Asymptotic Dynamics for Stochastic and Quantum Dynamics

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0307295Yau The dynamics of a quantum particle in a random media is governed by a random Schrodinger equation. In the kinetic limit, the phase space distribution of a solution to this equation converges to a linear Boltzmann equation. Beyond the kinetic scale, one expects that the Boltzmann equation is still correct in dimension three or higher provided that the diffusivity is renormalized. The first project aims to study this renormalization for time scales longer than the kinetic scale. The second project concerns the viscosity of the lattice gas models. Recently the PI has established the divergence rate for the diffusion coefficient for the asymmetric simple exclusion process in dimension two. The goal of this project is to extend this result to the dimension one case and to the lattice gas models with the Navier-Stokes equation as the formal limit. The fundamental question regarding the conductivity of electrons in semiconductors is how conduction occurs. As the technology is reaching the stage that the semi-classical theory no longer applies, the quantum effect will dominate and a mathematical rigorous theory is of fundamental importance. The first project aims to establish the quantum corrections of the standard semi-classical theory based on the Boltzmann equation. This will provide the first rigorous understanding of these quantum dynamical effects. The second project investigates the validity of the lattice gas models for fluids. These models, besides being widely used to simulate the Navier-Stokes equations, are important models for non-equilibrium statistical physics. The goal here is to establish the divergence rate of the viscosity for these models. This will give the first clue concerning how well the Navier-Stokes equation models the two-dimensional fluid.