The dynamics of a quantum system coupled to a classical environment and subject to constraints that drive it out of equilibrium are described. The evolution of the system is governed by the quantum-classical Liouville equation. Rather than evaluating the evolution of the mixed quantum-classical density operator, we derive exact equations of motion for the nonequilibrium average values of a set of operators or variables, along with correlation function expressions for the dissipative coefficients that enter these equations. These equations are obtained by requiring that the exact nonequilibrium averages are equal to local nonequilibrium averages that depend on auxiliary fields whose values satisfy evolution equations obtained using projection operator methods. The results are illustrated by deriving reaction-diffusion equations coupled to fluid hydrodynamic equations for a solution of quantum particles that can exist in two metastable states. Nonequilibrium steady states are discussed along with the reaction rate and diffusion correlation functions that characterize such states.
The dynamics of a quantum system coupled to a classical environment and subject to constraints that drive it out of equilibrium are described. The evolution of the system is governed by the quantum-classical Liouville equation. Rather than evaluating the evolution of the mixed quantum-classical density operator, we derive exact equations of motion for the nonequilibrium average values of a set of operators or variables, along with correlation function expressions for the dissipative coefficients that enter these equations. These equations are obtained by requiring that the exact nonequilibrium averages are equal to local nonequilibrium averages that depend on auxiliary fields whose values satisfy evolution equations obtained using projection operator methods. The results are illustrated by deriving reaction-diffusion equations coupled to fluid hydrodynamic equations for a solution of quantum particles that can exist in two metastable states. Nonequilibrium steady states are discussed along with the reaction rate and diffusion correlation functions that characterize such states.
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
A statistical mechanical formulation where nonequilibrium averages of selected quantum observables are given exactly as averages over local density operators is used to describe the dynamics and thermodynamics of a small quantum system coupled to reservoirs. The local density operators are functionals of constraint variables that play the role of thermodynamic forces that satisfy a set of equations of motion. A focus of the work is on the evolution of the average internal energy of the quantum subsystem that may be subject to external autonomous driving that does work on the system, and on entropy production in the system. The nonequilibrium statistical thermodynamics the quantum subsystem coupled to reservoirs, as well as that for the isolated subsystem and reservoirs, are discussed. When the subsystem is coupled to large quantum reservoirs, the effects of the quantum subsystem on the reservoirs can be captured through boundary conditions on the reservoir equations, and these boundary conditions are derived. The formulation is illustrated by considering the heat transfer through a system comprising a quantum system coupled to two heat reservoirs with different temperatures.
Quantum time correlation functions are often the principal objects of interest in experimental investigations of the dynamics of quantum systems. For instance, transport properties, such as diffusion and reaction rate coefficients, can be obtained by integrating these functions. The evaluation of such correlation functions entails sampling from quantum equilibrium density operators and quantum time evolution of operators. For condensed phase and complex systems, where quantum dynamics is difficult to carry out, approximations must often be made to compute these functions. We present a general scheme for the computation of correlation functions, which preserves the full quantum equilibrium structure of the system and approximates the time evolution with quantum-classical Liouville dynamics. Several aspects of the scheme are discussed, including a practical and general approach to sample the quantum equilibrium density, the properties of the quantum-classical Liouville equation in the context of correlation function computations, simulation schemes for the approximate dynamics and their interpretation and connections to other approximate quantum dynamical methods.
Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system.\nA promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system.\nImposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom.\nAny numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros.\n\nIn this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators.\nThe result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian.\nThis equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived.\nCalculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.
The study of dissipative systems holds significant interest in physics.This thesis aims to explore these phenomena in both classical and quantum regimes.In the first part, we utilize the system-reservoir approach to investigate the dynamics of two Brownian particles immersed in the same bath.Two methods are employed to address this problem: one with bilinear coupling and the other with nonlinear coupling between the particles and the bath.The extension of the system to include two particles with bilinear coupling yields unphysical results, such as free-particle motion for the relative coordinate and a lack of interaction between closely spaced particles.To address this issue, authors have been introduced an exponential coupling.In this work, we propose a method to reconcile both linear and nonlinear couplings.We demonstrate how to derive the same nonlinear dissipation rates starting from the bilinear Lagrangian, achieved through a modified spectral function that explicitly depends on the distance between the particles.Additionally, we implement a modified spectral function to mitigate anomalous diffusion observed in the standard nonlinear model, along with a phenomenological model describing hydrodynamic interaction between a pair of Brownian particles in a viscous fluid.In the quantum regime, we adopt the same system-reservoir approach to investigate the dissipative adaptation hypothesis proposed by Jeremy England.This hypothesis proposes a general thermodynamic mechanism that explains the self-organization of systems through the dissipation of work absorbed from an external drive.In the second part of this thesis, we explore the quantum dynamics of systems subjected to an external drive, evaluating the thermodynamic quantities of a self-organization process.To do so, we utilize a time-dependent spin-boson Hamiltonian that characterizes a particle subject to a metastable double potential with time-dependent parameters controlling the asymmetry of the wells.Our objective is to demonstrate that the asymmetric potential can localize the particle in the unstable side of the well and verify that this transition results in the highest energy absorption.In conclusion, we propose further investigations into the driven spin-boson model to establish a comprehensive theory of the system's evolution and its thermodynamic implications.
We use the so-called Liouville--von Neumann (LvN) approach to study the nonequilibrium quantum dynamics of time-dependent second order phase transitions. The LvN approach is a canonical method that unifies the functional Schr\"odinger equation for the quantum evolution of pure states and the LvN equation for the quantum description of mixed states of either equilibrium or nonequilibrium. As nonequilibrium quantum mechanical systems we study a time-dependent harmonic and an anharmonic oscillator and find the exact Fock space and density operator for the harmonic oscillator and the nonperturbative Gaussian Fock space and density operator for the anharmonic oscillator. The density matrix and the coherent, thermal, and coherent-thermal states are found in terms of their classical solutions, for which the effective Hamiltonians and equations of motion are derived. The LvN approach is further extended to quantum fields undergoing time-dependent second order phase transitions. We study an exactly solvable model with a finite smooth quench and find the two-point correlation functions. Because of the spinodal instability of long wavelength modes, the two-point correlation functions lead to the t1/4-scaling relation during the quench and the Cahn-Allen scaling relation t1/2 after completion of the quench. Further, after the finite quench the domain formation shows a time-lag behavior at the cubic power of the quench period. Finally we study the time-dependent phase transition of a self-interacting scalar field.