Microscopic Models for Vibrations in Mechanical Systems Under Equilibrium and Non-equilibrium Conditions

Abstract

Equilibrium Thermodynamics studies states of macroscopic objects that do not change in time when isolated from their environment. This requires chemical, mechanical and thermal equilibrium which together amount to thermodynamic equilibrium. A non-equilibrium state can be established putting the system in contact with more than one reservoir of heat, mass, or other physical quantities. The dynamical evolution of a system of particles representing a macroscopic bar in both equilibrium and non-equilibrium conditions is illustrated by means of a simple one-dimensional molecular dynamics model, illustrating how macroscopic phenomena may be qualitatively understood with microscopic toy models. In particular, a system of hard point-particles undergoing only binary collisions is considered. A conservative force is applied on one of the end particles to reproduce the cohesion of the bar. Non-equilibrium conditions are obtained by adding two deterministic thermostats acting on the first and on the last particle of the bar. In the equilibrium case, we determine the values of macroscopic and microscopic proprieties of the system, such as length, linear density, specific kinetic energy, average energy per particle, and position of the centre of mass of control groups located in different parts of the bar. In the non-equilibrium case, we focus on length oscillations, and we demonstrate their dependence on the characteristic parameters of the thermostats. Although highly idealized, this model reproduces an important qualitative aspect of metal bars: hardening.