Cold, thermal and oscillator closure of the atomic chain

We consider a simple microscopic model for a solid body and study the problematic nature of micro-macro transitions. The microscopic model describes the solid body by a many-particle system that develops according to Newton's equations of motion. We discuss various Riemannian initial value problems that lead to the propagation of waves. The initial value problems are solved directly from the microscopic equations of motion. Additionally, these equations serve to establish macroscopic field equations. The macroscopic field equations consist of conservation laws, which follow rigorously from the microscopic equations, and of closure relations which are completely determined by the distributions of the microscopic motion. In particular, we consider three kinds of closure relations which correspond to three different kinds of equilibrium. It turns out that closure relations cannot be given appropriately without relating them to the initial conditions, and that closure relations might change during the temporal development of the initial data, because the body undergoes several transitions between different states of local equilibrium. In those examples that we have considered, the macroscopic variables of mass density and temperature do not constitute a unique kind of microscopic motion in equilibrium. (author)