Shock Waves in an Augmented One-Dimensional Atom Chain

We derive here a simplified discrete one-dimensional (1D) model describing the main features of shock waves. In order to avoid expensive multidimensional simulations, 1D models are commonly used, but the existing ones often exhibit some spurious physically irrelevant behavior. Here we build a 1D model with perturbations arising from mean higher-dimensional behavior. The coupling of the system with a deterministic heat bath “à la“ Kac-Zwanzig allows us to derive a generalized Langevin equation for the system, without a priori fixing the temperature in the shocked region. This deterministic problem with several degrees of freedom is then reduced to a simpler stochastic problem with memory. Some numerical results are provided, that illustrate and confirm the qualitative correctness of the model.