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We investigate the hydrodynamical behavior of a system of random walks with zero-range interactions moving in a common 'Sinai-type' random environment on a one dimensional torus. The hydrodynamic equation found is a quasilinear SPDE with a 'rough' random drift term coming from a scaling of the random environment and a homogenization of the particle interaction. Part of the motivation for this work is to understand how the space-time limit of the particle mass relates to that of the known single particle Brox diffusion limit. In this respect, given the hydrodynamic limit shown, we describe formal connections through a two scale limit.

We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. 'Static' scaling limits of the shape functions, under these Gibbs measures, have been shown in the literature. The purpose of this article is to study corresponding, but less understood, 'dynamical' limits. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types of parabolic PDEs, depending on the energy structure.

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[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] 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 nonuniqueness for the standard compressible Euler system, with and without pressure, agagin by taking limits of Hamiltonian dynamics as the number of molecules increases to infinity. The examples come from rescaling of well-posed, deterministic systems of molecules that either collide elastically or interact via singular pair potentials. We also obtain 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. 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.