Heat Conduction and Fourier’s Law by Consecutive Local Mixing and Thermalization

We present a first-principles study of heat conduction in a class of models which exhibit a new multistep local thermalization mechanism which gives rise to Fourier’s law. Local thermalization in our models occurs as the result of binary collisions among locally confined gas particles. We explore the conditions under which relaxation to local equilibrium, which involves no energy exchange, takes place on time scales shorter than that of the binary collisions which induce local thermalization. The role of this mechanism in multiphase material systems such as aerogels is discussed.

Figure 1

Example of lattice billiard with rhombic tiling. The mobile disks are color coded from blue to red with growing kinetic energies. See text for details.

Figure 2

Thermal conductivity $\kappa$ in reduced units, computed by molecular dynamics from the heat exchange of a single particle with stochastically thermalized neighbors, and binary collision frequency ${\nu }_b$, as functions of ${\rho }_m-{\rho }_c$, compared to our theoretical calculations (6, 8) (solid line). The inset shows the ratios between ${\nu }_b$ and $\kappa$ and the right-hand side of (6, 8). The computations were performed with a rhombic cell in contact with stochastically thermalized cells on each sides, in which case $c_3=128{\rho }_c/(3l^2)$ and $c_4=256{\rho }_c^2/(3l^4)$ in Eq. (9). The sum of radii is here fixed to the value $\rho ={\rho }_f+{\rho }_m=9/25$.

Figure 3

Lyapunov exponents ${\lambda }_i$ versus their index $i$. The numbers in the legend correspond to the values of ${\rho }_m$. The first one, ${\rho }_m=0.04<{\rho }_c$, yields the positive exponent ${\lambda }_{+}$ associated to isolated cells, with all the particles at the same speed. The solid red line is the first half of the spectrum as predicted by Eq. (10). The inset shows ${\lambda }_i$ versus ${\rho }_m-{\rho }_c$ for the first half of the positive exponents, ${\lambda }_1,\dots ,{\lambda }_N$ and compares them to the asymptotic value (10) (dashed lines). The computations were performed with a rhombic channel of size $N=10$ cells, $\rho =9/25$.