Abstract

The majority of the fundamental processes of our natural world are described by differential equations. Some examples are the flow of fluids, the formation of crystals, the spread of infections, the diffusion of chemicals, etc. These examples are responsible for our interest in partial differential equations such as Hamilton-Jacobi equation, Euler equation, Navier-Stokes equation and Diffusion equation. An important open problem in statistical mechanics is the derivation of Euler and Navier-Stokes equations from the small scale dynamics governed by Newton's second law. Some of the variants of this problem has been treated recently for some stochastic particle systems. These systems are microscopically described by stochastic rules and macroscopically are governed by Euler or Navier-Stokes type equations. The primary goal of these lectures is to study the connection between the microscopic structure and macroscopic behavior of some of these models.