Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.

arXiv:2412.15316Dec 19, 20240 citations

Statistical mechanics of quantum spin systems

Derek W. Robinson

Communications in Mathematical PhysicsJun 1, 196765 citations

Entropy Beyond the Second Law Thermodynamics and statistical mechanics for equilibrium, non-equilibrium, classical, and quantum systems

Phil Attard

Entropy Beyond the Second Law presents a coherent formulation of all aspects of thermodynamics and statistical mechanics with entropy as the unifying theme. This includes formulating equilibrium theory and explaining the role of the Second Law in establishing the equilibrium state. It also establishes the foundations of non-equilibrium theory by generalizing the concept of entropy in an appropriate way. Finally, it addresses the relationship between classical and quantum statistical mechanics by invoking the collapse of the wave function into entropy states. A secondary purpose is to increase the understanding of entropy and the Second Law, and to correct misapprehensions and errors that have arisen over the years.Entropy Beyond the Second Law presents a coherent formulation of all aspects of thermodynamics and statistical mechanics with entropy as the unifying theme. This includes formulating equilibrium theory and explaining the role of the Second Law in establishing the equilibrium state. It also establishes the foundations of non-equilibrium theory by generalizing the concept of entropy in an appropriate way. Finally, it addresses the relationship between classical and quantum statistical mechanics by invoking the collapse of the wave function into entropy states. A secondary purpose is to increase the understanding of entropy and the Second Law, and to correct misapprehensions and errors that have arisen over the years.

Institute of Physics eBooksJan 1, 20182 citations

Statistical mechanical methods for quantum spin systems

真井上

Jan 1, 19880 citations

Entropy Production in Quantum Spin Systems

David Pierre Ruelle

arXiv:math-ph/0006006Communications in Mathematical PhysicsNov 1, 200190 citations

Remarks on Shannon's Statistical Inference and the Second Law in Quantum Statistical Mechanics

Kazuo Fujikawa

We comment on a formulation of quantum statistical mechanics, which incorporates the statistical inference of Shannon. Our basic idea is to distinguish the dynamical entropy of von Neumann, $H = -k Tr \hatρ\ln\hatρ$ , in terms of the density matrix $\hatρ(t)$ , and the statistical amount of uncertainty of Shannon, $S= -k \sum_{n}p_{n}\ln p_{n}$ , with $p_{n}=< n|\hatρ|n>$ in the representation where the total energy and particle numbers are diagonal. These quantities satisfy the inequality $S\geq H$ . We propose to interprete Shannon's statistical inference as specifying the {\em initial conditions} of the system in terms of $p_{n}$ . A definition of macroscopic observables which are characterized by intrinsic time scales is given, and a quantum mechanical condition on the system, which ensures equilibrium, is discussed on the basis of time averaging. An interesting analogy of the change of entroy with the running coupling in renormalization group is noted. A salient feature of our approach is that the distinction between statistical aspects and dynamical aspects of quantum statistical mechanics is very transparent.We comment on a formulation of quantum statistical mechanics, which incorporates the statistical inference of Shannon. Our basic idea is to distinguish the dynamical entropy of von Neumann, $H = -k Tr \hatρ\ln\hatρ$, in terms of the density matrix $\hatρ(t)$, and the statistical amount of uncertainty of Shannon, $S= -k \sum_{n}p_{n}\ln p_{n}$, with $p_{n}=< n|\hatρ|n>$ in the representation where the total energy and particle numbers are diagonal. These quantities satisfy the inequality $S\geq H$. We propose to interprete Shannon's statistical inference as specifying the {\em initial conditions} of the system in terms of $p_{n}$. A definition of macroscopic observables which are characterized by intrinsic time scales is given, and a quantum mechanical condition on the system, which ensures equilibrium, is discussed on the basis of time averaging. An interesting analogy of the change of entroy with the running coupling in renormalization group is noted. A salient feature of our approach is that the distinction between statistical aspects and dynamical aspects of quantum statistical mechanics is very transparent.

arXiv:cond-mat/0005496Journal of the Physical Society of JapanJan 1, 20023 citations

We comment on a formulation of quantum statistical mechanics, which incorporates the statistical inference of Shannon. Our basic idea is to distinguish the dynamical entropy of von Neumann,

H = -k Tr \hatρ\ln\hatρ

, in terms of the density matrix

\hatρ(t)

, and the statistical amount of uncertainty of Shannon,

S= -k \sum_{n}p_{n}\ln p_{n}

, with

p_{n}=< n|\hatρ|n>

in the representation where the total energy and particle numbers are diagonal. These quantities satisfy the inequality

S\geq H

. We propose to interprete Shannon's statistical inference as specifying the {\em initial conditions} of the system in terms of

p_{n}

. A definition of macroscopic observables which are characterized by intrinsic time scales is given, and a quantum mechanical condition on the system, which ensures equilibrium, is discussed on the basis of time averaging. An interesting analogy of the change of entroy with the running coupling in renormalization group is noted. A salient feature of our approach is that the distinction between statistical aspects and dynamical aspects of quantum statistical mechanics is very transparent.