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OALib

Entropy of State of Quantum System and Dynamic of This System

Malkhaz Mumladze

Published Jan 1, 2022• 1 citation
Publisher

Abstract

This work defined the concept of entropy for states of a quantum system and built a stochastic model of a dynamic closed quantum system.This paper also is considered new approaches to the concepts of statistics and spin of elementary particles.
DOI
10.4236/oalib.1109052

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We study a simplified Heisenberg spin model in order to clarify the idea of decoherence in closed quantum systems. For this purpose, we define a new concept: the decoherence function Ξ(t), which describes the dynamics of decoherence in the whole system, and which is linked with the total (von Neumann) entropy of all particles. As expected, decoherence is understood both as a statistical process that is caused by the dynamics of the system, and also as a matter of entropy. Moreover, the concept of decoherence time is applicable in closed systems and we have solved its behaviour in the Heisenberg model with respect to particle number N, density ρand spatial dimension D in a 1/r -type of potential. We have also studied the Poincare recurrences occurring in these types of systems: in an N=1000 particle system the recurrence time is close to the order of the age of the universe. This encourages us to conclude that decoherence is the solution for quantum-classical problems not only in practice, but also in principle.
Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
This book presents a number of basic concepts and mathematical techniques for analyzing quantum and classical dynamical systems on an equal basis, mainly but not solely aimed at quantifying randomizing dynamical behaviour. One of the central topics is the introduction of a quantum dynamical entropy based on successive observations of the system. This leads to a symbolic model in terms of a state on a quantum spin chain and its associated entropy that quantifies the complexity of the time evolution. Several model systems are analyzed, but more realistic dynamics and scaling limits are considered as well. Gradually the key concepts and mathematical results are presented and special attention is paid to their relevance for quantum dynamics. The book starts out with a reminder of standard Hilbert space quantum mechanics and then gently introduces the need for the more general and abstract C*-dynamical system setup. Basic examples like quantum spin chains, CAR- and CCR-formalism, and the irrational rotation algebra are described. Next the essential notions from ergodic theory and quantum irreversibility are provided along with the properties and use of von Neumann entropy as quantifiers for randomness. The last part of the book introduces a quantum dynamical entropy based on operational partitions: for classical dynamics the well-established KS-invariant is recovered, while for quantum dynamics a variety of systems are analyzed. The book ends with some open problems and speculations on the footprints of microscopic quantum dynamics on transport phenomena at the macroscopic scale.
We study a simplified Heisenberg spin model in order to clarify the idea of decoherence in closed quantum systems. For this purpose, we define a new concept: the coherence function Xi(t)\\Xi(t)Xi(t), which describes the dynamics of coherence in the whole system, and which is linked with the total coarse-grained (von Neumann) entropy of all particles. We discuss in some detail a general coherence theory and its elementary results. For example, in the particular setup, decoherence diagonalises reduced density matrices in all possible basis sets. As expected, decoherence is understood as a statistical process that is caused by the dynamics of the system, similar to entropy. Moreover, the concept of decoherence time is applicable in closed systems. However, in most cases, the decay of off-diagonal elements differs from the usual exp(−t/taud)\\exp(-t/\\tau_d)exp(−t/taud​) behaviour. We have solved the form of decoherence time taud\\tau_dtaud​ in an infinite Heisenberg model with respect to density rho\\rhorho, spatial dimension DDD and epsilon\\epsilonepsilon in a 1/repsilon1/r^{\\epsilon}1/repsilon -type of potential.