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AIP conference proceedings

On the approach to thermal equilibrium of macroscopic quantum systems

Sheldon Goldstein, Roderich Tumulka, Pedro L. Garrido, Joaquín Marro, F. de los Santos

Published Jan 1, 2011• 15 citations
Publisher

Abstract

In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro‐canonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E+δE. The thermal equilibrium macro‐state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is “close” to Heq. We argue that for “typical” Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann’s quantum ergodic theorem of 1929.

DOI
10.1063/1.3577619

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In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro‐canonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E+δE. The thermal equilibrium macro‐state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is “close” to Heq. We argue that for “typical” Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann’s quantum ergodic theorem of 1929.
In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro‐canonical “energy shell,” i.e., a subspace of the system’s Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E+δE. The thermal equilibrium macro‐state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is “close” to Heq. We argue that for “typical” Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann’s quantum ergodic theorem of 1929.
We consider an isolated macroscopic quantum system. Let H\mathcal{H}H be a microcanonical ``energy shell,'' i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between EEE and $E+\ensuremath{\delta}E$. The thermal equilibrium macrostate at energy EEE corresponds to a subspace Heq{\mathcal{H}}_{eq}Heq​ of H\mathcal{H}H such that dim Heq/dim H\text{dim}\text{ }{\mathcal{H}}_{eq}/\text{dim}\text{ }\mathcal{H}dim Heq​/dim H is close to 1. We say that a system with state vector $\ensuremath{\psi}∊\mathcal{H}$ is in thermal equilibrium if $\ensuremath{\psi}$ is ``close'' to Heq{\mathcal{H}}_{eq}Heq​. We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors ${\ensuremath{\psi}}_{0}$ evolve in such a way that ${\ensuremath{\psi}}_{t}$ is in thermal equilibrium for most times ttt. This result is closely related to von Neumann's quantum ergodic theorem of 1929.
We treat the problem of the approach to thermal equilibrium by only resorting to quantum dynamics of an isolated macroscopic system. Inspired by the two important works in 2009 and in 1929, we have noted that a condition we call "thermodynamic normality" for a macroscopic observable guarantees the approach to equilibrium (in the sense that a measurement of the observable at time ttt almost certainly yields a result close to the corresponding microcanonical average for a sufficiently long and typical ttt). A crucial point is that we make no assumptions on the initial state of the system, except that its energy is distributed close to a certain macroscopic value. We also present three (rather artificial) models in which the thermodynamic normality can be established, thus providing concrete examples in which the approach to equilibrium is rigorously justified. Note that this kind of results which hold for ANY initial state are never possible in classical systems. We are thus dealing with a mechanism which is peculiar to quantum systems. The present note is written in a self-contained (and hopefully readable) manner. It only requires basic knowledge in quantum physics and equilibrium statistical mechanics.
We study the nature of and approach to thermal equilibrium in isolated quantum systems. An individual isolated macroscopic quantum system in a pure or mixed state is regarded as being in thermal equilibrium if all macroscopic observables assume rather sharply the values obtained from thermodynamics. Of such a system (or state) we say that it is in macroscopic thermal equilibrium (MATE). A stronger requirement than MATE is that even microscopic observables (i.e., ones referring to a small subsystem) have a probability distribution in agreement with that obtained from the micro‐canonical, or equivalently the canonical, ensemble for the whole system. Of such a system we say that it is in microscopic thermal equilibrium (MITE). The distinction between MITE and MATE is particularly relevant for systems with many‐body localization (MBL) for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. However, if we consider superpositions of energy eigenfunctions (i.e., typical wave functions ψ) in an energy shell, then for generic macroscopic systems, including those with MBL, most ψ are in both MATE and MITE. We explore here the properties of MATE and MITE and compare the two notions, thereby elaborating on ideas introduced in . image
In the present note, we discuss a simple example of a macroscopic quantum many-body system in which the approach to thermal equilibrium from an arbitrary initial state in the microcanonical energy shell is proved without relying on any unproven assumptions. The model, which is equivalent to a free fermion chain, is designed to be a toy model for a weakly heat-conducting one-dimensional solid. We take a phenomenological point of view and perceive that the system is in thermal equilibrium when the measured coarse-grained energy distribution is uniform. The result on thermalization reported here is a variation (and an improvement) of our previous result on the irreversible expansion in a free fermion chain. As far as we know, this is the first concrete and rigorous realization of the philosophy on the foundation of equilibrium statistical mechanics proposed by von Neumann in 1929, and further developed recently by Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zangh\`ıand the present author, namely, to characterize thermal equilibrium from a macroscopic viewpoint and to make use of the strong ETH to control the long-time dynamics. This note will be the most technical part of my longer article on thermalization, "What is thermal equilibrium and how do we get there?". I am making this document public at this stage since I have already announced (and will announce) the results at some of my talks.
Abstract We say of an isolated macroscopic quantum system in a pure state ψ\psi ψ ψ that it is in macroscopic thermal equilibrium (MATE) if ψ\psi ψ ψ lies in or close to a suitable subspace Heq\mathcal {H}_\textrm{eq}Heq​ H eq of Hilbert space. It is known that every initial state ψ0\psi _0ψ0​ ψ 0 will eventually reach and stay there most of the time (“thermalize”) if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation HθfFH_\theta ^\textrm{fF}HθfF​ H θ fF of the Hamiltonian H0fFH_0^\textrm{fF}H0fF​ H 0 fF of N≫1N\gg 1N≫1 N ≫ 1 free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of H0fFH_0^\textrm{fF}H0fF​ H 0 fF . Here, we first point out that also for degenerate Hamiltonians all ψ0\psi _0ψ0​ ψ 0 thermalize if the ETH holds, i.e., if every eigenbasis lies in MATE, and we prove that this is the case for H0fFH_0^\textrm{fF}H0fF​ H 0 fF . Inspired by the fact that there is one eigenbasis of H0fFH_0^\textrm{fF}H0fF​ H 0 fF for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given H0H_0H0​ H 0 that the existence of one eigenbasis in MATE implies quite generally that most eigenbases of H0H_0H0​ H 0 lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, H=H0+λVH=H_0+\lambda VH=H0​+λV H = H 0 + λ V with λ≪1\lambda \ll 1λ≪1 λ ≪ 1 , for most perturbations V the perturbed Hamiltonian H satisfies ETH and all states thermalize.