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

Anomalous thermal conduction in one dimension: a quantum calculation

In this paper, we study the thermal conductivity of an anharmonically coupled chain of atoms. Numerical studies using classical dynamics have shown that the conductivity of a chain with nearest neighbor couplings diverges with chain length L as L(alpha); earlier studies found alpha approximately = 0.4 under a range of conditions, but a recent study on longer chains claims alpha = 1/3. Analytically, this problem has been studied by calculating the relaxation rate gamma(q) of the normal modes of vibration as a function of its wave vector q. Two theoretical studies of classical chains, one using the mode-coupling formulation and the other the Boltzmann equation method, led to gamma(q) proportional to q(5/3), which is consistent with alpha = 0.4. Here we study the problem for a quantum anharmonic chain with quartic anisotropy. We develop a low-temperature expansion for gamma(q) and find that, in the regime Dirac's constant omega(q) << k(B)T, gamma(q) is proportional to q(5/3)T2, where omega(q) is the frequency of the mode. In our analysis, the relaxation arises due to umklapp scattering processes. We further evaluate the thermal conductivity of the chain using the Kubo formula, which enables us to take into account the transport relaxation time through vertex corrections for the current-current correlator. This calculation also yields alpha = 0.4.

Effect of interaction range on phonon relaxation in Fermi-Pasta-Ulam beta chain

We study the effect of increasing the range of interactions on phonon relaxation in a chain of atoms with quartic anharmonicity. The study is motivated by recent numerical studies, showing that the value of the exponent alpha characterizing the divergence of conductivity with system size apparently depends on the presence of second neighbor couplings. We perform a quantum calculation of the wave-vector (q) dependent relaxation rate gamma(q) in the second order perturbation theory. The nonanalytic dependence of gamma(q) arises due to small-q singularity of the collision integral. We find that gamma(q) proportional to Aq(5/3) + Bq2. This gives rise to an asymptotic value alpha = 0.4, but the q2 terms lead to a higher apparent value of alpha at small sizes of the chain.

Equilibration and universal heat conduction in fermi-pasta-ulam chains

It is shown numerically that for Fermi-Pasta-Ulam (FPU) chains with alternating masses and heat baths at slightly different temperatures at the ends, the local temperature (LT) on small scales behaves paradoxically in steady state. This expands the long established problem of equilibration of FPU chains. A well-behaved LT appears to be achieved for equal mass chains; the thermal conductivity is shown to diverge with chain length N as N(1/3), relevant for the much debated question of the universality of one-dimensional heat conduction. The reason why earlier simulations have obtained systematically higher exponents is explained.

Heat conduction and diffusion of hard disks in a narrow channel

Using molecular dynamics we study heat conduction and diffusion of hard disks in one-dimensional narrow channels. When collisions preserve momentum the heat conduction kappa diverges with the number of disks N as kappa approximately N alpha (alpha approximately 1/3) . Such a behavior is seen both when the ordering of disks is fixed ("pen-case" model), and when they can exchange their positions. Momentum conservation results also in sound-wave effects that enhance diffusive behavior and on an intermediate time scale (that diverges in the thermodynamic limit) normal diffusion takes place even in the "pen-case" model. When collisions do not preserve momentum, kappa remains finite and sound-wave effects are absent.

Nonequilibrium properties of the one-dimensional hard-point gas system

We discuss the stability properties of a one-dimensional hard-point gas. We study the decay of the Loschmidt echo which describes the stability of the motion under system perturbations. We show a universal behavior in the echo decay which is intimately connected to the linear dynamical instability of the motion. In particular, in spite of such a weak instability, the asymptotic decay follows a simple exponential law.

Universality of one-dimensional heat conductivity

We show analytically that the heat conductivity of oscillator chains diverges with system size N as N(1/3), which is the same as for one-dimensional fluids. For long cylinders, we use the hydrodynamic equations for a crystal in one dimension. This is appropriate for stiff systems such as nanotubes, where the eventual crossover to a fluid only sets in at unrealistically large . Despite the extra equation compared to a fluid, the scaling of the heat conductivity is unchanged. For strictly one-dimensional chains, we show that the dynamic equations are those of a fluid at all length scales even if the static order extends to very large . The discrepancy between our results and numerical simulations on Fermi-Pasta-Ulam chains is discussed.

Anomalous heat conductivity induced by finite size and non-Markovian dynamics

Heat conduction in a one-dimensional non-Markovian damping channel between two heat baths separated by a finite distance is studied numerically. It is found that the Fourier heat law is not obeyed for a finite-size underdamped channel under a Gaussian white noise and the coefficient of heat conductivity is a nonmonotonic function of the channel length in the sub-Ohmic damping case. The key dynamic feature is that the system does not approach the stationary state when it arrives at the cold bath for the former, and the system exhibits different diffusive behaviors from ballistic diffusion to subdiffusion at initial and asymptotic periods of time for the latter. We evaluate a damping-dependent critical separation size between two heat baths above which the heat conductivity becomes independent of the separation.

Breakdown of hydrodynamics in a simple one-dimensional fluid

We investigate the behavior of a one-dimensional diatomic fluid under a shock wave excitation. We find that the properties of the resulting shock wave are in striking contrast with those predicted by hydrodynamic and kinetic approaches; e.g., the hydrodynamic profiles relax algebraically toward their equilibrium values. Deviations from local thermodynamic equilibrium are persistent, decaying as a power law of the distance to the shock layer. Nonequipartition is observed infinitely far from the shock wave, and the velocity-distribution moments exhibit multiscaling. These results question the validity of simple hydrodynamic theories to understand collective behavior in 1D fluids.

Simplest piston problem. I. Elastic collisions

We study the dynamics of three elastic particles in a finite interval where two light particles are separated by a heavy "piston." The piston undergoes surprisingly complex motion that is oscillatory at short time scales but seemingly chaotic at longer scales. The piston also makes long-duration excursions close to the ends of the interval that stem from the breakdown of energy equipartition. Many of these dynamical features can be understood by mapping the motion of three particles on the line onto the trajectory of an elastic billiard in a highly skewed tetrahedral region. We exploit this picture to construct a qualitative random walk argument that predicts a power-law tail, with exponent -3/2, for the distribution of time intervals between successive piston crossings of the interval midpoint. These predictions are verified by numerical simulations.

Analytic calculation of energy transfer and heat flux in a one-dimensional system

In the context of the problem of heat conduction in one-dimensional systems, we present an analytical calculation of the instantaneous energy transfer across a tagged particle in a one-dimensional gas of equal-mass, hard-point particles. From this, we obtain a formula for the steady-state energy flux, and identify and separate the mechanical work and heat conduction contributions to it. The nature of the Fourier law for the model, and the nonlinear dependence of the rate of mechanical work on the stationary drift velocity of the tagged particle, are analyzed and elucidated.

Strong shock waves and nonequilibrium response in a one-dimensional gas: a Boltzmann equation approach

We investigate the nonequilibrium behavior of a one-dimensional binary fluid on the basis of Boltzmann equation, using an infinitely strong shock wave as probe. Density, velocity, and temperature profiles are obtained as a function of the mixture mass ratio mu. We show that temperature overshoots near the shock layer, and that heavy particles are denser, slower, and cooler than light particles in the strong nonequilibrium region around the shock. The shock width omega(mu), which characterizes the size of this region, decreases as omega(mu) approximately mu(1/3) for mu-->0. In this limit, two very different length scales control the fluid structure, with heavy particles equilibrating much faster than light ones. Hydrodynamic fields relax exponentially toward equilibrium: phi(chi) approximately exp[-chi/lambda]. The scale separation is also apparent here, with two typical scales, lambda1 and lambda2, such that lambda1 approximately mu(1/2 as mu-->0, while lambda2, which is the slow scale controlling the fluid's asymptotic relaxation, increases to a constant value in this limit. These results are discussed in light of recent numerical studies on the nonequilibrium behavior of similar one-dimensional binary fluids.

Thermal conductivity and bulk viscosity in quartic oscillator chains

We propose a relation which predicts the low-frequency thermal conductivity of a one-dimensional (1D) system from the thermal conductivity and bulk viscosity at higher frequency. Our theory is based on the assumption that "ballistic" transport by sound waves dominates the heat transport. For a system with equal heat capacities (c(p) = c(v)) this relation is particularly simple. We test the prediction by simulating a chain of particles with quartic interparticle potentials under zero pressure conditions. As the frequency omega --> 0 the theory predicts that the energy current power spectrum diverges as omega(-1/2), not seen in previous simulations. Because we simulate very long chains to long times we do observe the crossover into this regime. The bulk viscosity of a 1D chain has been determined via simulation. It is found to be finite for our system, in contrast to the thermal conductivity which is infinite.

Heat conduction in a two-dimensional harmonic crystal with disorder

We study the problem of heat conduction in a mass-disordered two-dimensional harmonic crystal. Using two different stochastic heat baths, we perform simulations to determine the system size (L) dependence of the heat current (J). For white noise heat baths we find that J approximately 1/L(alpha) with alpha approximately equal to 0.59, while correlated noise heat baths give alpha approximately equal to 0.51. A special case with correlated disorder is studied analytically and gives alpha=3/2, which agrees also with results from exact numerics.

Brownian motion and diffusion: from stochastic processes to chaos and beyond

One century after Einstein's work, Brownian motion still remains both a fundamental open issue and a continuous source of inspiration for many areas of natural sciences. We first present a discussion about stochastic and deterministic approaches proposed in the literature to model the Brownian motion and more general diffusive behaviors. Then, we focus on the problems concerning the determination of the microscopic nature of diffusion by means of data analysis. Finally, we discuss the general conditions required for the onset of large scale diffusive motion.

Heat conduction in the Frenkel-Kontorova model

Heat conduction is an old yet important problem. Since Fourier introduced the law bearing his name almost 200 years ago, a first-principle derivation of this simple law from statistical mechanics is still lacking. Worse still, the validity of this law in low dimensions, and the necessary and sufficient conditions for its validity are far from clear. In this paper we will review recent works on heat conduction in a simple nonintegrable model called the Frenkel-Kontorova model. The thermal conductivity of this model has been found to be finite. We will study the dependence of the thermal conductivity on the temperature and other parameters of the model such as the strength and the periodicity of the external potential. We will also discuss other related problems such as phase transitions and finite-size effects. The study of heat conduction is not only of theoretical interest but also of practical interest. We will show various recent designs of thermal rectifiers and thermal diodes by coupling nonlinear chains together. The study of heat conduction in low dimensions is also important to the understanding of the thermal properties of carbon nanotubes.

Studies of thermal conductivity in Fermi-Pasta-Ulam-like lattices

The pioneering computer simulations of the energy relaxation mechanisms performed by Fermi, Pasta, and Ulam (FPU) can be considered as the first attempt of understanding energy relaxation and thus heat conduction in lattices of nonlinear oscillators. In this paper we describe the most recent achievements about the divergence of heat conductivity with the system size in one-dimensional (1D) and two-dimensional FPU-like lattices. The anomalous behavior is particularly evident at low energies, where it is enhanced by the quasiharmonic character of the lattice dynamics. Remarkably, anomalies persist also in the strongly chaotic region where long-time tails develop in the current autocorrelation function. A modal analysis of the 1D case is also presented in order to gain further insight about the role played by boundary conditions.

Normal and anomalous heat transport in one-dimensional classical lattices

We present analytic and numerical results on several models of one-dimensional (1D) classical lattices with the goal of determining the origins of anomalous heat transport and the conditions for normal transport in these systems. Some of the recent results in the literature are reviewed and several original "toy" models are added that provide key elements to determine which dynamical properties are necessary and which are sufficient for certain types of heat transport. We demonstrate with numerical examples that chaos in the sense of positivity of Lyapunov exponents is neither necessary nor sufficient to guarantee normal transport in 1D lattices. Quite surprisingly, we find that in the absence of momentum conservation, even ergodicity of an isolated system is not necessary for the normal transport. Specifically, we demonstrate clearly the validity of the Fourier law in a pseudo-integrable particle chain.

Controlling the heat flow: now it is possible

We discuss the problem of heat conduction in 1D nonlinear chains in relation to the dynamical properties of the system. We provide convincing numerical evidence for the validity of Fourier law of heat conduction in linear mixing systems. Therefore, deterministic diffusion and normal heat transport which are usually associated with full hyperbolicity, actually take place in systems without exponential instability. We then show that, acting on the parameter which controls the strength of the on site potential inside a segment of the chain, we induce a transition from conducting to insulating behavior in the whole system. The control of heat conduction by nonlinearity opens the possibility to propose new devices such as a thermal rectifier.

Anomalous heat conduction and anomalous diffusion in nonlinear lattices, single walled nanotubes, and billiard gas channels

We study anomalous heat conduction and anomalous diffusion in low-dimensional systems ranging from nonlinear lattices, single walled carbon nanotubes, to billiard gas channels. We find that in all discussed systems, the anomalous heat conductivity can be connected with the anomalous diffusion, namely, if energy diffusion is sigma(2)(t)=2Dt(alpha) (0<alpha< or =2), then the thermal conductivity can be expressed in terms of the system size L as kappa=cL(beta) with beta=2-2/alpha. This result predicts that a normal diffusion (alpha=1) implies a normal heat conduction obeying the Fourier law (beta=0), a superdiffusion (alpha>1) implies an anomalous heat conduction with a divergent thermal conductivity (beta>0), and more interestingly, a subdiffusion (alpha<1) implies an anomalous heat conduction with a convergent thermal conductivity (beta<0), consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our theoretical prediction.

Mode-coupling theory and molecular dynamics simulation for heat conduction in a chain with transverse motions

We study heat conduction in a 1D chain of particles with longitudinal as well as transverse motions. The particles are connected by 2D harmonic springs together with bending angle interactions. The problem is analyzed by mode-coupling theory and compared with molecular dynamics. We find very good, quantitative agreement for the damping of modes between a full mode-coupling theory and molecular dynamics result, and a simplified mode-coupling theory gives qualitative description of the damping. The theories predict generically that thermal conductance diverges as N(1/3) as the size N increases for systems terminated with heat baths at the ends. The N(2/5) dependence is also observed in molecular dynamics, which we attribute to crossover effect.

Intramolecular vibrational energy redistribution in bridged azulene-anthracene compounds: Ballistic energy transport through molecular chains

Intramolecular vibrational energy flow in excited bridged azulene-anthracene compounds is investigated by time-resolved pump-probe laser spectroscopy. The bridges consist of molecular chains and are of the type (CH(2))(m) with m up to 6 as well as (CH(2)OCH(2))(n) (n=1,2) and CH(2)SCH(2). After light absorption into the azulene S(1) band and subsequent fast internal conversion, excited molecules are formed where the vibrational energy is localized at the azulene side. The vibrational energy transfer through the molecular bridge to the anthracene side and, finally, to the surrounding medium is followed by probing the red edge of the azulene S(3) absorption band at 300 nm and/or the anthracene S(1) absorption band at 400 nm. In order to separate the time scales for intramolecular and intermolecular energy transfer, most of the experiments were performed in supercritical xenon where vibrational energy transfer to the bath is comparably slow. The intramolecular equilibration proceeds in two steps. About 15%-20% of the excitation energy leaves the azulene side within a short period of 300 fs. This component accompanies the intramolecular vibrational energy redistribution (IVR) within the azulene chromophore and it is caused by dephasing of normal modes contributing to the initial local excitation of the azulene side and extending over large parts of the molecule. Later, IVR in the whole molecule takes place transferring vibrational energy from the azulene through the bridge to the anthracene side and thereby leading to microcanonical equilibrium. The corresponding time constants tau(IVR) for short bridges increase with the chain length. For longer bridges consisting of more than three elements, however, tau(IVR) is constant at around 4-5 ps. Comparison with molecular dynamics simulations suggests that the coupling of these chains to the two chromophores limits the rate of intramolecular vibrational energy transfer. Inside the bridges the energy transport is essentially ballistic and, therefore, tau(IVR) is independent on the length.

Fourier law in the alternate-mass hard-core potential chain

We study energy transport in a one-dimensional model of elastically colliding particles with alternate masses m and M. In order to prevent total momentum conservation, we confine particles with mass M inside a cell of finite size. We provide convincing numerical evidence for the validity of Fourier law of heat conduction in spite of the lack of exponential dynamical instability. Comparison with previous results on similar models shows the relevance of the role played by total momentum conservation.

Intriguing heat conduction of a chain with transverse motions

We study heat conduction in a one-dimensional chain of particles with longitudinal as well as transverse motions. The particles are connected by two-dimensional harmonic springs together with bending angle interactions. Using equilibrium and nonequilibrium molecular dynamics, three types of thermal conducting behaviors are found: a logarithmic divergence with system sizes for large transverse coupling, 1/3 power law at intermediate coupling, and 2/5 power law at low temperatures and weak coupling. The results are consistent with a simple mode-coupling analysis of the same model. We suggest that the 1/3 power-law divergence should be a generic feature for models with transverse motions.

Heat conduction in a one-dimensional chain of hard disks with substrate potential

Heat conduction in a one-dimensional chain of equivalent rigid particles in the field of the external on-site potential is considered. The zero diameters of the particles correspond to the integrable case with the divergent heat conduction coefficient. By means of a simple analytical model it is demonstrated that for any nonzero particle size the integrability is violated and the heat conduction coefficient converges. The result of the analytical computation is verified by means of numerical simulation in a plausible diapason of parameters, and good agreement is observed.

Universality of anomalous one-dimensional heat conductivity

In one and two dimensions, transport coefficients may diverge in the thermodynamic limit due to long-time correlation of the corresponding currents. The effective asymptotic behavior is addressed with reference to the problem of heat transport in one-dimensional crystals, modeled by chains of classical nonlinear oscillators. Extensive accurate equilibrium and nonequilibrium numerical simulations confirm that the finite-size thermal conductivity diverges with system size L as kappa proportional to L alpha. However, the exponent alpha deviates systematically from the theoretical prediction alpha=1/3 proposed in a recent paper [O. Narayan and S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002)].

Lyapunov exponent of ion motion in microplasmas

Dynamical chaos is studied in the Hamiltonian motion of ions confined in a Penning trap and forming so-called microplasmas. The dynamical chaos of the ion motion is characterized by the maximum Lyapunov exponent. Results are reported on the dependence of this exponent on the energy of the system, on the number of ions, as well as on the geometry of the trap. Different dynamical regimes are characterized from the crystalline state to a strongly chaotic regime, and to quasiharmonic motion in the external potential of the trap. Across these regimes, the Lyapunov exponent increases, reaches a maximum value, and decreases as a function of energy. Besides, the maximum value of the Lyapunov exponent increases as a function of the number of ions.

Correlations and scaling in one-dimensional heat conduction

We examine numerically the full spatiotemporal correlation functions for all hydrodynamic quantities for the random collision model introduced recently. The autocorrelation function of the heat current, through the Kubo formula, gives a thermal conductivity exponent of 1/3 in agreement with the analytical prediction and previous numerical work. Remarkably, this result depends crucially on the choice of boundary conditions: for periodic boundary conditions (as opposed to open boundary conditions with heat baths) the exponent is approximately 1/2. All primitive hydrodynamic quantities scale with the dynamic critical exponent predicted analytically.

Anomalous heat conduction and anomalous diffusion in one-dimensional systems

We establish a connection between anomalous heat conduction and anomalous diffusion in one-dimensional systems. It is shown that if the mean square of the displacement of the particle is <Deltax(2)>=2Dt(alpha)(0<alpha</=2), then the thermal conductivity can be expressed in terms of the system size L as kappa=cL(beta) with beta=2-2/alpha. This result predicts that normal diffusion (alpha=1) implies normal heat conduction obeying the Fourier law (beta=0) and that superdiffusion (alpha>1) implies anomalous heat conduction with a divergent thermal conductivity (beta>0). More interestingly, subdiffusion (alpha<1) implies anomalous heat conduction with a convergent thermal conductivity (beta<0), and, consequently, the system is a thermal insulator in the thermodynamic limit. Existing numerical data support our results.

One-dimensional heat conductivity exponent from a random collision model

We obtain numerically the thermal conductivity of a quasi-one-dimensional classical chain of hard sphere particles as a function of the length of the chain, introducing a fresh model for this problem. The conductivity scales as a power law of the length over two decades, with an exponent very close to the analytical prediction of 1/3.

Heat conduction in one-dimensional lattices with on-site potential

The process of heat conduction in one-dimensional lattices with on-site potential is studied by means of numerical simulation. Using the discrete Frenkel-Kontorova, phi(4), and sinh-Gordon models we demonstrate that contrary to previously expressed opinions the sole anharmonicity of the on-site potential is insufficient to ensure the normal heat conductivity in these systems. The character of the heat conduction is determined by the spectrum of nonlinear excitations peculiar for every given model and therefore depends on the concrete potential shape and the temperature of the lattice. The reason is that the peculiarities of the nonlinear excitations and their interactions prescribe the energy scattering mechanism in each model. For sine-Gordon and phi(4) models, phonons are scattered at a dynamical lattice of topological solitons; for sinh-Gordon and for phi(4) in a different parameter regime the phonons are scattered at localized high-frequency breathers (in the case of phi(4) the scattering mechanism switches with the growth of the temperature).

Heat conductivity in linear mixing systems

We present analytical and numerical results on the heat conduction in a linear mixing system. In particular we consider a quasi-one-dimensional channel with triangular scatterers with internal angles which are irrational multiples of pi, and we show that the system obeys the Fourier law of heat conduction. Therefore, deterministic diffusion and normal heat transport which are usually associated with full hyperbolicity, actually take place in systems without exponential instability.

Anomalous heat conduction in one-dimensional momentum-conserving systems

We show that for one-dimensional fluids the thermal conductivity generically diverges with system size L as L(1/3), as a result of momentum conservation. Our results are consistent with the largest-scale numerical studies of two-component hard-particle systems. We suggest explanations for the apparent disagreement with studies on Fermi-Pasta-Ulam chains.