The problem of hydrodynamic scaling appears naturally in many contexts. We have large system that changes in time. The dynamics is dictated by an interaction that involves only components of the system that are close by. There are two distinct spatial scales. A microsopic sacle that determines the effective range of the interaction and a macroscopic scale in which the quantities of interest are measured. The dynamics has a few conserved quantities and one is interested in the evolution of local averages of conserved quantities in the macroscopic scale. Because of conservation these objects change slowly and time has to be speeded up to a macroscopic time scale. But then the rate of change of these conserved quantities depend on variables that are not necessarily conserved and due to the speed up of time change rapidly. The equations then do not close. One has to use some form of averaging or an ergodic theorem to replace the rapidly changing quantities by their mean values relative to certain invariant measures. These invariant or "Gibbs" measures are parametrized by the conserved quantities and this is used to close the equations. The prime example of this procedure is the formal derivation of Euler equations from the Hamiltonian equations for a classical particle system.
Whereas the classical particle motion is deterministic we shall be concerned with systems that have a stochastic or random component as well. The randomness is often helpful and makes the averaging principle easier.
We will look at a few examples. In some of them the scaling factor is the same in space and time and this leads to a non-linear hyperbolic system of conservation laws. In others the scaling is parabolic and leads to a non-linear diffusion equation for the object of interest. In the parabolic case there is a distinction between gradient and non-gradient models. The latter are far more complicated to analyse.
One common thread in all of our models it that the analysis depends on entropy, relative entropy and its production. Ideas from the theory of large deviations are used to control the probabilities in terms of entropy.