Information theory in quantum statistical mechanics

Francesco R. Ruggeri

In physics, there are deterministic theories such as classical mechanics and statistical theories such as classical statistical mechanics and quantum mechanics. Many interpretations are applied to statistical theories and there is sometimes debate as to the application of such theories in various physical scenarios. In a previous note, we considered cycling due to exp(iEt) in quantum mechanics and tried to argue that such cycling may also appear in statistical mechanics. We made use of the example of the ground state of the quantum oscillator which has a quantum mechanical density of the same form as the classical statistical mechanical result. In this note, we wish to consider the idea of cycling in statistical mechanics using some simple examples. In addition, we wish to also examine the notion of losing information to pass from a deterministic theory to a statistical one. Finally, we wish to consider some possible aspects of the classical statistical density in a potential. It seems that even in classical statistical mechanics, the picture presented by the ‘averaging’ may actually be quite different from the physical scenario which may contain deterministic behavior which is lost in the calculation of statistical results.

Dėnes Petz

E. T. Jaynes

One of the most successful physicists of the generation immediately before Hugh Everett III, E. T. Jaynes shared both Everett's interest in applications of information theory to physics and his foundational orientation. But whereas Everett focused on quantum mechanics, much of Jaynes' work concerned how to use information theory in thermodynamics. This document is a single page from Jaynes' paper "Information Theory and Statistical Mechanics," Physical Review, 106, 620-630 (1957). Based on the basic content of his penciled observations, Everett wrote to Jaynes with an analysis of the flaws in Jaynes paper. He enclosed a copy of his own relative states preprint. Jaynes wrote back at great length.

Herbert S. Green

Murali Krishna Pasupuleti

Abstract: This study proposes a unified framework that integrates topological quantum field theories with statistical mechanics to provide a comprehensive understanding of quantum states and dynamics. The problem of reconciling the quantum mechanical description of microscopic systems with macroscopic statistical behavior is addressed by exploring the relationship between topological invariants and thermodynamic properties in quantum systems. The methodology involves the development of a mathematical model that combines the abstract principles of topological quantum field theory with the probabilistic approaches of statistical mechanics, utilizing tensor calculus and advanced mathematical physics tools. Key results indicate that the proposed framework offers new insights into quantum phase transitions, critical phenomena, and the thermodynamic behavior of quantum systems, especially in systems exhibiting topological phases. The framework is further evaluated through a series of theoretical simulations, demonstrating its applicability to quantum information science, condensed matter physics, and quantum technology applications. The implications of this research are significant, providing a foundation for the development of new quantum technologies, such as quantum computing and quantum materials, and contributing to a deeper understanding of the connections between topological properties and statistical mechanics in quantum systems. Keywords: Topological Quantum Field Theory, Statistical Mechanics, Quantum States, Quantum Dynamics, Phase Transitions, Critical Phenomena, Quantum Computing, Quantum Materials, Thermodynamics, Topological Phases, Quantum Information Science.

Lev B. Levitin