Emergence of thermodynamic properties in quantum pure states. I. Foundations (original) (raw)
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2009
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin syst...
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.,
The second law of thermodynamics for pure quantum states
Eprint Arxiv 1303 6393, 2013
A version of the second law of thermodynamics states that one cannot lower the energy of an isolated system by a cyclic operation. We prove this law without introducing statistical ensembles and by resorting only to quantum mechanics. We choose the initial state as a pure quantum state whose energy is almost E_0 but not too sharply concentrated at energy eigenvalues. Then after an arbitrary unitary time evolution which follows a typical "waiting time", the probability of observing the energy lower than E_0 is proved to be negligibly small.
AIP Conference Proceedings, 2002
The Kullback-Leibler inequality is a way of comparing any two density matrices. A technique to set up the density matrix for a physical system is to use the maximum entropy principle, given the entropy as a functional of the density matrix, subject to known constraints. In conjunction with the master equation for the density matrix, these two ingredients allow us to formulate the second law of thermodynamics in its widest possible setting. Thus problems arising in both quantum statistical mechanics and quantum information can be handled. Aspects of thermodynamic concepts such as the Carnot cycle will be discussed. A model is examined to elucidate the role of entanglement in the Landauer erasure problem.
Quantum Mechanics as a Classical Theory XI: Thermodynamics and Equilibrium
arXiv: Quantum Physics, 1996
In this continuation paper the theory is further extended to reveal the connection between its formal aparatus, dealing with microscopic quantities, and the formal aparatus of thermodynamics, related to macroscopic properties of large systems. We will also derive the Born-Sommerfeld quantization rules from the formalism of the infinitesimal Wigner-Moyal transformations and, as a consequence of this result, we will also make a connection between the later and the path integral approach of Feynman. Some insights of the relation between quantum mechanics and equilibrium states will be given as a natural development of the interpretation of the above results.
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.
Perspective on quantum thermodynamics
2016
Classical thermodynamics is unrivalled in its range of applications and relevance to everyday life. It enables a description of complex systems,made up ofmicroscopic particles, in terms of a small number ofmacroscopic quantities, such aswork and entropy. As systems get ever smaller, fluctuations of these quantities become increasingly relevant, prompting the development of stochastic thermodynamics. Recently we have seen a surge of interest in exploring the quantum regime, where the origin offluctuations is quantum rather than thermal.Many questions, such as the role of entanglement and the emergence of thermalisation, lie wide open. Answering these questionsmay lead to the development of quantumheat engines and refrigerators, as well as to vitally needed simple descriptions of quantummany-body systems.
Quantum thermodynamics of a single particle system
arXiv (Cornell University), 2018
Classical thermodynamics is built with the concept of equilibrium states. However, it is less clear how equilibrium thermodynamics emerges through dynamics that follows the principle of quantum mechanics. In this paper, we develop a theory to study nonequilibrium thermodynamics of quantum systems which is applicable to arbitrary small systems, even for a single particle system, in contact with a reservoir. We generalize the concept of temperature beyond equilibrium that depends on the details of quantum states of the system and their dynamics. This nonequilibrium theory for quantum thermodynamics unravels (1) the emergence of classical thermodynamics from quantum dynamics of a single particle system in the weak system-reservoir coupling regime, without introducing any hypothesis on equilibrium state; (2) the breakdown of classical thermodynamics in the strong coupling regime, induced by non-Markovian memory dynamics; and (3) the occurrence of negative temperature associated with a dynamical quantum phase transition. The corresponding dynamical criticality provides the border separating the classical and quantum thermodynamics, and it may also reveal the origin of universe inflation. The third law of thermodynamics, allocated in the deep quantum realm, is also proved in this theory.
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.
Origin of the Canonical Ensemble: Thermalization with Decoherence
Journal of the Physical Society of Japan, 2009
We solve the time-dependent Schrödinger equation for the combination of a spin system interacting with a spin bath environment. In particular, we focus on the time development of the reduced density matrix of the spin system. Under normal circumstances we show that the environment drives the reduced density matrix to a fully decoherent state, and furthermore the diagonal elements of the reduced density matrix approach those expected for the system in the canonical ensemble. We show one exception to the normal case is if the spin system cannot exchange energy with the spin bath. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the canonical ensemble is a state that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics.