Transport Phenomena at a Critical Point -- Thermal Conduction in the Creutz Cellular Automaton --

Nature of energy transport around a critical point is studied in the Creutz cellular automaton. Fourier heat law is confirmed to hold in this model by a direct measurement of heat flow under a temperature gradient. The thermal conductivity is carefully investigated near the phase transition by the use of the Kubo formula. As the result, the thermal conductivity is found to take a finite value at the critical point contrary to some previous works. Equal-time correlation of the heat flow is also analyzed by a mean-field type approximation to investigate the temperature dependence of thermal conductivity. A variant of the Creutz cellular automaton called the Q2R is also investigated and similar results are obtained.Comment: 27 pages including 14figure

Heat conduction in one dimensional nonintegrable systems

Two classes of 1D nonintegrable systems represented by the Fermi-Pasta-Ulam (FPU) model and the discrete $\phi^4$ model are studied to seek a generic mechanism of energy transport in microscopic level sustaining macroscopic behaviors. The results enable us to understand why the class represented by the $\phi^4$ model has a normal thermal conductivity and the class represented by the FPU model does not even though the temperature gradient can be established.Comment: 4 Revtex Pages, 4 Eps figures included, to appear in Phys. Rev. E, March 200

A Symmetry Property of Momentum Distribution Functions in the Nonequilibrium Steady State of Lattice Thermal Conduction

We study a symmetry property of momentum distribution functions in the steady state of heat conduction. When the equation of motion is symmetric under change of signs for all dynamical variables, the distribution function is also symmetric. This symmetry can be broken by introduction of an asymmetric term in the interaction potential or the on-site potential, or employing the thermal walls as heat reservoirs. We numerically find differences of behavior of the models with and without the on-site potential.Comment: 13 pages. submitted to JPS

Can disorder induce a finite thermal conductivity in 1D lattices?

We study heat conduction in one dimensional mass disordered harmonic and anharmonic lattices. It is found that the thermal conductivity $\kappa$ of the disordered anharmonic lattice is finite at low temperature, whereas it diverges as $\kappa \sim N^{0.43}$ at high temperature. Moreover, we demonstrate that a unique nonequilibrium stationary state in the disordered harmonic lattice does not exist at all.Comment: 4 pages with 4 eps figure

Finite thermal conductivity in 1D models having zero Lyapunov exponents

Heat conduction in three types of 1D channels are studied. The channels consist of two parallel walls, right triangles as scattering obstacles, and noninteracting particles. The triangles are placed along the walls in three different ways: (a) periodic, (b) disordered in height, and (c) disordered in position. The Lyapunov exponents in all three models are zero because of the flatness of triangle sides. It is found numerically that the temperature gradient can be formed in all three channels, but the Fourier heat law is observed only in two disordered ones. The results show that there might be no direct connection between chaos (in the sense of positive Lyapunov exponent) and the normal thermal conduction.Comment: 4 PRL page

Dynamics of oscillator chains

The Fermi\u2013Pasta\u2013Ulam (FPU) nonlinear oscillator chain has proved to be a seminal system for investigating problems in nonlinear dynamics. First proposed as a nonlinear system to elucidate the foundations of statistical mechanics, the initial lack of confirmation of the researchers expectations eventually led to a number of profound insights into the behavior of high-dimensional nonlinear systems. The initial numerical studies, proposed to demonstrate that energy placed in a single mode of the linearized chain would approach equipartition through nonlinear interactions, surprisingly showed recurrences. Although subsequent work showed that the origin of the recurrences is nonlinear resonance, the question of lack of equipartition remained. The attempt to understand the regularity bore fruit in a profound development in nonlinear dynamics: the birth of soliton theory. A parallel development, related to numerical observations that, at higher energies, equipartition among modes could be approached, was the understanding that the transition with increasing energy is due to resonance overlap. Further numerical investigations showed that time-scales were also important, with a transition between faster and slower evolution. This was explained in terms of mode overlap at higher energy and resonance overlap at lower energy. Numerical limitations to observing a very slow approach to equipartition and the problem of connecting high-dimensional Hamiltonian systems to lower dimensional studies of Arnold diffusion, which indicate transitions from exponentially slow diffusion along resonances to power-law diffusion across resonances, have been considered. Most of the work, both numerical and theoretical, started from low frequency (long wavelength) initial conditions. Coincident with developments to understand equipartition was another program to connect a statistical phenomenon to nonlinear dynamics, that of understanding classical heat conduction. The numerical studies were quite different, involving the excitation of a boundary oscillator with chaotic motion, rather than the excitation of the entire chain with regular motion. Although energy transitions are still important, the inability to reproduce exactly the law of classical heat conduction led to concern for the generiticity of the FPU chain and exploration of other force laws. Important concepts of unequal masses, and \u201canti-integrability,\u201d i.e. isolation of some oscillators, were considered, as well as separated optical and acoustic modes that could only communicate through very weak interactions. The importance of chains that do not allow nonlinear wave propagation in producing the Fourier heat conduction law is now recognized. A more recent development has been the exploration of energy placed on the FPU or related oscillator chains in high-frequency (short wavelength) modes and the existence of isolated structures (breathers). Breathers are found as solutions to partial differential equations, analogous to solitons at lower frequency. On oscillator chains, such as the FPU, energy initially in a single high-frequency mode is found, at higher energies, to self-organize in oscillator space to form compact structures. These structures are \u201cchaotic breathers,\u201d i.e. not completely stable, and disintegrate on longer time-scales. With the significant progress in understanding this evolution, we now have a rather complete picture of the nonlinear dynamics of the FPU and related oscillator chains, and their relation to a wide range of concepts in nonlinear dynamics. This chapter\u2019s purpose is to explicate these many concepts. After a historical perspective the basic chaos theory background is reviewed. Types of oscillators, numerical methods, and some analytical results are considered. Numerical results of studies of equipartition, both from low-frequency and high-frequency modes, are presented, together with numerical studies of heat conduction. These numerical studies are related to analytical calculations and estimates of energy transitions and time-scales to equipartition