Simulations of Classical Three-Body Thermalization in One Dimension

One-dimensional systems, such as nanowires, or electrons moving along strong magnetic field lines, have peculiar thermalization physics. The binary collision of point-like particles, typically the dominant process for reaching thermal equilibrium in higher dimensional systems, cannot thermalize a 1D system. We study how dilute classical 1D gases thermalize through three-body collisions. We consider a system of identical classical point particles with pairwise repulsive inverse power-law potential V_ij ∝ 1/|x_i − x_j|ⁿ, or the pairwise Lennard-Jones potential. Using Monte Carlo methods, we compute a collision kernel and use it in the Boltzmann equation to evolve a perturbed thermal state with temperature T toward equilibrium. We explain the shape of the kernel and its dependence on the system parameters. Additionally, we implement molecular dynamics simulations of a many-body gas and show agreement with the Boltzmann evolution in the low density limit. For the inverse power-law potential, the rate of thermalization is proportional to ρ²T^{1/2 − 1/n} where ρ is the number density, and the corresponding proportionality constant decreases with increasing n.