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Mori Projected Dynamics On A Quantum System

Rachel Harte Nasto

Published Apr 30, 2007• 2 citations
OpenAlex

Abstract

In this thesis we discuss Mori Projected Time Dynamics in a quantum mechanical system. As a precursor to calculating the time derivative of a mixed state of the system we examine the derivation of the Mori-Zwanzig formalism and different ways of calculating the time dynamics of various quantum systems. We consider the exact calculation of the time derivative of a mixed state. We then calculate the same time evolution using Mori Theory and compare the two results. From the general calculation of the Mori Equation we were able to perform a series of simple tests to compare Mori Theory to the known result. We discovered that in each of the three simple cases the Mori Equation and the direct calculation of the derivative give the same result, but in the more complicated situations the two calculations differed. This result leads us to believe that the Mori Equation is an accurate way of calculating the derivative of a mechanical variable in a quantum system.

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In this thesis we discuss Mori Projected Time Dynamics in a quantum mechanical system. As a precursor to calculating the time derivative of a mixed state of the system we examine the derivation of the Mori-Zwanzig formalism and different ways of calculating the time dynamics of various quantum systems. We consider the exact calculation of the time derivative of a mixed state. We then calculate the same time evolution using Mori Theory and compare the two results. From the general calculation of the Mori Equation we were able to perform a series of simple tests to compare Mori Theory to the known result. We discovered that in each of the three simple cases the Mori Equation and the direct calculation of the derivative give the same result, but in the more complicated situations the two calculations differed. This result leads us to believe that the Mori Equation is an accurate way of calculating the derivative of a mechanical variable in a quantum system.
The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.
We present a formalism that explicitly unifies the commonly used Nakajima-Zwanzig approach for reduced density matrix dynamics with the more versatile Mori theory in the context of nonequilibrium dynamics. Employing a Dyson-type expansion to circumvent the difficulty of projected dynamics, we obtain a self-consistent equation for the memory kernel which requires only knowledge of normally evolved auxiliary kernels. To illustrate the properties of the current approach, we focus on the spin-boson model and limit our attention to the use of a simple and inexpensive quasi-classical dynamics, given by the Ehrenfest method, for the calculation of the auxiliary kernels. For the first time, we provide a detailed analysis of the dependence of the properties of the memory kernels obtained via different projection operators, namely, the thermal (Redfield-type) and population based (NIBA-type) projection operators. We further elucidate the conditions that lead to short-lived memory kernels and the regions of parameter space to which this program is best suited. Via a thorough analysis of the different closures available for the auxiliary kernels and the convergence properties of the self-consistently extracted memory kernel, we identify the mechanisms whereby the current approach leads to a significant improvement over the direct usage of standard semi- and quasi-classical dynamics.
The ability to efficiently and accurately calculate equilibrium time correlation functions of many-body condensed phase quantum systems is one of the outstanding problems in theoretical chemistry. The Nakajima-Zwanzig-Mori formalism coupled to the self-consistent solution of the memory kernel has recently proven to be highly successful for the computation of nonequilibrium dynamical averages. Here, we extend this formalism to treat symmetrized equilibrium time correlation functions for the spin-boson model. Following the first paper in this series [A. Montoya-Castillo and D. R. Reichman, J. Chem. Phys. 144, 184104 (2016)], we use a Dyson-type expansion of the projected propagator to obtain a self-consistent solution for the memory kernel that requires only the calculation of normally evolved auxiliary kernels. We employ the approximate mean-field Ehrenfest method to demonstrate the feasibility of this approach. Via comparison with numerically exact results for the correlation function Czz(t)=Re⟨σz(0)σz(t)⟩, we show that the current scheme affords remarkable boosts in accuracy and efficiency over bare Ehrenfest dynamics. We further explore the sensitivity of the resulting dynamics to the choice of kernel closures and the accuracy of the initial canonical density operator.
The problem of describing the dynamics of quantum systems generated by a nonlocal-in-time interaction is considered. It is shown that the use of the Feynman approach to quantum theory in combination with the canonical approach allows one to extend quantum dynamics to describe the time evolution in the case of such interactions. In this way, using only the current concepts of quantum theory, a generalized equation of motion for state vectors is derived. In the case where the fundamental interaction generating the dynamics in a system is local in time, this equation is equivalent to the Schrödinger equation. Explicit examples are given for an exactly solvable model. The proposed formalism is shown to provide a new insight into the problem of the description of nonlocal interactions in quantum field theory. It is shown that such a property of the equation of motion, such as nonlocality in time, may be important for describing hadron-hadron interactions at low and intermediate energies.
Abstract Explaining the emergence of stochastic irreversible macroscopic dynamics from time-reversible deterministic microscopic dynamics is one of the key problems in philosophy of physics. The Mori-Zwanzig (MZ) projection operator formalism, which is one of the most important methods of modern nonequilibrium statistical mechanics, allows for a systematic derivation of irreversible transport equations from reversible microdynamics and thus provides a useful framework for understanding this issue. However, discussions of the MZ formalism in philosophy of physics tend to focus on simple variants rather than on the more sophisticated ones used in modern physical research. In this work, I will close this gap by studying the problems of probability and irreversibility using the example of Grabert’s time-dependent projection operator formalism. This allows to better understand how general proposals for understanding probability in statistical mechanics, namely (a) quantum approaches and (b) almost-objective probabilities, can be accomodated in the MZ formalism. Moreover, I will provide a detailed physical analysis, based on the MZ formalism, of various proposals from the philosophical literature, such as (a) Robertson’s theory of justifying coarse-graining via autonomous macrodynamics, (b) Myrvold’s problem of explaining autonomous macrodynamics, and (c) Wallace’s simple dynamical conjecture.