Mathematical Sciences: Scaling Limits of Nongradient Dynamics

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The very first question concerning stochastic dynamics is the rate of relaxation to the equilibrium. One can roughly divide these rates into two categories: the exponential relaxation and the power law relaxation. The main objective of this research is to study dynamics with the power law decay. The first goal is to derive the macroscopic scaling limiting equations rigorously and then to study various corrections to these equations. In the past few years, several methods have been developed for these problems. One of the major restrictions to these methods is a technical condition called the gradient condition. The investigator will study several nongradient models, in particular, the hydrodynamical limit and the rate of relaxation to equilibrium of lattice gas at finite temperature. A related problem for lattice gas with random field will also be considered. Finally, the investigator will study problems related to a time scale longer than the hydrodynamical scale. These are particularly important because the incompressible Navier-Stokes equation is believed to be the long time (diffusive) scaling limit of Hamiltonian systems. A basic problem concerning dynamical system is to determine the rate of approach to the equilibrium. In this research the investigator will study dynamics with slow relaxation to equilibrium, which most natural processes follow. Major goals are to determine the precise rate of relaxation to the equilibrium of these processes and to derive macroscopic equations governing the macroscopic behavior of relaxations and then to study various corrections to these equations. In the past few years, several methods have been developed for these problems for several types of models. But some major restrictions of these methods prohibit the study of more realistic models. The investigator will study several models to extend the current technology and to address some key technical difficulties. Models of interest include: the lattice gases at finite temperature, the effect of random impurities in a lattice gas, and the discrete velocity model which simulates the incompressible Navier-Stokes equation.