Nonequilibrium Energy Profiles for a Class of 1-D Models

Abstract

As a paradigm for heat conduction in 1 dimension, we propose a class of models represented by chains of identical cells, each one of which containing an energy storage device called a "tank". Energy exchange among tanks is mediated by tracer particles, which are injected at characteristic temperatures and rates from heat baths at the two ends of the chain. For stochastic and Hamiltonian models of this type, we develop a theory that allows one to derive rigorously -under physically natural assumptions -macroscopic equations for quantities related to heat transport, including mean energy profiles and tracer densities. Concrete examples are treated for illustration, and the validity of the Fourier Law in the present context is discussed.

Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems

“…From the left reservoir at density n > and temperature T > , particles are injected into the system at a rate j > . Observe that the mean energy of the particles injected into the system is not T > but 3 2 T > (see e.g., [1]). Thus, the left reservoir injects energy into the system at a rate q > given by…”

“…In [1], a stochastic approximation of the mechanical model was considered. In this approximation it was assumed that what happens in one cell is independent of the state of the other cells.…”

“…The case when the α X Y are independent of the local fields, but different from 1 2 corresponds to a persistent random walk and will be studied in the next section. More general and realistic laws will be investigated in the last sections.…”

“…Close to equilibrium, one can model the Hamiltonian system by persistent random walks. In this case the system remains of gradient type and thus, the theory of [1] is still applicable with corrections of the order of 1/N with N being the size of the system.…”

“…In each cell, there is a rotating disc, with which these particles interact, and this is the only interaction in the model. It was shown in [1] that when the cells are weakly coupled, to a good approximation, the jump rates of particles and the energy-exchange rates from cell to cell follow linear profiles. Here, we refine that study by analyzing higher-order effects which are induced by the presence of external gradients for situations in which memory effects, typical of Hamiltonian dynamics, cannot be neglected.…”