Heat conduction and entropy production in a one-dimensional hard-particle gas

Energy diffusion in two-dimensional momentum-conserving nonlinear lattices: Lévy walk and renormalized phonon

The energy diffusion process in a few two-dimensional Fermi-Pasta-Ulam-type lattices is numerically simulated via the equilibrium local energy spatiotemporal correlation. Just as the nonlinear fluctuating hydrodynamic theory suggested, the diffusion propagator consists of a bell-shaped central heat mode and a sound mode extending with a constant speed. The profiles of the heat and sound modes satisfy the scaling properties from a random-walk-with-velocity-fluctuation process very well. An effective phonon approach is proposed, which expects the frequencies of renormalized phonons as well as the sound speed with quite good accuracy. Since many existing analytical and numerical studies indicate that heat conduction in such two-dimensional momentum-conserving lattices is divergent and the thermal conductivity κ increases logarithmically with lattice length, it is expected that the mean-square displacement of energy diffusion grows as tlnt. Discrepancies, however, are noticeably observed.

Room temperature second sound in cumulene

Second sound is known as the thermal transport regime occurring in a wave-like fashion, usually identified in a limited number of materials only at cryogenic temperatures. Here we show that second sound in a μm-long carbon chain (cumulene) might occur even at room temperature. To this aim, we calibrate a many-body force field on the first principles calculated phonon dispersion relations of cumulene and, through molecular dynamics, we mimic laser-induced transient thermal grating experiments. We provide evidence that by tuning temperature as well as the space modulation of its initial profile we can reversibly drive the system from a wave-like to a diffusive-like thermal transport. By following three different theoretical methodologies (molecular dynamics, the Maxwell-Cattaneo-Vernotte equation, and heat transport microscopic theory) we estimate for cumulene a second sound velocity in the range of 2.4-3.2 km s-1.

Blast in a One-Dimensional Cold Gas: From Newtonian Dynamics to Hydrodynamics

A gas composed of a large number of atoms evolving according to Newtonian dynamics is often described by continuum hydrodynamics. Proving this rigorously is an outstanding open problem, and precise numerical demonstrations of the equivalence of the hydrodynamic and microscopic descriptions are rare. We test this equivalence in the context of the evolution of a blast wave, a problem that is expected to be at the limit where hydrodynamics could work. We study a one-dimensional gas at rest with instantaneous localized release of energy for which the hydrodynamic Euler equations admit a self-similar scaling solution. Our microscopic model consists of hard point particles with alternating masses, which is a nonintegrable system with strong mixing dynamics. Our extensive microscopic simulations find a remarkable agreement with Euler hydrodynamics, with deviations in a small core region that are understood as arising due to heat conduction.

Too Close to Integrable: Crossover from Normal to Anomalous Heat Diffusion

Energy transport in one-dimensional chains of particles with three conservation laws is generically anomalous and belongs to the Kardar-Parisi-Zhang dynamical universality class. Surprisingly, some examples where an apparent normal heat diffusion is found over a large range of length scales were reported. We propose a novel physical explanation of these intriguing observations. We develop a scaling analysis that explains how this may happen in the vicinity of an integrable limit, such as, but not only, the famous Toda model. In this limit, heat transport is mostly supplied by quasiparticles with a very large mean free path ℓ. Upon increasing the system size L, three different regimes can be observed: a ballistic one, an intermediate diffusive range, and, eventually, the crossover to the anomalous (hydrodynamic) regime. Our theoretical considerations are supported by numerical simulations of a gas of diatomic hard-point particles for almost equal masses and of a weakly perturbed Toda chain. Finally, we discuss the case of the perturbed harmonic chain, which exhibits a yet different scenario.

Fractional-order heat conduction models from generalized Boltzmann transport equation

The relationship between fractional-order heat conduction models and Boltzmann transport equations (BTEs) lacks a detailed investigation. In this paper, the continuity, constitutive and governing equations of heat conduction are derived based on fractional-order phonon BTEs. The underlying microscopic regimes of the generalized Cattaneo equation are thereafter presented. The effective thermal conductivity κ_eff converges in the subdiffusive regime and diverges in the superdiffusive regime. A connection between the divergence and mean-square displacement 〈|Δx|²〉 ∼ t^γ is established, namely, κ_eff ∼ t^γ-1, which coincides with the linear response theory. Entropic concepts, including the entropy density, entropy flux and entropy production rate, are studied likewise. Two non-trivial behaviours are observed, including the fractional-order expression of entropy flux and initial effects on the entropy production rate. In contrast with the continuous time random walk model, the results involve the non-classical continuity equations and entropic concepts. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Anomalous energy diffusion in two-dimensional nonlinear lattices

Heat transport in one-dimensional (1D) momentum-conserved lattices is generally assumed to be anomalous, thus yielding a power-law divergence of thermal conductivity with system length. However, whether heat transport in a two-dimensional (2D) system is anomalous or not is still up for debate because of the difficulties involved in experimental measurements or due to the insufficiently large simulation cell size. Here we simulate energy and momentum diffusion in the 2D nonlinear lattices using the method of fluctuation correlation functions. Our simulations confirm that energy diffusion in the 2D momentum-conserved lattices is anomalous and can be well described by the Lévy-stable distribution. As is expected, we verify that 2D nonlinear lattices with on-site potentials exhibit normal energy diffusion, independent of the dimension. Contrary to the hypothesis of a 1D system, we further clarify that anomalous heat transport in the 2D momentum-conserved system cannot be corroborated by the momentum superdiffusion any longer. Our findings offer some valuable insights into mechanisms of thermal transport in 2D system.

Lévy walks on finite intervals: A step beyond asymptotics

A Lévy walk of order β is studied on an interval of length L, driven out of equilibrium by different-density boundary baths. The anomalous current generated under these settings is nonlocally related to the density profile through an integral equation. While the asymptotic solution to this equation is known, its finite-L corrections remain unstudied despite their importance in the study of anomalous transport. Here a perturbative method for computing such corrections is presented and explicitly demonstrated for the leading correction to the asymptotic transport of a Lévy walk of order β=5/3, which represents a broad universal class of anomalous transport models. Surprisingly, many other physical problems are described by similar integral equations, to which the method introduced here can be directly applied.

Observing golden-mean universality class in the scaling of thermal transport

We address the issue of whether the golden-mean [ψ=(sqrt[5]+1)/2≃1.618] universality class, as predicted by several theoretical models, can be observed in the dynamical scaling of thermal transport. Remarkably, we show strong evidence that ψ appears to be the scaling exponent of heat mode correlation in a purely quartic anharmonic chain. This observation seems to somewhat deviate from the previous expectation and we explain it by the unusual slow decay of the cross correlation between heat and sound modes. Whenever the cubic anharmonicity is included, this cross correlation gradually dies out and another universality class with scaling exponent γ=5/3, as commonly predicted by theories, seems recovered. However, this recovery is accompanied by two interesting phase transition processes characterized by a change of symmetry of the potential and a clear variation of the dynamic structure factor, respectively. Due to these transitions, an additional exponent close to γ≃1.580 emerges. All this evidence suggests that, to gain a full prediction of the scaling of thermal transport, more ingredients should be taken into account.

One-dimensional superdiffusive heat propagation induced by optical phonon-phonon interactions

One-dimensional anomalous heat propagation is usually characterized by a Lévy walk superdiffusive spreading function with two side peaks located on the fronts due to the finite velocity of acoustic phonons. In the case when the acoustic phonons vanish, e.g., due to the phonon-lattice interactions such that the system's momentum is not conserved, the side peaks will disappear and a normal Gaussian diffusive heat-propagating behavior will be observed. Here we show that there exists another new type of superdiffusive, non-Gaussian heat propagation but without side peaks in a typical nonacoustic, momentum-nonconserving system. It implies that thermal transport in this system disobeys the Fourier law, in clear contrast with the existing theoretical predictions. The underlying mechanism is related to an effect of optical phonon-phonon interactions. These findings may open a new avenue for further exploring thermal transport in low dimensions.

Energy Current Cumulants in One-Dimensional Systems in Equilibrium

A recent theory based on fluctuating hydrodynamics predicts that one-dimensional interacting systems with particle, momentum, and energy conservation exhibit anomalous transport that falls into two main universality classes. The classification is based on behavior of equilibrium dynamical correlations of the conserved quantities. One class is characterized by sound modes with Kardar-Parisi-Zhang scaling, while the second class has diffusive sound modes. The heat mode follows Lévy statistics, with different exponents for the two classes. Here we consider heat current fluctuations in two specific systems, which are expected to be in the above two universality classes, namely, a hard particle gas with Hamiltonian dynamics and a harmonic chain with momentum conserving stochastic dynamics. Numerical simulations show completely different system-size dependence of current cumulants in these two systems. We explain this numerical observation using a phenomenological model of Lévy walkers with inputs from fluctuating hydrodynamics. This consistently explains the system-size dependence of heat current fluctuations. For the latter system, we derive the cumulant-generating function from a more microscopic theory, which also gives the same system-size dependence of cumulants.

Fourier heat conduction as a strong kinetic effect in one-dimensional hard-core gases

For a one-dimensional (1D) momentum conserving system, intensive studies have shown that generally its heat current autocorrelation function (HCAF) tends to decay in a power-law manner and results in the breakdown of the Fourier heat conduction law in the thermodynamic limit. This has been recognized to be a dominant hydrodynamic effect. Here we show that, instead, the kinetic effect can be dominant in some cases and leads to the Fourier law for finite-size systems. Usually the HCAF undergoes a fast decaying kinetic stage followed by a long slowly decaying hydrodynamic tail. In a finite range of the system size, we find that whether the system follows the Fourier law depends on whether the kinetic stage dominates. Our Rapid Communication is illustrated by the 1D hard-core gas models with which the HCAF is derived analytically and verified numerically by molecular dynamics simulations.

Scattering of lattice solitons and decay of heat-current correlation in the Fermi-Pasta-Ulam-α-β model

As is well known, solitons can be excited in nonlinear lattice systems; however, whether, and if so, how, this kind of nonlinear excitation can affect the energy transport behavior is not yet fully understood. Here we study both the scattering dynamics of solitons and heat transport properties in the Fermi-Pasta-Ulam-α-β model with an asymmetric interparticle interaction. By varying the asymmetry degree of the interaction (characterized by α), we find that (i) for each α there exists a momentum threshold for exciting solitons from which one may infer an α-dependent feature of probability of presentation of solitons at a finite-temperature equilibrium state and (ii) the scattering rate of solitons is sensitively dependent on α. Based on these findings, we conjecture that the scattering between solitons will cause the nonmonotonic α-dependent feature of heat conduction. Fortunately, such a conjecture is indeed verified by our detailed examination of the time decay behavior of the heat current correlation function, but it is only valid for an early time stage. Thus, this result may suggest that solitons can have only a relatively short survival time when exposed in a thermal environment, eventually affecting the heat transport in a short time.

Heat Transport via Low-Dimensional Systems with Broken Time-Reversal Symmetry

We consider heat transport via systems with broken time-reversal symmetry. We apply magnetic fields to the one-dimensional charged particle systems with transverse motions. The standard momentum conservation is not satisfied. To focus on this effect clearly, we introduce a solvable model. We exactly demonstrate that the anomalous transport with a new exponent can appear. We numerically show the violation of the standard relation between the power-law decay in the equilibrium correlation and the diverging exponent of the thermal conductivity in the open system.

Heat perturbation spreading in the Fermi-Pasta-Ulam-β system with next-nearest-neighbor coupling: Competition between phonon dispersion and nonlinearity

We employ the heat perturbation correlation function to study thermal transport in the one-dimensional Fermi-Pasta-Ulam-β lattice with both nearest-neighbor and next-nearest-neighbor couplings. We find that such a system bears a peculiar phonon dispersion relation, and thus there exists a competition between phonon dispersion and nonlinearity that can strongly affect the heat correlation function's shape and scaling property. Specifically, for small and large anharmoncities, the scaling laws are ballistic and superdiffusive types, respectively, which are in good agreement with the recent theoretical predictions; whereas in the intermediate range of the nonlinearity, we observe an unusual multiscaling property characterized by a nonmonotonic delocalization process of the central peak of the heat correlation function. To understand these multiscaling laws, we also examine the momentum perturbation correlation function and find a transition process with the same turning point of the anharmonicity as that shown in the heat correlation function. This suggests coupling between the momentum transport and the heat transport, in agreement with the theoretical arguments of mode cascade theory.

Anomalous temperature-dependent heat transport in one-dimensional momentum-conserving systems with soft-type interparticle interaction

We numerically investigate the heat transport problem in a one-dimensional momentum-conserving lattice with a soft-type (ST) anharmonic interparticle interaction. It is found that with the increase of the system's temperature, while the introduction of ST anharmonicity softens phonons and decreases their velocities, this type of nonlinearity like its hard type (HT) counterpart, can still not be able to fully damp the longest wavelength phonons. Therefore, a usual anomalous temperature dependence of heat transport with certain scaling properties similarly to those shown in the Fermi-Pasta-Ulam-β-like systems with HT interactions can be seen. Our detailed examination from simulations verifies this temperature-dependent behavior well.

One-Dimensional Self-Organization and Nonequilibrium Phase Transition in a Hamiltonian System

Self-organization and nonequilibrium phase transitions are well known to occur in two- and three-dimensional dissipative systems. Here, instead, we provide numerical evidence that these phenomena also occur in a one-dimensional Hamiltonian system. To this end, we calculate the heat conductivity by coupling the two ends of our system to two heat baths at different temperatures. It is found that when the temperature difference is smaller than a critical value, the heat conductivity increases with the system size in power law with an exponent considerably smaller than 1. However, as the temperature difference exceeds the critical value, the system's behavior undergoes a transition and the heat conductivity tends to diverge linearly with the system size. Correspondingly, an ordered structure emerges. These findings suggest a new direction for exploring the transport problems in one dimension.

A violation of universality in anomalous Fourier's law

Since the discovery of long-time tails, it has been clear that Fourier's law in low dimensions is typically anomalous, with a size-dependent heat conductivity, though the nature of the anomaly remains puzzling. The conventional wisdom, supported by renormalization-group arguments and mode-coupling approximations within fluctuating hydrodynamics, is that the anomaly is universal in 1d momentum-conserving systems and belongs in the Lévy/Kardar-Parisi-Zhang universality class. Here we challenge this picture by using a novel scaling method to show unambiguously that universality breaks down in the paradigmatic 1d diatomic hard-point fluid. Hydrodynamic profiles for a broad set of gradients, densities and sizes all collapse onto an universal master curve, showing that (anomalous) Fourier's law holds even deep into the nonlinear regime. This allows to solve the macroscopic transport problem for this model, a solution which compares flawlessly with data and, interestingly, implies the existence of a bound on the heat current in terms of pressure. These results question the renormalization-group and mode-coupling universality predictions for anomalous Fourier's law in 1d, offering a new perspective on transport in low dimensions.

Pressure-induced recovery of Fourier's law in one-dimensional momentum-conserving systems

We report the two typical models of normal heat conduction in one-dimensional momentum-conserving systems. They show the Arrhenius and the non-Arrhenius temperature dependence. We construct the two corresponding phenomenologies, transition-state theory of thermally activated dissociation and the pressure-induced crossover between two fixed points in fluctuating hydrodynamics. Compressibility yields the ballistic fixed point, whose scaling is observed in Fermi-Pasta-Ulam (FPU) β lattices.

Heat conduction and energy diffusion in momentum-conserving one-dimensional full-lattice ding-a-ling model

The ding-a-ling model is a kind of half lattice and half hard-point-gas (HPG) model. The original ding-a-ling model proposed by Casati et al. does not conserve total momentum and has been found to exhibit normal heat conduction behavior. Recently, a modified ding-a-ling model which conserves total momentum has been studied and normal heat conduction has also been claimed. In this work, we propose a full-lattice ding-a-ling model without hard point collisions where total momentum is also conserved. We investigate the heat conduction and energy diffusion of this full-lattice ding-a-ling model with three different nonlinear inter-particle potential forms. For symmetrical potential lattices, the thermal conductivities diverges with lattice length and their energy diffusions are superdiffusive signaturing anomalous heat conduction. For asymmetrical potential lattices, although the thermal conductivity seems to converge as the length increases, the energy diffusion is definitely deviating from normal diffusion behavior indicating anomalous heat conduction as well. No normal heat conduction behavior can be found for the full-lattice ding-a-ling model.

Thermoelectricity of interacting particles: a numerical approach

A method for computing the thermopower in interacting systems is proposed. This approach, which relies on Monte Carlo simulations, is illustrated first for a diatomic chain of hard-point elastically colliding particles and then in the case of a one-dimensional gas with (screened) Coulomb interparticle interaction. Numerical simulations up to N>10^{4} particles confirm the general theoretical arguments for momentum-conserving systems and show that the thermoelectric figure of merit increases linearly with the system size.

Probing local equilibrium in nonequilibrium fluids

We use extensive computer simulations to probe local thermodynamic equilibrium (LTE) in a quintessential model fluid, the two-dimensional hard-disks system. We show that macroscopic LTE is a property much stronger than previously anticipated, even in the presence of important finite-size effects, revealing a remarkable bulk-boundary decoupling phenomenon in fluids out of equilibrium. This allows us to measure the fluid's equation of state in simulations far from equilibrium, with an excellent accuracy comparable to the best equilibrium simulations. Subtle corrections to LTE are found in the fluctuations of the total energy which strongly point to the nonlocality of the nonequilibrium potential governing the fluid's macroscopic behavior out of equilibrium.

Scaling laws and bulk-boundary decoupling in heat flow

When driven out of equilibrium by a temperature gradient, fluids respond by developing a nontrivial, inhomogeneous structure according to the governing macroscopic laws. Here we show that such structure obeys strikingly simple scaling laws arbitrarily far from equilibrium, provided that both macroscopic local equilibrium and Fourier's law hold. Extensive simulations of hard disk fluids confirm the scaling laws even under strong temperature gradients, implying that Fourier's law remains valid in this highly nonlinear regime, with putative corrections absorbed into a nonlinear conductivity functional. In addition, our results show that the scaling laws are robust in the presence of strong finite-size effects, hinting at a subtle bulk-boundary decoupling mechanism which enforces the macroscopic laws on the bulk of the finite-sized fluid. This allows one to measure the marginal anomaly of the heat conductivity predicted for hard disks.

Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain

Recent work has developed a nonlinear hydrodynamic fluctuation theory for a chain of coupled anharmonic oscillators governing the conserved fields, namely, stretch, momentum, and energy. The linear theory yields two propagating sound modes and one diffusing heat mode, all three with diffusive broadening. In contrast, the nonlinear theory predicts that, at long times, the sound mode correlations satisfy Kardar-Parisi-Zhang scaling, while the heat mode correlations have Lévy-walk scaling. In the present contribution we report on molecular dynamics simulations of Fermi-Pasta-Ulam chains to compute various spatiotemporal correlation functions and compare them with the predictions of the theory. We obtain very good agreement in many cases, but also some deviations.

Equilibrium time-correlation functions for one-dimensional hard-point systems

As recently proposed, the long-time behavior of equilibrium time-correlation functions for one-dimensional systems are expected to be captured by a nonlinear extension of fluctuating hydrodynamics. We outline the predictions from the theory aimed at the comparison with molecular dynamics. We report on numerical simulations of a fluid with a hard-shoulder potential and of a hard-point gas with alternating masses. These models have in common that the collision time is zero and their dynamics amounts to iterating collision by collision. The theory is well confirmed, with the twist that the nonuniversal coefficients are still changing at longest accessible times.

Thermal conductivity of molecular chains with asymmetric potentials of pair interactions

We provide molecular-dynamics simulation of heat transport in one-dimensional molecular chains with different interparticle pair potentials. We show that the thermal conductivity is finite in the thermodynamic limit in chains with the potentials that allow for bond dissociation. The Lennard-Jones, Morse, and Coulomb potentials are such potentials. The convergence of the thermal conductivity is provided by phonon scattering on the locally strongly stretched loose interatomic bonds at low temperature and by the many-particle scattering at high temperature. On the other hand, chains with a confining pair potential, which does not allow for bond dissociation, possess anomalous thermal conductivity, diverging with the chain length. We emphasize that chains with a symmetric or asymmetric Fermi-Pasta-Ulam potential or with combined potentials, containing a parabolic and/or a quartic confining potential, all exhibit anomalous heat transport.

Finite-size effects on current correlation functions

We study why the calculation of current correlation functions (CCFs) still suffers from finite-size effects even when the periodic boundary condition is taken. Two important one-dimensional, momentum-conserving systems are investigated as examples. Intriguingly, it is found that the state of a system recurs in the sense of microcanonical ensemble average, and such recurrence may result in oscillations in CCFs. Meanwhile, we find that the sound mode collisions induce an extra time decay in a current so that its correlation function decays faster (slower) in a smaller (larger) system. Based on these two unveiled mechanisms, a procedure for correctly evaluating the decay rate of a CCF is proposed, with which our analysis suggests that the global energy CCF decays as ∼ t(-2/3) in the diatomic hard-core gas model and in a manner close to ∼ t(-1/2) in the Fermi-Pasta-Ulam-β model.

Fourier's law from a chain of coupled planar harmonic oscillators under energy-conserving noise

We study the transport of heat along a chain of particles interacting through a harmonic potential and subject to heat reservoirs at its ends. Each particle has two degrees of freedom and is subject to a stochastic noise that produces infinitesimal changes in the velocity while keeping the kinetic energy unchanged. This is modeled by means of a Langevin equation with multiplicative noise. We show that the introduction of this energy-conserving stochastic noise leads to Fourier's law. By means of an approximate solution that becomes exact in the thermodynamic limit, we also show that the heat conductivity κ behaves as κ = aL/(b + λL) for large values of the intensity λ of the energy-conserving noise and large chain sizes L. Hence, we conclude that in the thermodynamic limit the heat conductivity is finite and given by κ = a/λ.

Mechanical control of heat conductivity in molecular chains

We discuss a possibility to control heat conductivity in molecular chains by means of external mechanical loads. To illustrate such possibilities we consider first well-studied one-dimensional chain with degenerate double-well potential of the nearest-neighbor interaction. We consider varying lengths of the chain with fixed number of particles. Number of possible energetically degenerate ground states strongly depends on the overall length of the chain, or, in other terms, on average length of the link between neighboring particles. These degenerate states correspond to mechanical equilibria; therefore, one can say that formation of such structures mimics a process of plastic deformation. We demonstrate that such modification of the chain length can lead to quite profound (almost fivefold) reduction of the heat conduction coefficient. Even more profound effect is revealed for a model with a single-well nonconvex potential. It is demonstrated that in a certain range of constant external forcing, this model becomes effectively double-well and has a multitude of possible states of equilibrium for fixed value of the external load. Due to this degeneracy, the heat-conduction coefficient can be reduced by two orders of magnitude. We suggest a mechanical model of a chain with periodic double-well potential, which allows control of the heat transport. The models considered may be useful for description of heat transfer in biological macromolecules and for control of the heat transport in microsystems. The possibility of the heat transport control in more realistic three-dimensional systems is illustrated by simulation of a three-dimensional model of polymer α-helix. In this model, the mechanical stretching also brings about the structural inhomogeneity and, in turn, to essential reduction of the heat conductivity.

Dissipation and entropy production in deterministic heat conduction of quasi-one-dimensional systems

We explore the consequences of a deterministic microscopic thermostat-reservoir contact mechanism. With different temperature reservoirs at each end of a two-dimensional system, a heat current is produced and the system has an anomalous thermal conductivity. The microscopic form for the local heat flux vector is derived and both the kinetic and potential contributions are calculated. The total heat flux vector is shown to satisfy the continuity equation. The properties of this nonequilibrium steady state are studied as functions of system size and temperature gradient, identifying key scaling relations for the local fluid properties and separating bulk and boundary effects. The local entropy density calculated from the local equilibrium distribution is shown to be a very good approximation to the entropy density calculated directly from the velocity distribution even for systems that are far from equilibrium. The dissipation and kinetic entropy production and flux are compared quantitatively and the differing mechanisms discussed within the Bhatnagar-Gross-Krook approximation. For equal-temperature reservoirs the entropy production near the reservoir walls is shown to be proportional to the local phase space contraction calculated from the tangent space dynamics. However, for unequal temperatures, the connection between local entropy production and local phase space contraction is more complicated.

Fourier's law from a chain of coupled anharmonic oscillators under energy-conserving noise

We analyze the transport of heat along a chain of particles interacting through anharmonic potentials consisting of quartic terms in addition to harmonic quadratic terms and subject to heat reservoirs at its ends. Each particle is also subject to an impulsive shot noise with exponentially distributed waiting times whose effect is to change the sign of its velocity, thus conserving the energy of the chain. We show that the introduction of this energy-conserving stochastic noise leads to Fourier's law. The behavior of the heat conductivity for small intensities of the shot noise and large system sizes is found to obey a finite-size scaling relation. We also show that the heat conductivity is not constant but is an increasing monotonic function of temperature.

Exact solution of a Lévy walk model for anomalous heat transport

The Lévy walk model is studied in the context of the anomalous heat conduction of one-dimensional systems. In this model, the heat carriers execute Lévy walks instead of normal diffusion as expected in systems where Fourier's law holds. Here we calculate exactly the average heat current, the large deviation function of its fluctuations, and the temperature profile of the Lévy walk model maintained in a steady state by contact with two heat baths (the open geometry). We find that the current is nonlocally connected to the temperature gradient. As observed in recent simulations of mechanical models, all the cumulants of the current fluctuations have the same system-size dependence in the open geometry. For the ring geometry, we argue that a size-dependent cutoff time is necessary for the Lévy walk model to behave like mechanical models. This modification does not affect the results on transport in the open geometry for large enough system sizes.

Three unequal masses on a ring and soft triangular billiards

The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the masses ratio between particles and softness ratio of the particles interaction. The transition from soft to hard interactions can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Toda-like interaction and the error function.

Exact results for anomalous transport in one-dimensional hamiltonian systems

Anomalous transport in one-dimensional translation invariant hamiltonian systems with short range interactions is shown to belong in general to the Kardar-Parisi-Zhang universality class. Exact asymptotic forms for density-density and current-current time correlation functions and their Fourier transforms are given in terms of the Prähofer-Spohn scaling functions, obtained from their exact solution for the polynuclear growth model. The exponents of corrections to scaling are found as well, but not so the coefficients. Mode coupling theories developed previously are found to be adequate for weakly nonlinear chains but in need of corrections for strongly anharmonic interparticle potentials. A simple condition is given under which Kardar-Parisi-Zhang behavior does not apply, sound attenuation is only logarithmically superdiffusive, and heat conduction is more strongly superdiffusive than under Kardar-Parisi-Zhang behavior.

Nonequilibrium stationary state of a harmonic crystal with alternating masses

We analyze the nonequilibrium steady states (NESS) of a one-dimensional harmonic chain of N atoms with alternating masses connected to heat reservoirs at unequal temperatures. We find that the temperature profile defined through the local kinetic energy T(j)≡<pj2>/mj oscillates with period two in the bulk of the system. Depending on boundary conditions, either the heavier or the lighter particles in the bulk are hotter. We obtain explicit integral expressions for the bulk temperature profile and steady state current in the limit N→∞. These depend on whether N is odd or even. We also study similar temperature oscillations in the NESS of systems with noise in the dynamics. These die out as N→∞.

Universality classes for phonon relaxation and thermal conduction in one-dimensional vibrational systems

We study phonon relaxation in chains of particles coupled through polynomial-type pair-interaction potentials and obeying quantum dynamics. We present detailed calculations for the sixth-order potential and find that the wave-vector-dependent relaxation rate follows a power-law behavior, Γ(q)∼q(δ), with δ=5/3, which is identical to that of the fourth-order potential. We argue through diagrammatic analysis that this is a generic feature of even-power potentials. Our earlier analysis has shown that δ=3/2 when the leading-order term in the nonlinear potential is odd, suggesting that there are two universality classes for the phonon relaxation rates dependent on a simple property of the potential. This implies that the thermal conductivity κ which diverges as a function of chain size N as κ∝N(α) also has two universal behaviors, in that α=1-1/δ as follows from a finite-size argument. We support these arguments by numerical calculations of conductivity for chains obeying classical dynamics for polynomial potentials of some even and odd powers.

Boltzmann equations for a binary one-dimensional ideal gas

We consider a time-reversal invariant dynamical model of a binary ideal gas of N molecules in one spatial dimension. By making time-asymmetric assumptions about the behavior of the gas, we derive Boltzmann and anti-Boltzmann equations that describe the evolution of the single-molecule velocity distribution functions for an ensemble of such systems. We show that for a special class of initial states of the ensemble one can obtain an exact expression for the N-molecule velocity distribution function, and we use this expression to rigorously prove that the time-asymmetric assumptions needed to derive the Boltzmann and anti-Boltzmann equations hold in the limit of large N. Our results clarify some subtle issues regarding the origin of the time asymmetry of Boltzmann's H theorem.

Separation of trajectories and its relation to entropy for intermittent systems with a zero Lyapunov exponent

One-dimensional intermittent maps with stretched exponential δx(t)∼δx(0)e(λ(α)t(α)) separation of nearby trajectories are considered. When t→∞ the standard Lyapunov exponent λ=∑(i=0)(t-1)ln|M'(x(i))|/t is zero (M' is a Jacobian of the map). We investigate the distribution of λ(α)=∑(i=0)(t-1)ln|M'(x(i))|/t(α), where α is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of λ(α) is determined by the infinite invariant density. Using semianalytical arguments we calculate the infinite invariant density for the Pomeau-Manneville map, and with it we obtain excellent agreement between numerical simulation and theory. We show that α(λ(α)) is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that (λ(α)) and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.

Different thermal conductance of the inter- and intrachain interactions in a double-stranded molecular structure

A double-stranded system, modeled by a Frenkel-Kontorova lattice, is studied through nonequilibrium molecular dynamics simulations. We have investigated the thermal conductance influenced by the intrachain interaction as well as by the interchain interaction. It is found that the intrachain interaction always enhance the thermal conductance. The interchain interaction, however, has a positive effect on the thermal conductance in the case of strong nonlinear potential, and has a negative effect on the thermal conductance in the case of weak nonlinear potential. This phenomenon can be explained by the transition of thermal transport mode and the phonon band shift of the particles. It is suggested that the inter- and intrachain interactions present different thermal properties in double-stranded lattices.

Crystallization of hard disks induced by a temperature gradient

While uniform temperature has no effect on equilibrium properties of hard-core systems, its gradient might substantially change their behavior. In particular, in hard-disk system subject to temperature difference ΔT disks are repelled from the hot boundary of the system and accumulate at the cold one. Using event-driven molecular dynamics simulations we show that for sufficiently large ΔT or coverage ratio ρ∗, crystal forms at the cold boundary. In this spatially inhomogeneous system a significant decrease of diffusivity of disks clearly marks the stationary interface between liquid and crystal. Such a behavior is also supported through calculation of the radial distribution function and the bond order parameter. Simulations show that for this nonequilibrium system the equipartition of energy holds and velocity obeys the Boltzmann distribution.

Heat conduction and reversed thermal diode: the interface effect

The important role of interface collisions on the thermal-diode effect, of the two-segment lattices is studied. In the high-average temperature region, it is found that the thermal-diode effect may be significantly weaken and even annihilated. In the low-temperature region, where the thermal diode is inhibited in the collisionless case, an interesting reversed thermal diode is achieved. These behaviors are interpreted in terms of phonon-band mixing induced by interface collisions. The regime where a reversed thermal diode can be observed by resorting to the dependence of the heat current on the average temperature, and a critical temperature exists. The results proposed in this paper reveal that thermal-diode effect can be qualitatively influenced if the interface collisions could not be neglected.

Continuum and lattice heat currents for oscillator chains

We show that two commonly used definitions for the heat current give different results-through the Kubo formula-for the heat conductivity of oscillator chains. The difference exists for finite chains, and is expected to be important more generally for small structures. For a chain of N particles that are tethered at the ends, the ratio of the heat conductivities calculated with the two currents differs from unity by O(1/N). For a chain held at constant pressure, the difference from unity decays more slowly, and is consistent with O(1/Neta) with 1>eta>0.5.

Pesin-type identity for intermittent dynamics with a zero Lyaponov exponent

Pesin's identity provides a profound connection between the Kolmogorov-Sinai entropy h_{KS} and the Lyapunov exponent lambda. It is well known that many systems exhibit subexponential separation of nearby trajectories and then lambda=0. In many cases such systems are nonergodic and do not obey usual statistical mechanics. Here we investigate the nonergodic phase of the Pomeau-Manneville map where separation of nearby trajectories follows deltax_{t}=deltax_{0}e;{lambda_{alpha}t;{alpha}} with 0<alpha<1. The limit distribution of lambda_{alpha} is the inverse Lévy function. The average lambda_{alpha} is related to the infinite invariant density, and most importantly to entropy. Our work gives a generalized Pesin's identity valid for systems with an infinite invariant density.

Heat conduction in the disordered Fermi-Pasta-Ulam chain

We address the question of the effect of disorder on heat conduction in an anharmonic chain with interactions given by the Fermi-Pasta-Ulam (FPU) potential. In contrast to the conclusions of an earlier paper [Phys. Rev. Lett. 86, 63 (2001)], which found that disorder could induce a finite thermal conductivity at low temperatures, we find no evidence of a finite-temperature transition in conducting properties. Instead, we find that at low temperatures, small system-size transport properties are dominated by disorder but the asymptotic system size dependence of current is given by the usual FPU result J approximately 1N;{23} . We also present interesting results on the binary-mass ordered FPU chain.

Universality of thermal conduction in vibrating chains for a class of potentials

We present calculations for the relaxation rates of phonons in one-dimensional chains in which atoms interact with a class of pairwise potentials which are anharmonic with odd powers. The calculations are based on a self-consistent procedure for second order processes and lead to integral equations for the wave-vector-dependent on-shell relaxation rate Gammaq for phonons. For the cubic anharmonicity, one finds that for small q, Gammaq proportional, q3/2. The self-consistent procedure is extended to potentials with higher odd powers and one finds that the leading order behavior is still Gammaq proportional, q3/2+O(q2). With the assumption that the transport relaxation rate has the same wave-vector dependence, this result implies that the thermal conductivity, kappa diverges with the chain size, N, as kappa proportional, N1/3 for this class of potentials. Thus, our calculations provide a microscopic basis for one class of universal behavior.

Origin of chaos in soft interactions and signatures of nonergodicity

The emergence of chaotic motion is discussed for hard-point like and soft collisions between two particles in a one-dimensional box. It is known that ergodicity may be obtained in hard-point like collisions for specific mass ratios gamma=m(2)/m(1) of the two particles and that Lyapunov exponents are zero. However, if a Yukawa interaction between the particles is introduced, we show analytically that positive Lyapunov exponents are generated due to double collisions close to the walls. While the largest finite-time Lyapunov exponent changes smoothly with gamma , the number of occurrences of the most probable one, extracted from the distribution of finite-time Lyapunov exponents over initial conditions, reveals details about the phase-space dynamics. In particular, the influence of the integrable and pseudointegrable dynamics without Yukawa interaction for specific mass ratios can be clearly identified and demonstrates the sensitivity of the finite-time Lyapunov exponents as a phase-space probe. Being not restricted to two-dimensional problems such as Poincaré sections, the number of occurrences of the most probable Lyapunov exponents suggests itself as a suitable tool to characterize phase-space dynamics in higher dimensions. This is shown for the problem of two interacting particles in a circular billiard.

Dynamic screening in a two-species asymmetric exclusion process

The dynamic scaling properties of the one-dimensional Burgers equation are expected to change with the inclusion of additional conserved degrees of freedom. We study this by means of one-dimensional (1D) driven lattice gas models that conserve both mass and momentum. The most elementary version of this is the Arndt-Heinzel-Rittenberg (AHR) process, which is usually presented as a two-species diffusion process, with particles of opposite charge hopping in opposite directions and with a variable passing probability. From the hydrodynamics perspective this can be viewed as two coupled Burgers equations, with the number of positive and negative momentum quanta individually conserved. We determine the dynamic scaling dimension of the AHR process from the time evolution of the two-point correlation functions, and find numerically that the dynamic critical exponent is consistent with simple Kardar-Parisi-Zhang- (KPZ) type scaling. We establish that this is the result of perfect screening of fluctuations in the stationary state. The two-point correlations decay exponentially in our simulations and in such a manner that in terms of quasiparticles, fluctuations fully screen each other at coarse grained length scales. We prove this screening rigorously using the analytic matrix product structure of the stationary state. The proof suggests the existence of a topological invariant. The process remains in the KPZ universality class but only in the sense of a factorization, as (KPZ)2. The two Burgers equations decouple at large length scales due to the perfect screening.