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arXiv:2404.04533

Heat flows from hot to cold: A simple rigorous example of thermalization in an isolated macroscopic quantum system

Hal Tasaki

Published Apr 6, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2404.04533

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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.
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.
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.
Focusing on isolated macroscopic systems, described either in terms of a quantum mechanical or a classical model, our two key questions are: In how far does an initial ensemble (usually far from equilibrium and largely unknown in detail) evolve towards a stationary long-time behavior ("equilibration")? In how far is this steady state in agreement with the microcanonical ensemble as predicted by Statistical Mechanics ("thermalization")? In the first part of the paper, a recently developed quantum mechanical treatment of the problem is briefly summarized, putting particular emphasis on the realistic modeling of experimental measurements and non-equilibrium initial conditions. Within this framework, equilibration can be proven under very weak assumptions about those measurements and initial conditions, while thermalization still requires quite strong additional hypotheses. In the second part, an analogous approach within the framework of classical mechanics is developed and compared with the quantum case. In particular, the assumptions to guarantee classical equilibration are now rather strong, while thermalization then follows under relatively weak additional conditions.
By applying the theory recently developed by Roos, Sugimoto, Teufel, Tumulka, and Vogel [1], we construct a rigorous example of an isolated macroscopic quantum system in which every initial state chosen from a microcanonical energy shell exhibits thermalization solely by the unitary time evolution determined by the Hamiltonian. We consider the Hamiltonian H^L\hat{H}_LH^L​ of the standard two-dimensional ferromagnetic Ising model with the plus boundary conditions and perturb it with a small self-adjoint operator λLV^Lλ_L\hat{V}_LλL​V^L​ drawn randomly from the space of self-adjoint operators on the whole Hilbert space. The Roos-Sugimoto-Teufel-Tumulka-Vogel theory then guarantees that, for all but an exponentially small fraction of the random perturbations, the model exhibits strong ETH (energy eigenstate thermalization hypothesis), i.e., every energy eigenstate in a microcanonical energy shell is in thermal equilibrium in the sense of large deviation, as precisely stated in the main text. It is then standard that the strong ETH implies the following result for thermalization. Suppose that the initial state Ψ(0)⟩Ψ(0)\rangleΨ(0)⟩ is chosen from the microcanonical energy shell and may be very far from thermal equilibrium. Then the time-evolved state ∣Ψ(t)⟩=e−i(H^L+λLV^L)t∣Ψ(0)⟩|Ψ(t)\rangle=e^{-i(\hat{H}_L+λ_L\hat{V}_L)t}|Ψ(0)\rangle∣Ψ(t)⟩=e−i(H^L​+λL​V^L​)t∣Ψ(0)⟩ is in thermal equilibrium at sufficiently long and typical times ttt, in the sense that the measurement result of the magnetization density almost certainly coincides with the corresponding spontaneous magnetization. When we state ``typical'' and ``almost certainly'' above, we mean that the corresponding exceptional fractions are exponentially small in the system size LLL.