We establish conditions on the Hamiltonian evolution of interacting molecules that imply hydrodynamic equations at the limit of infinitely many molecules and show that these conditions are satisfied whenever the solutions of the classical equations for N interacting molecules obey uniform in N bounds.
We show that this holds when the initial conditions are bounded and the molecule interaction is weak enough at the initial time.
We then obtain hydrodynamic equations that coincide with Maxwell's.
We then construct explicit examples of spontaneous energy generation and non-uniqueness for the standard compressible Euler system, with and without pressure, again by taking limits of Hamiltonian dynamics as the number of molecules increases to infinity.
The examples come from rescalings of well-posed, deterministic systems of molecules that either collide elastically or interact via singular pair potentials.
We also obtain the Percus macroscopic equation as the limit of a sequence of single systems of N hard rods with the number of hard rods going to infinity, with the use of ensembles.
Finally, we establish the strict convexity of the pressure as a thermodynamic limit for continuous systems.
As a result we show the existence of a local bijection between macroscopic density, velocity, and energy on one hand and thermodynamic parameters on the other, for continuous systems.