Scaling Limits for Stochastic and Quantum Dynamics

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9703752 Yau The incompressible Navier-Stokes equation have been proved as the hydrodynamical limit equation of stochastic lattice gas models in dimension 3. The first project investigates the following three related topics: (i) the scaling limit of tagged particles, (ii) the equilibrium fluctuation and (iii) the appropriate time scale for dimensions 1 and 2. The first problem emphasizes the individual behavior of a typical particle instead of the collective behavior of all the particles. The second problem is a question about the central limit theorem. The third problem addresses the time scale in dimensions 1 and 2, which is conjectured to be very different from the diffusive scale in dimension 3. The second project studies the relaxation rates of the Kawasaki dynamics of Gibbs states in infinite volume. This is believed to be a power law. The method is based on the Poincare inequalities (spectral gap), logarithmic Sobolev inequalities and some entropy estimates. The third project concerns the scaling limit of a quantum particle in a random potential. There are two limiting cases: the low density limit and the weak coupling limit. In both cases, the phase space density of the quantum evolution defined through the Wigner transform or the coherent state is expected to converge weakly to a linear Boltzmann equation with collision kernel given by the quantum scattering cross section. This research addresses in particular the following two questions: (1) What is the typical behavior of an individual particle in a fluid? Though the collective behavior of fluid has been studied intensively, the important question of the propagation of individual particles in the fluid has not. The project will study this question with stochastic models which are believed to capture the essential behavior of the fluid. (2) How does a wave travel in random media. It is believed that with high disorder, a wave in random media becomes loca lized. The important question is to analyze the low disorder region when a wave propagates. This can be considered as a model for conduction of current in a semiconductor, or propagation of radio waves or seismic waves. The practical equation governing all these diverse phenomena is the Boltzmann equation. This study will try to validate the Boltzmann equation from more basic models involving wave equations in random media and to understand its next order corrections or fluctuations.