Shock waves in an augmented one-dimensional atom chain

We derive here a simplified discrete one-dimensional model describing some important features of shock waves. In order to avoid expensive multidimensional simulations, one-dimensional models are commonly used, but the existing ones often exhibit some spurious physically irrelevant behaviour. Here we build a one-dimensional model with perturbations arising from mean higher-dimensional behaviour. The coupling of the system with a deterministic heat bath in the Kac-Zwanzig fashion 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, which illustrate and confirm the qualitative correctness of the model.