Macroscopic and microscopic thermal equilibrium (original) (raw)

Thermal Equilibrium of a Macroscopic Quantum System in a Pure State

Physical Review Letters, 2015

We consider the notion of thermal equilibrium for an individual closed macroscopic quantum system in a pure state, i.e., described by a wave function. The macroscopic properties in thermal equilibrium of such a system, determined by its wave function, must be the same as those obtained from thermodynamics, e.g., spatial uniformity of temperature and chemical potential. When this is true we say that the system is in macroscopic thermal equilibrium (MATE). Such a system may however not be in microscopic thermal equilibrium (MITE). The latter requires that the reduced density matrices of small subsystems be close to those obtained from the microcanonical, equivalently the canonical, ensemble for the whole system. 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. We note however that for generic macroscopic systems, including those with MBL, most wave functions in an energy shell are in both MATE and MITE. For a classical macroscopic system, MATE holds for most phase points on the energy surface, but MITE fails to hold for any phase point.

On the approach to thermal equilibrium of macroscopic quantum systems

Nucleation and Atmospheric Aerosols, 2011

We consider an isolated, macroscopic quantum system. Let 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 E and E + δE. The thermal equilibrium macro-state at energy E corresponds to a subspace H eq of H such that dim H eq / dim H is close to 1. We say that a system with state vector ψ ∈ H is in thermal equilibrium if ψ is "close" to H eq. We show that for "typical" Hamiltonians with given eigenvalues, all initial state vectors ψ 0 evolve in such a way that ψ t is in thermal equilibrium for most times t. This result is closely related to von Neumann's quantum ergodic theorem of 1929.

Approach to thermal equilibrium of macroscopic quantum systems

2010

We consider an isolated, macroscopic quantum system. Let 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 E and E + δE. The thermal equilibrium macro-state at energy E corresponds to a subspace H eq of H such that dim H eq / dim H is close to 1. We say that a system with state vector ψ ∈ H is in thermal equilibrium if ψ is "close" to H eq . We show that for "typical" Hamiltonians with given eigenvalues, all initial state vectors ψ 0 evolve in such a way that ψ t is in thermal equilibrium for most times t. This result is closely related to von Neumann's quantum ergodic theorem of 1929.

Quantum thermodynamics of single particle systems

Scientific Reports, 2020

thermodynamics is built with the concept of equilibrium states. However, it is less clear how equilibrium thermodynamics emerges through the dynamics that follows the principle of quantum mechanics. in this paper, we develop a theory of quantum thermodynamics that is applicable for arbitrary small systems, even for single particle systems coupled with a reservoir. We generalize the concept of temperature beyond equilibrium that depends on the detailed dynamics of quantum states. We apply the theory to a cavity system and a two-level system interacting with a reservoir, respectively. The results unravels (1) the emergence of thermodynamics naturally from the exact quantum dynamics in the weak system-reservoir coupling regime without introducing the hypothesis of equilibrium between the system and the reservoir from the beginning; (2) the emergence of thermodynamics in the intermediate system-reservoir coupling regime where the Born-Markovian approximation is broken down; (3) the breakdown of thermodynamics due to the long-time non-Markovian memory effect arisen from the occurrence of localized bound states; (4) the existence of dynamical quantum phase transition characterized by inflationary dynamics associated with negative dynamical temperature. the corresponding dynamical criticality provides a border separating classical and quantum worlds. The inflationary dynamics may also relate to the origin of big bang and universe inflation. And the third law of thermodynamics, allocated in the deep quantum realm, is naturally proved. In the past decade, many efforts have been devoted to understand how, starting from an isolated quantum system evolving under Hamiltonian dynamics, equilibration and effective thermodynamics emerge at long times 1-5. On the other hand, the investigations of open quantum systems initiate interests on the issue of quantum thermodynamics taking place under the quantum evolution of open systems 6-20. The questions of how thermodynamics emerges from quantum dynamics, how do quantum systems dynamically equilibrate and thermalize, and whether thermalization is always reachable in quantum regime, are central and fundamental to research for quantum thermodynamics. However, a general theory of quantum thermodynamics that has conceptually no ambiguity in answering the above questions has not yet been obtained, because investigations in addressing above questions inevitably take various assumptions and approximations. In this paper, we will attempt to answer these questions by rigorously solving the quantum dynamics based on the exact master equation we developed recently for a large class of open quantum systems 21-25. Recall that thermodynamics is built with the hypothesis of equilibrium 26. Macroscopic systems at equilibrium are fully described by the relation between the internal energy E and a set of other extensive parameters, the entropy S, the volume V, and the particle number N i of different components i = 1, 2, ••• , magnetic moment M, etc.,

Emergence of equilibrium thermodynamic properties in quantum pure states. I. Theory

2009

Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and the dynamical aspects in the description of isolated quantum systems. Like in the classical statistical mechanics, the equilibrium distribution of any property is identified on the basis of the time evolution of the considered system. As a consequence equilibrium properties of quantum system appear to depend on the details of the initial state due to the abundance of constants of the motion in ...

On the Distribution of the Wave Function for Systems in Thermal Equilibrium

Journal of Statistical Physics, 2006

For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution µ of its wave function, a normalized vector belonging to its Hilbert space H . While ρ itself does not determine a unique µ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which µ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in H , denoted GAP (ρ). We do this using a suitable projection of the Gaussian measure on H with covariance ρ. We establish some nice properties of GAP (ρ) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix ρ β = (1/Z) exp(−βH). GAP (ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on H are often used.

The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace

2014

We study the problem of the approach to equilibrium in a macroscopic quantum system in an abstract setting. We prove that, for a typical choice of "nonequilibrium subspace", any initial state (from the energy shell) thermalizes, and in fact does so very quickly, on the order of the Boltzmann time τ B := h/(k B T). This apparently unrealistic, but mathematically rigorous, conclusion has the important physical implication that the moderately slow decay observed in reality is not typical in the present setting. The fact that macroscopic systems approach thermal equilibrium may seem puzzling, for example, because it may seem to conflict with the time-reversibility of the microscopic dynamics. According the present result, what needs to be explained is, not that macroscopic systems approach equilibrium, but that they do so slowly. Mathematically our result is based on an interesting property of the maximum eigenvalue of the Hadamard product of a positive semi-definite matrix and a random projection matrix. The recent exact formula by Collins for the integral with respect to the Haar measure of the unitary group plays an essential role in our proof.

Dynamics of an elementary quantum system in environments out of thermal equilibrium

EPL (Europhysics Letters), 2012

We study the internal dynamics of an elementary quantum system placed close to a body held at a temperature different from that of the surrounding radiation. We derive general expressions for lifetime and density matrix valid for bodies of arbitrary geometry and dielectric permittivity. Out of equilibrium, the thermalization process and steady states become both qualitatively and quantitatively significantly different from the case of radiation at thermal equilibrium. For the case of a three-level atom close to a slab of finite thickness, we predict the occurrence of population inversion and an efficient cooling mechanism for the quantum system, whose effective internal temperature can be driven to values much lower than both involved temperatures. Our results show that non-equilibrium configurations provide new promising ways to control the state of an atomic system.

Information-theoretic equilibrium and observable thermalization

A crucial point in statistical mechanics is the definition of the notion of thermal equilibrium, which can be given as the state that maximises the von Neumann entropy, under the validity of some constraints. Arguing that such a notion can never be experimentally probed, in this paper we propose a new notion of thermal equilibrium, focused on observables rather than on the full state of the quantum system. We characterise such notion of thermal equilibrium for an arbitrary observable via the maximisation of its Shannon entropy and we bring to light the thermal properties that it heralds. The relation with Gibbs ensembles is studied and understood. We apply such a notion of equilibrium to a closed quantum system and show that there is always a class of observables which exhibits thermal equilibrium properties and we give a recipe to explicitly construct them. Eventually, an intimate connection with the Eigenstate Thermalisation Hypothesis is brought to light. To understand under which conditions thermodynamics emerges from the microscopic dynamics is the ultimate goal of statistical mechanics. However, despite the fact that the theory is more than 100 years old, we are still discussing its foundations and its regime of applicability. The ordinary way in which thermal equilibrium properties are obtained, in statistical mechanics, is through a complete characterisation of the thermal form of the state of the system. One way of deriving such form is by using Jaynes principle 1-4 , which is the constrained maximisation of von Neumann entropy S vN = − Trρ logρ. Jaynes showed that the unique state that maximises S vN (compatibly with the prior information that we have on the system) is our best guess about the state of the system at the equilibrium. The outcomes of such procedure are the so-called Gibbs ensembles. In the following we argue that such a notion of thermal equilibrium, de facto is not experimentally testable because it gives predictions about all possible observables of the system, even the ones which we are not able to measure. To overcome this issue, we propose a weaker notion of thermal equilibrium, specific for a given observable. The issue is particularly relevant for the so-called "Pure states statistical mechanics" 5-19 , which aims to understand how and in which sense thermal equilibrium properties emerge in a closed quantum system, under the assumption that the dynamic is unitary. In the last fifteen years we witnessed a revival of interest in these questions , mainly due to remarkable progresses in the experimental investigation of isolated quantum systems 20-25. The high degree of manipulability and isolation from the environment that we are able to reach nowadays makes possible to experimentally investigate such questions and to probe the theoretical predictions. The starting point of Jaynes' derivation of statistical mechanics is that S vN is a way of estimating the uncertainty that we have about which pure state the system inhabits. Unfortunately we know from quantum information theory that it does not address all kind of ignorance we have about the system. Indeed, it is not the entropy of an observable (though the state is observable); its conceptual meaning is not tied to something that we can measure. This issue is intimately related with the way we acquire information about a system, i.e. via measurements. The process of measuring an observable on a quantum system allows to probe only the diagonal part of the density matrix λ ρ λ i i , when this is written in the observable eigenbasis λ { } i. For such a reason, from the experimental point of view, it is not possible to assess whether a many-body quantum system is at thermal equilibrium (e.g. Gibbs state ρ G): the number of observables needed to probe all the density matrix elements is too big. In any experimentally reasonable situation we have access only to a few (sometimes just one or two) observables. It is 1 tomic ann aser sicss arennon aaoratorr niiersitt of OOforr arrs oaa OOforr O1 333. Centre for uantum eccnooooiess ationaa niiersitt of innaporee 1177433 innapore. 3 Department of Physics, National niiersitt of innapore cience Driie 3 1171 innapore. 4 Center for Quantum Information, Institute for Interiscippinar Information ciences sinua niiersit 1000844 eiiin ina. * These authors contributed eeua to tis wor. orresponence an reeuests for materias souu e aressee to .. emai: faio.ana pppsics.oo.ac.uuu receiiee: 13 Octooer 016 acceptee: 31 anuarr 017 Puuuissee: 07 arcc 017 OPEN

Degenerate observables and the many Eigenstate Thermalization Hypotheses

2018

Under unitary time evolution, expectation values of physically reasonable observables often evolve towards the predictions of equilibrium statistical mechanics. The eigenstate thermalization hypothesis (ETH) states that this is also true already for individual energy eigenstates. Here we aim at elucidating the emergence of ETH for observables that can realistically be measured due to their high degeneracy, such as local, extensive or macroscopic observables. We bisect this problem into two parts, a condition on the relative overlaps and one on the relative phases between the eigenbases of the observable and Hamiltonian. We show that the relative overlaps are completely unbiased for highly degenerate observables and demonstrate that unless relative phases conspire to cumulative effects this makes such observables verify ETH. Through connecting the degeneracy of observables and entanglement of the energy eigenstates this result elucidates potential pathways to equilibration in a fully general way. "Pure state quantum statistical mechanics"[1-5] aims at understanding under which conditions the use of tools from statistical mechanics can be justified based on the first principles of standard quantum mechanics with as few extra assumptions as possible. To explain the emergence of thermalization it combines three approaches: Typicality arguments [6-13], the dynamical equilibration approach [14-21] and the Eigenstate Thermalization Hypothesis (ETH) [22-34]. According to the first one, systems appear to be in equilibrium because, in a precise sense, most states are in equilibrium. Alternatively, according to the second approach apparent equilibration of observables and whole subsystems emerges because initial states of large many-body systems overlap with many energy eigenstates and therefore explore a large part of Hilbert space during their evolution, almost all the while being almost indistinguishable from a static equilibrium state. ETH, on the other hand, is a hypothesis about properties of individual eigenstates of sufficiently complicated quantum many-body systems which was suggested by various results in quantum chaos theory and it ad-duces the appearance of thermalization during such equi-libration to an underlying chaotic behavior. The basic idea is that, for large system sizes and in sufficiently complicated quantum many-body systems, the energy eigen-states can be so entangled that when we look at their overlaps with the basis of a physical observable they can be effectively described by random variables. If the ETH is fulfilled, it guarantees thermalization whenever equi-libration happens because of the mechanisms described above. Depending on how broad one wants the class of initial states that thermalize to be, the fulfillment of the ETH is also a necessary criterion for thermalization [5, 35]. The ETH is sometimes criticized for its lack of pre-dictive power, as it leaves open at least three important questions: what precisely are "physical observables"; what makes a system "sufficiently complicated" to expect that ETH applies; how long will it take for such observ-ables to reach thermal expectation values [18, 21]. For this reason, a lot of effort has been focused on numerical investigations that validate the ETH in specific Hamilto-nian models and for various observables, often including local ones. The ETH is generally found to hold in non-integrable systems that are not many-body localized and equilibration towards thermal expectation values usually happens on reasonable times scales [18, 20, 21, 34]. Recently [36] it has been shown that for any Hamilto-nian there is always a large number of observables which satisfy ETH. They have been dubbed "Hamiltonian Unbi-ased Observables" (HUO) and admit an algorithmic construction. Unfortunately this still leaves open when concrete physically relevant observables satisfy the ETH. In this letter we make progress in this direction. Building on the connection between HUOs and ETH, we present a theorem which can be used as a tool to investigate the emergence of ETH. In order to show how it can be used, we present three applications: local observables, extensive observables, and macro-observables. We will give precise definitions for each of them later. The paper is organized as follows. First we setup the notation, recall different formulations of the ETH and clarify which one we will be using throughout the paper. We continue with a brief digression on physical observables and degeneracies and recall the concepts of Hamiltonian unbiased basis and observables. We then present our main result, which elucidates the question under which conditions highly degenerate observables are HUO and discuss consequences of it for local observ-ables, extensive observables, and a certain type of macro