Quantum coarse-grained entropy and thermalization in closed systems

Šafránek, Dominik; Aguirre, Anthony

doi:10.1103/physreva.99.012103

Cited by 43 publications

(75 citation statements)

References 79 publications

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“…(2) with Eq. (14). We will see that, under an assumption specified later, the proposed Boltzmann entropy S(ψ) satisfies the properties (S1)-(S3) with respect to the ensemble of pure states given by the PDF P (β,V,N ) N (ψ), that is the normal distribution model for the canonical density operator ρ can (β, V, N) associated with the thermodynamic state (β, V, N).…”

Section: Boltzmann Entropy For Quantum Field Systemsmentioning

confidence: 81%

“…(2) with Eq. (14) and to show its potential for characterizing nonequilibrium dynamical processes including thermalization. Validation of the assumption (A1) and detailed analysis of j ′ |U k |j and ETH for some specific field systems are beyond the scope of the present study.…”

Section: Discussionmentioning

confidence: 99%

“…The present study is inspired by the author's previous work on the construction of entropy as the function of state in classical field systems [12]. Following the previous work, the construction of the Boltzmann entropy is considered in the wavevector space and operators that shift the field in the wavevector space are used.Recently,Šafránek, Deutsch and Aguirre [13,14] proposed a decomposition of the total Hilbert space based on a coarse-graining in position space to construct a Boltzmann entropy for systems of many quantum particles. (Hereafter, Refs.…”

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confidence: 99%

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Boltzmann entropy for quantum field systems

Yoshida

2020

Phys. Rev. A

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A way to construct Boltzmann entropy, i.e., the entropy as a function of microscopic pure state, for quantum field systems is proposed. Operators that shift the field in wavevector space are used in the construction. By employing an assumption, it is shown that, for almost all states in the ensemble of pure states corresponding to a thermodynamic state, the value of the proposed Boltzmann entropy coincides with that of the thermodynamic entropy for the thermodynamic state. For general self-interacting fields, the Boltzmann entropy evolves with time under Hamiltonian dynamics, so that it is capable of characterizing the thermalization of isolated quantum field systems. Rigol, Dunjko and Olshanii[8]. From thermodynamic point of view, the entropy is the function that characterize the thermalization processes. It generally increases under adiabatic thermalization processes. So it is of interest to define entropy as a function of pure state, i.e., the Boltzmann entropy, which satisfies typicality and its ensemble average coincide with the entropy defined for the ensemble, i.e., the Gibbs entropy. In VN29, the Boltzmann entropy is defined by using the abstract formalism of the decomposition of the total Hilbert space. (Note that it is different from the well-known von Neumann entropy defined for the density operator.) Although the existence of many appropriate decompositions of the total Hilbert is guaranteed, the prescription of the appropriate decompositions for specific systems were not given.The aim of this paper is to construct a Boltzmann entropy, a function of pure state, for quantum systems on lattice, or systems of quantum field, and to show that it satisfies some properties that are desirable for the entropy. The present study is inspired by the author's previous work on the construction of entropy as the function of state in classical field systems [12]. Following the previous work, the construction of the Boltzmann entropy is considered in the wavevector space and operators that shift the field in the wavevector space are used.Recently,Šafránek, Deutsch and Aguirre [13,14] proposed a decomposition of the total Hilbert space based on a coarse-graining in position space to construct a Boltzmann entropy for systems of many quantum particles. (Hereafter, Refs. [13,14] are referred to asŠDA19. ) Our study shares interest and aim withŠDA19 to some extent but the two studies developed to construct different types of Boltzmann entropy. Comparison between the two studies is given in Sec. 6. This paper is organized as follows. We start with the general formalism of constructing the Boltzmann entropy in Sec. 2. Then, we introduce a normal distribution model for ensemble of pure states in Sec. 3. Next, We give the setting of the quantum field systems in Sec. 4. After these preparations, we construct a Boltzmann entropy for the quantum field systems and examine its properties using the normal distribution model for ensemble of pure states in Sec. 5. We conclude with some discussion on the results in Sec. 6.

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Section: Boltzmann Entropy For Quantum Field Systemsmentioning

confidence: 81%

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

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Boltzmann entropy for quantum field systems

Yoshida

2020

Phys. Rev. A

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“…in each of these theories. Some authors (e.g., von Neumann (1929); Safranek et al (2017Safranek et al ( , 2018) have proposed averaging S qB (ν) with weights P ν ψ 2 , but that would yield an average entropy, not the entropy (see also Section 4.1 and our discussion of (66) in Section 9.2).…”

Section: Entropy In Quantum Mechanicsmentioning

confidence: 99%

Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics

Lebowitz

Tumulka

Zanghı̀

2020

Statistical Mechanics and Scientific Explanation

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The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.

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“…These notions are certainly related, and some of the relations are fairly clear; but some are quite obscure or ambiguous, both conceptually and mathematically. 1 Over the past several years the authors and others have developed the framework of observational entropy [1][2][3][4][5][6][7][8]. Starting as a quantum version of Boltzmann entropy, 2 observational entropy has evolved into a way to mathematically and conceptually unify many of these disparate concepts.…”

Section: Introductionmentioning

confidence: 99%

A Brief Introduction to Observational Entropy

2021

Self Cite

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In the past several years, observational entropy has been developed as both a (timedependent) quantum generalization of Boltzmann entropy, and as a rather general framework to encompass classical and quantum equilibrium and non-equilibrium coarse-grained entropy. In this paper we review the construction, interpretation, most important properties, and some applications of this framework. The treatment is self-contained and relatively pedagogical, aimed at a broad class of researchers.

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Quantum coarse-grained entropy and thermalization in closed systems

Cited by 43 publications

References 79 publications

Boltzmann entropy for quantum field systems

Boltzmann entropy for quantum field systems

Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics

A Brief Introduction to Observational Entropy

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