Heat conduction in a chain of colliding particles with stiff repulsive potential

Abstract

One-dimensional billiard, i.e. a chain of colliding particles with equal masses, is well-known example of completely integrable system. Billiards with different particles are generically not integrable, but still exhibit divergence of a heat conduction coefficient (HCC) in thermodynamic limit. Traditional billiard models imply instantaneous (zero-time) collisions between the particles. We lift this condition and consider the heat transport in a chain of stiff colliding particles with power-law potential of the nearest-neighbor interaction. The instantaneous collisions correspond to the limit of infinite power in the interaction potential; for finite powers, the interactions take nonzero time. This modification of the model leads to profound physical consequence -- probability of multiple, in particular, triple particle collisions becomes nonzero. Contrary to the integrable billiard of equal particles, the modified model exhibits saturation of the heat conduction coefficient for large system size. Moreover, identification of scattering events with the triple particle collisions leads to simple definition of characteristic mean free path and kinetic description of the heat transport. This approach allows prediction both of temperature and density dependencies for the HCC limit values. The latter dependence is quite counterintuitive - the HCC is inversely proportional to the particle density in the chain. Both predictions are confirmed by direct numeric simulations.