# Browsing by Author "Cao, Yu"

###### Results Per Page

###### Sort Options

Item Open Access Analytical and Numerical Study of Lindblad Equations(2020) Cao, YuLindblad equations, since introduced in 1976 by Lindblad, and by Gorini, Kossakowski, and Sudarshan, have received much attention in many areas of scientific research. Around the past fifty years, many properties and structures of Lindblad equations have been discovered and identified. In this dissertation, we study Lindblad equations from three aspects: (I) physical perspective; (II) numerical perspective; and (III) information theory perspective.

In Chp. 2, we study Lindblad equations from the physical perspective. More specifically, we derive a Lindblad equation for a simplified Anderson-Holstein model arising from quantum chemistry. Though we consider the classical approach (i.e., the weak coupling limit), we provide more explicit scaling for parameters when the approximations are made. Moreover, we derive a classical master equation based on the Lindbladian formalism.

In Chp. 3, we consider numerical aspects of Lindblad equations. Motivated by the dynamical low-rank approximation method for matrix ODEs and stochastic unraveling for Lindblad equations, we are curious about the relation between the action of dynamical low-rank approximation and the action of stochastic unraveling. To address this, we propose a stochastic dynamical low-rank approximation method. In the context of Lindblad equations, we illustrate a commuting relation between the dynamical low-rank approximation and the stochastic unraveling.

In Chp. 4, we investigate Lindblad equations from the information theory perspective. We consider a particular family of Lindblad equations: primitive Lindblad equations with GNS-detailed balance. We identify Riemannian manifolds in which these Lindblad equations are gradient flow dynamics of sandwiched Rényi divergences. The necessary condition for such a geometric structure is also studied. Moreover, we study the exponential convergence behavior of these Lindblad equations to their equilibria, quantified by the whole family of sandwiched Rényi divergences.

Item Open Access On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamicsCao, Yu; Lu, Jianfeng; Wang, LihanWe provide a new explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance. To achieve this, we first prove a Poincar\'{e}-type inequality with Gibbs measure in space and Gaussian measure in momentum. Our new estimate provides a more explicit and simpler expression of decay rate; moreover, when the potential is convex with Poincar\'{e} constant $m \ll 1$, our new estimate offers the decay rate of $\mathcal{O}(\sqrt{m})$ after optimizing the choice of friction coefficient, which is much faster compared to $\mathcal{O}(m)$ for the overdamped Langevin dynamics.