Bulk Properties of Anharmonic Chains in Strong Thermal Gradients: Non-Equilibrium φ Theory

We study nonequilibrium properties of a one-dimensional lattice Hamiltonian with quartic interactions in strong thermal gradients. Nonequilibrium temperature profiles, T (x), are found to develop significant curvature and boundary jumps. From the determination of the bulk thermal conductivity, we develop a quantitative description of T (x) including the jumps. PACs numbers: 05.70.Ln, 05.60.-k, 44.10.+i, 02.70.Ns keywords: Non-equilibrium steady state, thermal conductivity, long-time tails, Green-Kubo, anharmonic chains, Fourier’s law. ∗ E–mail: [email protected] E–mail: [email protected] 1 The description of transport in physical systems is usually relegated to the near equilibrium regime, where Green-Kubo theory can be used. When one strays from this into systems far from equilibrium, far less is known about the statistical mechanics, or the behavior of transport coefficients [1]. While the physics of non-equilibrium systems is certainly of broad interest, many basic problems have yet to be fully understood. Of particular interest are the non-equilibrium stationary states and the nature of the statistical mechanics which characterize it. One approach to understanding this problems is to place systems in thermal gradients and examine the long time behavior of observables. In one dimension, this has been done in a variety of problems where both finite and divergent transport coefficients were measured. Typically the transport coefficients are found to diverge when the Hamiltonian conserves total momentum. This is typical of systems where the interactions depend only upon differences xi − xj , such as the FPU and Toda chains [1–3]. When an ‘on site’ potential, V (xi), is also present, the coherent propagation of long wavelength modes is suppressed resulting in finite conductivity [4]. This is the case in the Frenkel-Kontorova, ding-a-ling, Lorenz and other models [5–8]. Bulk behavior is also known in higher dimensions as well [9]. In this letter, we investigate the non-equilibrium steady state properties and thermal transport of lattice Hamiltonians with quartic interactions in 1 spatial dimension. Our system is the discrete version of φ scalar field theory, a proto-typical field theory that has broad application. In contrast to systems such as the FPU β model [2,4,10], which have divergent thermal conductivities, we find a well defined bulk limit for our non-equilibrium results. This we attribute to the on-site nature of the φ interaction. Certain properties of this theory have been studied in the past, which include the Hamiltonian dynamics and phase transitions, as well as the ergodic properties [11]. However, the thermal conductivity has not been determined, and further, there is yet no understanding of the role of boundary jumps and its inter-relation with the shape of the non-equilibrium thermal profile. We present a quantitative analysis of this effect which describes the behavior near and far from thermal equilibrium. A full description of the temperature profiles far from equilibrium is 2 shown to require a description of the boundary jumps. Our model system is the 1-dimensional Hamiltonian H ′ = 1 2 L