Heat conduction and Fourier's law in a class of many particle dispersing   billiards
  We consider the motion of many confined billiard balls in interaction and discuss their transport and chaotic properties. In spite of the absence of mass transport, due to confinement, energy transport can take place through binary collisions between neighbouring particles. We explore the conditions under which relaxation to local equilibrium occurs on time scales much shorter than that of binary collisions, which characterize the transport of energy, and subsequent relaxation to local thermal equilibrium. Starting from the pseudo-Liouville equation for the time evolution of phase-space distributions, we derive a master equation which governs the energy exchange between the system constituents. We thus obtain analytical results relating the transport coefficient of thermal conductivity to the frequency of collision events and compute these quantities. We also provide estimates of the Lyapunov exponents and Kolmogorov-Sinai entropy under the assumption of scale separation. The validity of our results is confirmed by extensive numerical studies. 
Pierre Gaspard and Thomas Gilbert