Stochastic and Quantum Dynamics of Large Systems

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0072098Yau The Euler equations were derived from microscopic Newtonian dynamics heuristically by Morrey in the sixties and rigorously by Olla-Varadhan-Yau recently (assuming the Boltzmann hypothesis). The first part of the project proposes to change the underlying dynamics to quantum mechanics. The aim is to show that the Euler equations obtained in the quantum case are similar to the classical ones except all physical quantities should be computed quantum mechanically. The approach will be based on the relative entropy method and certain extension of large deviation theory to the quantum case. The second part of the project concerns the dynamics of an electron interacting with a lattice structure, modeled by a phonon field. Therefore, its dynamics is governed by a Schrodinger equation of an electron coupled to a scalar quantum field. The goal is to prove that the phase space density of the electron converges to a Boltzmann equation in the weak coupling limit. The underlying dynamics for electrons in a material are governed by quantum mechanics, or more precisely the Schrodinger equations. Since the Schrodinger equations are difficult to solve no matter analytically or numerically, most models in use nowadays for transport behavior of electrons are semi-classical ones based on classical particles and diffusions. Although these semi-classical models are valid approximations for large distance behavior, they neglect crucial quantum effects in small scales. The aim of this project is to study the transport behavior starting from the first principle---the Schrodinger equations. In particular, the validity of the Euler and Boltzmann equations will be established from the Schrodinger equations. This is the first step toward understanding the quantum effects in transport behavior. The long term goal is to understand quantum corrections to these equations.