Heat Conductivity from Molecular Chaos Hypothesis in Locally Confined Billiard Systems

We study the transport properties of a large class of locally confined Hamiltonian systems, in which neighboring particles interact through hard-core elastic collisions. When these collisions become rare and the systems large, we derive a Boltzmann-like equation for the evolution of the probability densities. We solve this equation in the linear regime and compute the heat conductivity from a Green-Kubo formula. The validity of our approach is demonstrated by comparing our predictions with the results of numerical simulations performed on a new class of high-dimensional defocusing chaotic billiards.

Figure 1

Typical trajectories of the square-strings model, displayed for increasing values of $a$, color coded (shaded) from blue (dark gray) to red (gray) according to their energies. Small and large arrows indicate initial and final velocities, respectively. For large $a$, the system appears to be near-integrable, reflecting the rarity of interactions, but is nevertheless fully chaotic.

Figure 2

Nonequilibrium temperature profiles of the square-strings model with $a=2.08$, for increasing values of $N=5,10,\dots ,50$. The black curve is the stationary solution of the heat equation. The inset displays the corresponding measurements of $⟨\kappa (T_i)/\nu (T_i)⟩$. The infinite $N$ extrapolation, $\kappa /\nu =1.0037$, is the approximate heat conductivity reported in Table .