and a lowerbound on the decay of current correlation function is given [40]. In Ref. [20] it is argued also such models shouldbelong to the KPZ class.One related distinctive feature of anomalous transport is that the temperature profiles in nonequilibrium steadystates are nonlinear, even for vanishing applied temperature gradients [32, 41]. There is indeed a close connectionwith fractional heat equation, that has been demonstrated and discussed in the recent literature [10, 42].
A. The importance of being small
As mentioned in the previous section, theoretical inferences on the problem of heat transport in anharmonic chainsstems from the basic assumption that one should compute any relevant quantity in the limits L → ∞ and t →∞ , performed just in this order. On the other hand, in any numerical simulation or in real low-dimensional heatconductors, such as nanowires, carbon nanotubes, polymers or even thin fibers, we have to deal with finite sizeand finite time corrections. These can be taken under control in a linear response regime if the mean-free path ofpropagating excitations, λ , and their mean interaction time, τ , are such that λ (cid:28) L and τ (cid:28) t . It is a matter offact that, when dealing with models of anharmonic chains, such a control is often not granted, mainly because ofnonlinear effects. This is may be a very relevant problem also for interpreting possible experimental verifications ofanomalous transport in real systems and also for designing nanomaterials, exhibiting deviations from the standarddiffusive conductivity.As a matter of fact, very severe finite-size effects invariably arise when trying to check predictions numerically. Veryoften, the estimate of the relevant exponents γ or δ systematically deviates from the expected values, and sometimesare even claimed to depend on parameters [43–46]. If universality is (as we do believe) to hold, this effects should bedue to subleading corrective terms to the asymptotics that are still relevant on the scales accessible to simulations.Besides those issues, other unexpected effects arise. For instance, for the FPUT [47], Toda [43] and the Kac-vanMoerbecke [38] chains perturbed by conservative noise, the exponent γ increases with the noise strength. Besides theproblem of evaluating the precise exponents, this is pretty surprising since it suggest that a larger stochasticity in themodel makes the system more diffusive, at least for finite systems.Another instance where finite size corrections are “amplified” by nonlinear effects is the case of anharmonic chainswith asymmetric potential , i.e. with V ( x ) (cid:54) = V ( − x ) as the FPUT model with α (cid:54) = 0. As already shown in Fig.1,both equilibrium and out of equilibrium measurements of the heat conductivity in the presence of an applied thermalgradient are usually consistent with KPZ scaling. However, in other temperature regimes, Fourier’s law appears tohold, i.e. thermal conductivity is constant over a large range of sizes [48]. This was traced back to the relativelylong relaxation time of mass inhomogeneities induced by the asymmetry of the interaction potential and acting asscatterers of phonons [48]. Actually, it was later shown in [49–51] that this is a strong finite-size effect, since it persistsfor relatively large values of L and t . Yet, the expected theoretical prediction of a diverging heat conductivity can berecovered in simulations performed for sufficiently large values of L and t . It should be pointed out that all of thesespeculations rely on numerical results, while a theoretical approach capable to provide estimates of the combinationof nonlinear with finite-size corrections to the hydrodynamics would be useful. Indeed, fluctuating hydrodynamicsprovide some prediction: subleading corrections to the leading asymptotic decay Eq. (2) can be very large and decayvery slowly [20].Finite size-effects can have other facets. A remarkable example is the Discrete Nonlinear Schroedinger equation, awell-known model for atomic condensates in periodic optical lattices. The model has two conserved quantities (energyand number density) and display normal diffusive transport [14]. However, at very low temperatures there appearsa further almost conserved quantity (the phase difference among oscillators) and, for a finite chain and long timesthe dynamics is the the same of a generic anharmonic model, leading to KPZ scaling of correlations and anomaloustransport [52, 53]. Further unexpected features have been reported also in [54]. In this paper the authors study thisproblem for a the FPUT- β model (i.e. Eq. (1) with V = V F P UT and α = 0) with an additional local, also called”pinning”, potential of the form U ( q ) = 12 L (cid:88) n =1 q n . (3)This term breaks translational invariance making energy the sole conserved quantity. By varying the nonlinear coupling β one observes a crossover from a ballistic transport, typical of an integrable model, to an anomalous diffusive regimeruled by and exponent of the time correlation function, which corresponds to a value of γ ∼ .
2. The crossover occursin the parameter region 0 . < β <
1. Numerical simulations performed for a chain of a few thousands of oscillatorsshow that further increasing β seems to yield an increasing γ . The overall outcome challenges the basic theoreticalargument, which predicts that an anharmonic chain equipped by a local potential should exhibit normal diffusion.For the sake of completeness it is worth mentioning that in this paper the model under scrutiny is compared with theso-called φ -model, where the nonlinear term in Hamiltonian (1), i.e. β ( q n +1 − q n ) , is substituted with βq n . Also forthis model one observes, in the same parameter region, a crossover from a ballistic regime to an anomalous diffusiveregime, but for β > γ ∼
0, i.e. the expected diffusive behavioris recovered.All of these results have a logical interpretation only if we invoke, once again, the role of finite size correctionscombined with nonlinearity. Actually, in the φ -model there is no way to argue that a ballistic regime should beobserved for any finite, even if small, value of β . The ballistic behavior observed in both models for β < . β >
1. A problem that one should investigatesystematically is the dependence on β of the chain length and of the integration time necessary to recover standarddiffusive transport, at least in the crossover region 0 . < β <
1, where one can expect to perform proper numericalanalysis in an accessible computational time.
B. Chimeras of ballistic regimes
In the light of what discussed in the previous section one should not be surprised to encounter further nonlinearchain models, equipped with a pinning potential, which exhibit a regime of ballisitic transport of energy, compatiblewith a linearly divergent heat conductivity, κ ( L ) ∼ L . Again, one could conjecture that this is due to the puzzlingcombination of nonlinearity and finite size effects, although, as we are going to discuss, the emerging scenario is moreintricate and interesting than the former statement could foretell.As a preliminary remark we want to recall that ballistic transport is the typical situation of an integrable Hamiltonianchain, whose prototypical example is the harmonic lattice, with V ( x ) = x in Hamiltonian (1). It is worth pointingout that the addition of the harmonic pinning term (3) again keeps the harmonic chain integrable. In this perspective,it seems reasonable to find that for sufficiently small nonlinearities both the FPUT- β and the φ chains, as discussedin the Section II.B, may exhibit a seemingly ballistic regime for β < . V ( x ) = e − x + x −
1: in this model heat transport is ballistic due to the finite–speed propagationof solitons (rather than phonons, as in the harmonic chain). Toda solitons are localized nonlinear excitations whichare known to interact with each other by a non–dissipative diffusion mechanism: the soliton experiences a randomsequence of spatial shifts while it moves through the lattice and interacts with other excitations without exchangingmomentum [56]. In fact, the calculation of the transport coefficients by the Green-Kubo formula indicates the presenceof a finite Onsager coefficient, which corresponds to a diffusive process on top of the dominant ballistic one [57, 58].When the pinning term (3) is at work the Toda chain becomes chaotic, as one can easily conclude by measuringthe spectrum of Lyapunov exponents [55]. This notwithstanding, not only the energy, but also the ”center of mass” h c = 12 ( L (cid:88) n =1 q n ) + 12 ( L (cid:88) n =1 p n ) is a conserved quantity. The peculiarity of the quadratic pinning potential (3) is testified by the observation thatif one turns it into a quartic one, i.e. U ( q i ) = (cid:80) Ln =1 q n , the quantity h c is no more conserved. Nonequilibriumsimulations of the Toda chain with the addition of (3), where heat reservoirs at different temperature, T > T act atits boundaries, yield a ”flat” temperature profile at T = ( T + T ) / T and T (i.e., Fourier’s law) one has to simulatethe dynamics of very large chains over very long times: typically L ∼ O (10 ) and t ∼ O (10 ), when all the parametersof the model are set to unit.Equilibrium measurements based on the Green-Kubo relation, i.e. on the behavior of the energy current correlator(2), provide further interesting facets of this scenario. By comparing the Toda chain with quadratic and quarticpinning potentials one observes in the latter case clear indications of a diffusive regime, i.e. a finite heat conductivity,and a practically negligible influence of finite size corrections, while in the former case the power spectrum (i.e. theFourier transform of (2)) is found to exhibit a peculiar scaling regime (with a power − / α and β have been chosen in such a way to correspond to a Taylor series expansion of the Toda chain)with the addition of (3) is found to converge to a plateau, in the absence of any precursor of a power-law scaling.Further details about the unexpected transport regimes encountered in the Toda chain equipped with the quadraticpinning can be found in [59]. One should point out that many of them are still waiting for a convincing theoreticalinterpretation. C. Anomalous transport in the presence of a magnetic field
Recent contributions [60, 61] have tackled the important problem of heat transport in chains of charged oscillators inthe presence of a magnetic field B . The model at hand is a one-dimensional polymer, that allows for transverse motionof the oscillators interacting via a harmonic potential. For B = 0 the exponent of the energy current correlator is δ = , thus indicating the presence of anomalous transport and a divergent heat conductivity κ ( L ) ∼ L . Byswitching on B the first basic consequence is the breaking of translation invariance, so that the total momentum isno more conserved. This notwithstanding, the total pseudo-momentum is conserved, but the hydrodynamics of themodel is definitely modified. Actually, numerical and analytic estimates indicate that the exponent δ may turn to avalue different from . In particular, in [60] two different cases were considered: the one where oscillators have thesame charge and the one where oscillators have alternate charges of sign ( − n , n being the integer index numberingoscillators along the chain. It can be easily shown that in the former case the sound velocity is null and the energycorrelator exhibits a thermal peak centered at the origin and spreading in time. Conversely, in the latter case thesound velocity has a finite, B -dependent value and the thermal peak of the energy correlator is coupled to soundmodes propagating through the chain. In the case of finite sound velocity (alternate charges) the exponent ruling thedivergence of the heat conductivity with the system size is found to remain the same obtained for B = 0, i.e. γ = .This is not surprising, since also for B = 0 the sound velocity in the model is finite. In the case of equally chargedbids, on the other hand, it appears a new exponent γ = , which corresponds to a universality class, different from allthe others encountered in anomalous transport in nonlinear chains of oscillators. An important remark on this newexponent is that, in the absence of a finite sound velocity, the identification of the exponents δ and γ , introduced inSec.II, is no more correct. In fact, in this case the value of the exponent δ is found to be very close to . Rigorousestimates of all of these exponents and also for the d = 2 and d = 3 versions of the charged polymer model have beenobtained through the asymptotics of the corresponding Green-Kubo integrals, where the deterministic dynamics hasbeen substituted with a stochastic version that conserves the same quantities [61] . For what concerns the differentone–dimensional cases, these rigorous estimates agree with the previous findings, while in d = 2 and in d = 3 theexpected logarithmic divergence and a finite heat conductivity have been singled out, respectively. D. The case of long-range interactions
Long-range forces, slowly decaying in the relative distance between particles are well-studied in statistical mechanics.They characterize a wide domain of physical situations, e.g. self-gravitating systems, plasmas, interacting vortices influids, capillary effects of colloids at an interface, chemo-attractant dynamics, cold atoms in optical lattices, colloidalactive particles etc. Several unusual features are known: ensemble inequivalence, long-living metastable states andanomalous energy diffusion [62, 63], inhomogeneous stationary states [64], lack of thermalization on interaction witha single external bath [65], etc. Moreover perturbations can spread with infinite velocities, leading to qualitativedifference with respect to their short-ranged counterparts [66, 67].The study of heat transport in chains with long-range interactions has been tackled only recently [68–72]. Themain question is to what extent are the anomalous properties changed when the spatial range of interaction amongoscillators increases. In two recent papers [73, 74] this problem has been investigated for Hamiltonian chains with along range potential of the form V = 12 N ( α ) N (cid:88) i =1 N (cid:88) i (cid:54) = j v ( q i − q j ) | i − j | α , (4)where the generalized Kac factor N ( α ) = 1 N N (cid:88) i =1 N (cid:88) j (cid:54) = i | i − j | α . (5) FIG. 2 A pictorial representations of heat transfer processes for long-range interacting chain, in the limit case α = ∞ (left)and α = 0 (right). Oscillators in contact with thermal reservoirs are contained in the rectangular boxes while the bulk ones arerepresented in the ellipse. The relevant transport channels are represented by black lines (adapted with permission from [74]Copyright (2019) by American Physical Society). guarantees the extensivity of the Hamiltonian [62, 63]. In particular, the long-range versions of both the rotor chain v ( x ) = V XY ( x ) and the FPUT- β model, v ( x ) = x + x have been investigated. The reason of this twofold choiceis that their nearest-neighbor versions (i.e., α → + ∞ ) of the former model exhibits standard diffusion of energy, whilethe latter is characterized by anomalous diffusion.For what concerns the rotor chain, nonequilibrium measurements, with thermal reservoirs at different temperature, T > T , acting at the chain ends, show that when α >
1, the resulting temperature profile interpolates linearlybetween T and T . Despite the long range nature of the interaction this is a strong indication that a standarddiffusive process still governs energy transport through the rotor chain, as in the limit α → + ∞ . Conversely, when α < T = ( T + T ) / α → + . It is important to point out that, at variance with the situation of chimeras ofintegrable models discussed in Section II.B, such flat temperature profiles have nothing to do with integrability, butthey are driven by the dominance of a ”parallel” energy transport mechanism, that connects the heat baths at thechain boundaries with each other directly through each individual rotors in the bulk of the chain. Energy transportalong the chain is practically immaterial and the overall process is mediated by rotors, that have to compromisebetween the two different temperatures imposed by the reservoirs. A sketch of this mechanism is presented in figure2. For small α each lattice site in the bulk is directly coupled to both thermal baths and its temperature sets to theaverage ( T + T ) / α <
1, a similar scenario seems to characterize the FPUT model : flat temperature profiles are observedalso in this case and one can verify that the same parallel transport mechanism described for rotors is at work. Onthe other hand, the scenario is definitely more complicated for α >
1. Careful numerical studies, exploring finite sizeeffects, indicate an overall scenario where an anomalous diffusion mechanism sets in, characterized by an exponent γ ,that is expected to increase up to the one of the quartic FPUT model in the limit α → + ∞ . But one is faced witha first surprise for α = 2, where a flat temperature profile is restored, although, as numerics clearly indicates, themechanism of transport along the chain certainly dominates over the parallel transport process. In the light of whatdiscussed in the Section II.B this appear as a possible manifestation of a chimera ballistic regime, although there isno simple argument allowing us to invoke a relation of this special case with an integrable approximation, if any. Thefact that the case α = 2 is characterized by a somehow ”weaker nonintegrability” is confirmed also for a related model[70, 72]. This can be traced back to the fact that in this case the lattice supports a special type of free-tail localizedexcitations (travelling discrete breathers) that enhance energy transfer [72].A complementary approach is the analysis of space-time scaling of equilibrium correlations, that in the short-rangecase yields useful information by the dynamical exponent z [20]. A numerical study of the structure factors of theFPUT model [74], shows that for α > z depends on α in a way that is certainly differentform the one that could be expected on the basis of the theory of L´evy processes. Moreover, upon adding the cubicterm ( q i − q j ) to the potential one recovers the same dependence of z on α , up to α = 5. This is again a surprise,because in the limit α → + ∞ the cubic and quartic versions of the FPUT-model should converge to different values of z . It is a matter of fact that no theoretical explanation is, at present, available to cope with this challenging scenario:in particular no hydrodynamic description is, to the best of our knowledge, available. E. Anomalous transport via the Multiparticle Collision Method
So far we have discussed the case of lattice models. To test the generality of the results and their universality, itis important to consider more general low-dimensional many-body systems like interacting fluids or even plasmas.Although molecular dynamics would be the natural choice, it is computationally convenient to consider effectivestochastic processes capable to mimic particle interaction through random collisions. A prominent example is theMulti-Particle-Collision (MPC) simulation scheme [75]. which has been proposed to simulate the mesoscopic dynamicsof polymers in solution, as well as colloidal and complex fluids. Another application is the modeling parallel heattransport in edge tokamak plasma [76]. Indeed, in the regimes of interest of magnetic fusion devices, large temperaturegradients will build up along the field line joining the hot plasma region (hot source), and the colder one close to thewall (that act as a sink). Besides this motivation, we mention here some results that are pertinent to the problem ofanomalous transport.In brief, the MPC method consists in partitioning the set of N p point- particles into N c disjoint cells. Within eachcell, the local center of mass coordinates and velocity are computed and a rotation of particle velocities around a ran-dom axis in the cell’s center of mass frame is performed. The rotation angles are fixed by imposing that the conservedquantities (energy, momentum) are locally preserved. Then, all particles are then propagated freely. Physical detailsof the interaction can be easily included phenomenologically, for instance introducing energy-dependent collision rates[18]. Interaction with external reservoirs can be also included by imposing Maxwellian distributions of velocity andchemical potentials on the thermostatted cells [77].For the case of a one-dimensional MPC fluid, since the conservation laws are the same as say, the FPUT model,we expect it to belong to the same KPZ universality class of anomalous transport. Indeed, numerical measurementsof dynamical scaling fully confirm this prediction [78]. This result is rather robust. The same type of anomalies havebeen shown to hold also for quasi-one-dimensional MPC dynamics, namely in the case of a fluid confined in a boxwith a large enough aspect ratio [18]. F. Anomalous heat transport in material science
The impact of the discovery of anomalous heat transport in anharmonic chains has triggered the search for such animportant physical effect in real low-dimensional materials. There is nowadays a vast literature, where these kind ofphenomena has been predicted and experimentally observed and part of this growing research field has been sketchedin [9]. Here we just want to illustrate two recent contributions which allow the reader to appreciate the relevance ofthis phenomenon for nanowires and polymers. We want to point out that an overview on the recent literature in bothfields is contained in the bibliography of these recent contributions.In [79] the problem of the lattice thermal conductivity in Si-Ge nanowires has been tackled by solving the Boltzmanntransport equation. More precisely, the authors used a Monte Carlo algorithm for sampling the phonon mean free path,combined with phenomenological ingredients concerning a suitable representation of realistic boundary conditions. Itis quite remarkable that they find evidence of a heat transport mechanism ruled by a L´evy walk dynamics of phononflights through the lattice structure. In particular, the phonon mean free paths are found to be characterized by aheavy-tailed distribution, which is associated to an anomalous diffusive behaviour, characterized by a size-divergentheat conductivity κ ( L ) ∼ L . . This behavior has been checked for system sizes in the range 10 nm < L < µm .Note that the phonon mean free path is orders of magnitude smaller than this size range. It is important to point outthat this scenario is robust for different alloy compositions, where the Ge component varies in the range [6% , γ = .In [80] atomistic simulations have been performed for poly(3,4-ethylenedioxythiophene), termed PEDOT, a conju-gated polymer which is considered to be of interest in view of its tunable and large electrical conductivity, transparencyand air stability [81]. The authors simulated this polymer model in d = 1 and in d = 3, both in equilibrium andnonequilibrium setups. More precisely, equilibrium measurements have been performed by estimating the depen-dence of the heat conductivity κ on the system size L by the Green-Kubo formula, where one has to estimate theasymptotic decay in time of the correlator of the total energy current (2). The outcome of this analysis has beencompared with the numerics obtained in nonequilibrium conditions. The setup adopted in this case is based on atransient measurement of the effective heat diffusivity ¯ κ . The two halves of the system are initially prepared in twothermalized states at different temperatures T and T . By running molecular dynamics simulation one can measure¯ κ as a function of L during the transient evolution to thermal equilibrium state at temperature ( T + T ) /
2. Moreprecisely, the estimate of ¯ κ relies on the fit of the time-dependent temperature difference in the two regions (for detailssee Eq.(9) in [80]). Finally, the thermal conductivity is obtained by the formula κ = ¯ κ ρ c V where ρ is the polymermass density and c V its specific heat. The authors have obtained consistent results showing that for the polymerchain anomalous diffusion is obeserved, κ ( L ) ∼ L γ with γ (cid:39) , while for the polymer crystal standard diffusion,i.e. a size-independent finite thermal conductivity κ , is recovered. These results are quite remarkable, because theyprovide a very clear confirmation of the role played by space dimension in determining anomalous transport effects.On the other hand the authors point out that the exponent γ (cid:39) does not agree with the expected theoretical value , since the phenomenological AMBER nonlinear potential adopted for the PEDOT model is certainly asymmetric,i.e. dominated by a leading cubic nonlinearity. Simulations of the polymer chain have been performed for quite largesystem sizes, namely 0 , µm < L < . µm . This notwithstanding, we cannot exclude that, as discussed in theprevious section, the combination of finite size effects and nonlinearity might be at work also in this case, yielding apower-law divergence of κ ( L ) that should be compatible with a symmetric phenomenological potential.These findings should be compared with those obtained for simpler models: for instance, in the case mentioned inSec. II.C, the exponent γ = has been found in a chain model with a quadratic interaction potential between thebeads [60] (notice that the possibility of displacements in both the horizontal and the vertical directions make thismodel non–integrable, at variance with the harmonic chain). On the other hand, a three-dimensional anharmonicchain with cubic and quartic interaction has been shown to belong instead to the KPZ class with γ = 1 / III. COUPLED TRANSPORT
The purpose of this section is to discuss the relevance of anomalous diffusion in coupled transport. In particular,we shall focus on steady-state transport and use for concreteness the language of thermoelectricty [2, 86], where thecoupled flows are charge and heat flow (other examples where the flow coupled to heat is particle or magnetizationflow could be discussed similarly). Moreover, we shall limit our discussion to power production, even though manyof the results and open problems highlighted below could be readily extended to refrigeration. Thermoelectricity isa steady state heat engine. Relevant quantities to characterize the performance of a generic heat engine, operatingbetween a hot reservoir at temperature T h and a cold one at T c , are: • The efficiency η = W/Q h , where W is the output work and Q h the heat extracted from the hot reservoir. Forcyclic as well as for steady-state heat engines, the Carnot efficiency η C = 1 − T c /T h is an upper bound for theefficiency η . • The output power P . It is common belief that an engine reaching the Carnot efficiency would require a quasi-static transformation, i.e. infinite cycle-time, implying vanishing power. For steady-state engines this argumentis replaced by the one that finite currents would imply dissipation, thus forbidding Carnot efficiency for non-zeropower. Hence, it is important to consider the power-efficiency trade-off. This is a key problem in the field offinite-time thermodynamics [87], in relation to the fundamental thermodynamic bounds on the performance ofheat engines as well as to the practical purpose of designing engines that, for a given output power, work atthe maximum possible efficiency. For classical cyclic heat engines, whose interactions with heat bath can bedescribed by a Markov process, it was proved [88] that the mean power P has an upper bound P ≤ AT c η ( η c − η ) , (6)where A is a system-specific prefactor [see also [89] for an analogous linear-response result within the frameworkof stochastic thermodynamics [90]]. While at first sight this bound implies that P → η → η c (and ofcourse when η → A diverges as the efficiency approaches the Carnot value [91]. • The fluctuations in the power output around its mean value P . Indeed, large fluctuations render heat enginesunreliable. Especially for heat engines at the nanoscale, one expects power fluctuations due, e.g., to thermal0noise, which are not negligible in comparison with the mean output power. In general, one would like to obtainhigh efficiency (as close as possible to the Carnot efficiency), large power, and small fluctuation. However,a trade-off between these three quantities has been proved [92] for a broad class of steady-state heat engines(including machines described by suitable rate equations or modeled with an overdamped Langevin dynamics): P ηη C − η T c ∆ P ≤ , (7)where the (steady-state) power fluctuations are given by∆ P ≡ lim t →∞ [ P ( t ) − P ] t, (8)where P ( t ) is the mean delivered power up to time t . Since P ( t ) converges for t → ∞ to P as 1 / √ t , an additionalfactor of t in (8) is needed to obtain a finite limit for ∆ P . Eq. (7) tells us that efficiency close to Carnot andhigh power entail large fluctuations. We note that the bound (8) has been recently generalized to periodicallydriven systems [93]. A. Linear response
Within linear response, the relationship between currents and generalized forces is linear [94, 95]. In particular, forthermoelectric transport we have j e = L ee F e + L eu F h ,j u = L ue F e + L uu F h , (9)where j e is the electric current density, j u is the energy current density, and the conjugated generalized forces are F e = −∇ ( µ/eT ) and F h = ∇ (1 /T ), where µ is the electrochemical potential and e is the electron charge. Thecoefficients L ab ( a, b = e, u ) are known as kinetic or Onsager coefficients; we will denote L the Onsager matrix withmatrix elements L ab . Note that the (total) energy current j u = j h + ( µ/e ) j e is the sum of the heat current j h and theelectrochemical potential energy current ( µ/e ) j e .The Onsager coefficients must satisfy two fundamental constraints. First, the second law of thermodynamics, thatis, the positivity of the entropy production rate, ˙ s = F e j e + F u j u ≥
0, implies L ee , L uu ≥ , L ee L uu −
14 ( L eu + L ue ) ≥ . (10)Second, for systems with time-reversal symmetry, Onsager derived fundamental relations, known as Onsager reciprocalrelations: L eu = L ue .The kinetic coefficients L ab are related to the familiar thermoelectric transport coefficients: the electrical conduc-tivity σ , the thermal conductivity κ , the thermopower (or Seebeck coefficient) S , and the Peltier coefficient Π: σ = − e (cid:18) j e ∇ µ (cid:19) ∇ T =0 = L ee T , (11) κ = − (cid:18) j h ∇ T (cid:19) j e =0 = 1 T det L L ee , (12) S = − e (cid:18) ∇ µ ∇ T (cid:19) j e =0 = 1 T (cid:18) L eu L ee − µe (cid:19) , (13)Π = (cid:18) j h j e (cid:19) ∇ T =0 = L ue L ee − µe . (14)For systems with time reversal symmetry, due to the Onsager reciprocal relations Π = T S .1The thermoelectric performance is governed by the thermoelectric figure of merit ZT = σS κ . (15)Thermodynamics imposes a lower bound on the figure of merit: ZT ≥
0. Moreover, the thermoelectric conversionefficiency is a monotonous increasing function of ZT , with η = 0 at ZT = 0 and η → η C in the limit ZT → ∞ .Nowadays, most efficient thermoelectric devices operate at around ZT ≈
1. On the other hand, it is generally acceptedthat
ZT > − S, σ, κ are generally interdependent.For instance in metals σ and κ are proportional according to the Wiedemann-Franz law, and the thermopower issmall, and these facts make metals poor thermoelectric materials. It is therefore of great importance to understandtha physical mechanisms that might allow to independently control the above transport coefficients. B. Anomalous transport and efficiency
The theoretical discussion of the role of anomalous (thermal) diffusion in thermoelectric transport is based on theGreen-Kubo formula. Such a formula expresses the Onsager coefficients in terms of dynamic correlation functions of thecorresponding currents, computed at thermodynamic equilibrium. When the current-current correlations (cid:104) j a (0) j b ( t ) (cid:105) ( (cid:104) . (cid:105) denotes the canonical average at a given temperature T ) do not decay after time averaging, then by definitionthe corresponding Drude weight D ab = lim t →∞ lim Λ →∞ t (cid:90) t dt (cid:104) j a (0) j b ( t ) (cid:105) (16)is different from zero. Here Ω is the system’s volume and Λ the system’s size along the direction of the currents.It has been shown [96–99] that a non-zero Drude weight D ab is a signature of ballistic transport, namely in thethermodynamic limit the corresponding kinetic coefficient L ab diverges linearly with the system size. Non-zero Drudeweights can be related to the existence of relevant conserved quantities, which determine a lower bound to D ab [100, 101]. By definition, a constant of motion Q is relevant if it is not orthogonal to the currents under consideration,in thermoelectricity (cid:104) j e Q (cid:105) (cid:54) = 0 and (cid:104) j u Q (cid:105) (cid:54) = 0.With regard to thermoelectric efficiency, a theoretical argument [102] predicts that, for systems with a singlerelevant conserved quantity, as it is the case for non-integrable systems with elastic collisions (momentum-conservingsystems), the figure of merit ZT diverges at the thermodynamics limit, so that the Carnot efficiency is attainedin that limit. Indeed, for systems where total momentum along the direction of the currents is the only relevantconstant of motion, as a consequence of ballistic transport the Onsager coefficients L ab ∝ Λ. Therefore, the electricalcurrent is ballistic, σ ∼ L ee ∼ Λ, while the thermopower is asymptotically size-independent, S ∼ L eu /L ee ∼ Λ . Onthe other hand, for such systems the ballistic contribution to det L is expected to vanish [102]. Hence the thermalconductivity κ ∼ det L /L ee grows sub-ballistically, κ ∼ Λ γ , with γ <
1. Since σ ∼ Λ and S ∼ Λ , we can concludethat ZT ∼ Λ − γ . That is, ZT diverges in the thermodynamic limit Λ → ∞ .This result has been illustrated in several models: in a diatomic chain of hard-point colliding particles [102] (seeFig. 3), in a two-dimensional system [77], with the dynamics simulated by the multiparticle collision dynamics methodmentioned in Section II.E [75], and in a one-dimensional gas of particles with screened (nearest neighbour) Coulombinteraction [104]. In all these (classical) models, collisions are elastic and the only relevant constant of motion is thecomponent of momentum along the direction of the charge and heat flows. In the numerical simulations, openingsconnect the system with two electrochemical reservoirs. The left ( L ) and right ( R ) reservoirs are modeled as ideal gases,at temperature T γ and electrochemical potential µ γ ( γ = L, R ). A stochastic model of the reservoirs [105, 106] is used:whenever a particle of the system crosses the opening, which separates the system from the left or right reservoir, it isremoved. Particles are injected into the system through the openings, with rates and energy distribution determinedby temperature and electrochemical potential (see, e.g. , [2]).We now show that systems with anomalous (thermal) diffusion allow a much better power-efficiency trade-offthan achievable by means of non-interacting systems or more generally by any system that can be described by thescattering theory.We first briefly discuss noninteracting systems. In this case, we can write the charge current following the Landauer-B¨uttiker scattering theory [107], adapted to classical physics: j e = eh (cid:90) ∞ d(cid:15) [ f L ( (cid:15) ) − f R ( (cid:15) )] T ( (cid:15) ) , (17)2 (a) N (b) S (c) N N (d) N ZT N FIG. 3 Electrical conductivity σ (a), Seebeck coefficient S (b), thermal conductivity κ (c) and thermoelectric figure of merit ZT (d) as a function of the mean number N of particles inside the system, for the one-dimensional, diatomic hard-point gas.Parameter values: masses m = 1 and M = 3, T = 1, and µ = 0, k B = e = 1, system lenhgth Λ equal to the number of particles.The figure is adapted with permission from [103] (Copyright (2018) by American Physical Society). where f γ ( (cid:15) ) = e − β γ ( (cid:15) − µ γ ) is the Maxwell-Boltzmann distribution function for reservoir γ and T ( (cid:15) ) is the transmissionprobability for a particle with energy (cid:15) to go from one reservoir to the other: 0 ≤ T ( (cid:15) ) ≤
1. Similarly, we obtain theheat current from reservoir γ as J h,γ = 1 h (cid:90) ∞ d(cid:15) ( (cid:15) − µ γ )[ f L ( (cid:15) ) − f R ( (cid:15) )] T ( (cid:15) ) . (18)For a given output power P = (∆ µ/e ) J e (∆ µ = µ R − µ L > µ L = 0), thetransmission function that maximizes the efficiency of the heat engine, η ( P ) = P/J h,L ( P, J h,L > T L > T R )was determined in [103], closely following the method developed for the quantum case in [108, 109]. The optimaltransmission function is a boxcar function, T ( (cid:15) ) = 1 for (cid:15) < (cid:15) < (cid:15) and T ( (cid:15) ) = 0 otherwise. Here (cid:15) = ∆ µ/η C isobtained from the condition f L ( (cid:15) ) = f R ( (cid:15) ), and corresponds to the special value of energy for which the flow ofparticles from left to right is the same as the flow from right to left. Thus, if particles only flow at energy (cid:15) , the flowcan be considered as “reversible”, in a thermodynamic sense. The energy (cid:15) and ∆ µ are determined numerically inthe optimization procedure [103, 108, 109]. The maximum achievable power is obtained when (cid:15) → ∞ : P (st)max = A π h k B (∆ T ) , (19)where ∆ T = T L − T R , A ≈ . P/P (st)max (cid:28) η ( P ) ≤ η (st)max ( P ) = η C (cid:32) − B (cid:115) T R T L PP (st)max (cid:33) , (20)where B ≈ . P → δ = (cid:15) − (cid:15) →
0. That is, we recover the celebrated delta-energy filtering mechanismfor Carnot efficiency [110–112]. Hence, the Carnot limit corresponds to the above mentioned reversible, zero powerflow of particles.It is clear that selecting transmission over a small energy window reduces power production. We can then expect thata different mechanism to reach Carnot efficiency might be more favorable for power production. Such expectations areborne out by numerical data for the above described interacting momentum-conserving systems. For these systems,the Carnot efficiency can be reached without delta-energy filtering [113], and the power-efficiency trade-off can beimproved. In Fig. 4, we show, for a given ∆ T and different system sizes, η/η C as a function of P/P max . Thesecurves have two branches. Indeed, they are obtained by increasing ∆ µ from zero, where trivially P = 0, up to the3 N = N = N = N = max C P/P
FIG. 4 Relative efficiency η/η C versus normalized power P/P max , at different systems sizes. The dotted, dashed, and dot-dashed curves show the linear response predictions, at the ZT ( N ) value corresponding to the given system size Λ = N . Thesolid line is the linear response result for ZT = ∞ (i.e., N = ∞ ). Model an parameter values as Fig. 3, with ∆ T = 0 . T L = 1 . T R = 0 . stopping value, where again P = 0. In the latter case, power vanishes because the electrochemical potential differencebecomes too high to be overcome by the temperature difference. Power first increases with ∆ µ , up to its maximumvalue P = P max , and then decreases, leading to a two-branch curve. Note that, despite the relatively high value of∆ T /T = 0 .
2, the numerical results are in rather good agreement with the universal linear curves, which only dependson the figure of merit ZT [103]. Not surprisingly, such agreement improves with increasing the system size, since |∇ T | = ∆ T /N decreases when N increases. In Fig. 4, we also show the limiting curve corresponding to ZT = ∞ ,obtained in momentum-conserving models in the thermodynamic limit N → ∞ . The upper branch of this curve isthe universal linear response upper bound to efficiency for a given power P . For P/P max (cid:28)
1, this bound reads asfollows: η lr ( P ) = η C (cid:18) − PP max (cid:19) , (21)which is a much less restrictive bound than bound (20), obtained above from scattering theory. Note that, using thelinear response result P max ∝ (∆ T ) , from (21) we obtain P ∝ ∆ T ( η C − η ). Accordingly, when η ≈ η C ∝ ∆ T , wefind the same dependence as in bound (6), that was obtained in a rather different context. C. Open problems
Several open questions about the role of anomalous transport in coupled transport remain, notably: • As discussed in Sec. II.D, heat transport in the presence of long-range interactions has been recently investigated.However, the role of the range of interactions on coupled transport, and in particular on the power-efficiencytrade-off, is unknown. • While momentum-conserving systems greatly improve the power-efficiency trade-off with respect to the nonin-teracting case, it is not known how they would behave with respect to the bound (7), which simultaneouslyinvolves efficiency, power, and fluctuations. That bound was obtained within the framework of stochastic ther-modynamics [90], with the transition rates between the system’s states which obey the local detailed balanceprinciple, without precise modeling of the underlying particle-particle interactions. On the other hand, in themodels described in this short review paper stochasticity is confined to the baths and the internal system’sdynamics plays a crucial role. This allowed us to discuss the impact of constants of motion and anomaloustransport on the efficiency of heat-to-work conversion and could be relevant also in relation to the bound (7). • While the results of this section have been corroborated by numerical simulations of several classical one-and two-dimensional systems, their extension to the quantum case remains as a challenging task for futureinvestigations.4 • The discussion of this section neglected phonons. However, besides their fundamental interest, the resultspresented here could be of practical relevance in very clean systems where the elastic mean free path of theconducting particles is much longer than the length scale associated with elastic particle-particle collisions, forinstance in high-mobility two-dimensional electron gases at very low temperatures. Phonon-free thermoelec-tricity (more precisely, thermodiffusion) has been experimentally realized in the context of cold atoms, first forweakly interacting particles [114] and more recently in a regime with strong interactions [115]. In this lattercase, a strong violation of the Wiedemann-Franz law has been observed. Such violations could not be explainedby the Landauer-B¨uttiker scattering theory. It would be interesting to investigate whether in such systems,where a high thermoelectric efficiency has been observed, the non-interacting bound on efficiency for a givenpower could be outperformed.
IV. OVERVIEW
In spite of the significant progress made over the last decades, the study of anomalous heat transport in nonlinearsystems still remains a challenging research field. While this review focused on some promising directions in classicalsystems, a main avenue for future investigations should undoubtedly be sought in the quantum domain. At thequantum level, anomalous heat transport is considerably less understood than in the classical case due to bothconceptual and practical difficulties. The same definition of thermodynamic observables like temperature, heat,and work, or of the concept of local equilibrium, is problematic in nanoscale systems. For instance, in solid-statenanodevices we can have structures smaller than the length scale over which electrons relax to a local equilibrium due toelectron-electron or electron-phonon interactions. Consequently, quantum interference effects, quantum correlations,and quantum fluctuation effects should be taken into account [2]. In particular, many-body localization provides amechanism for thermalization to fail in strongly disordered systems, with anomalous transport in the vicinity of thetransition between many-body localized and ergodic phases [116, 117].From a practical viewpoint, one has to face the computational complexity of simulating many-body open quantumsystems, with the size of the Hilbert space growing exponentially with the number of particles. Notwithstandingthese difficulties, a time-dependent density matrix renormalization group method allows the computation of transportproperties of integrable and nonintegrable quantum spin chains driven by local (Lindblad) operators acting close totheir boundaries [118]. Sizes up to n ∼
100 spins can be simulated, much larger than the n ∼
20 spins achievablewith other methods like Monte Carlo wavefunction approaches [119]. The obtained results confirm the relevanceof constants of motion on transport properties, with integrable systems that exhibit ballistic heat transport, whilefor quantum chaotic systems heat transport is normal (Fourier’s law, see [120, 121]). In passing, we note that formagnetization transport in some integrable models like XXZ one can obtain diffusive behavior (Fick’s law, see [118]).From a thermodynamic perspective, the use of local Lindblad operators is problematic. With the exception ofquantum chaotic systems, such operators do not drive the system to a grand-canonical state [122]. Furthermore, theuse of local Lindblad baths may result in apparent violations of the second law of thermodynamics [123]. GlobalLindblad dissipators are free from such problems and can be used to simulate heat transport, see e.g. [124], but arenot practical, in that limited to very small system sizes. Furthermore, it is crucial for the description of quantumheat engines in the extreme case where the working medium may even consist of a single two-level quantum system,to take into account medium-reservoir quantum correlations as well as non-Markovian effects, not included in thestandard, weak-coupling Lindblad description of quantum open systems. For first steps in this challenging direction,see [125–127]. The investigation of anomalous heat transport in such regime is terra incognita . Author Contributions
All Authors contributed equally to the present work.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationshipsthat could be construed as a potential conflict of interest.5
Acknowledgements
SL and RL acknowledge partial support from project MIUR-PRIN2017
Coarse-grained description for non-equilibrium systems and transport phenomena (CO-NEST) n. 201798CZL and the Eurofusion Enabling ResearchENR-MFE19.CEA-06
Emissive divertor and thank P. Di Cintio and S. Iubini for their unvaluable help.
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