An exact stochastic model for the thermalization of quantum states is proposed. The model has various physically appealing properties. The dynamics are characterized by an underlying Schrödinger evolution, together with a nonlinear term driving the system toward an asymptotic equilibrium state and a stochastic term reflecting fluctuations. There are two free parameters, one of which can be identified with the heat bath temperature, while the other determines the characteristic time scale for thermalization. Exact expressions are derived for the evolutionary dynamics of the system energy, the system entropy, and the associated density operator.
An exact stochastic model for the thermalization of quantum states is proposed. The model has various physically appealing properties. The dynamics are characterized by an underlying Schrödinger evolution, together with a nonlinear term driving the system toward an asymptotic equilibrium state and a stochastic term reflecting fluctuations. There are two free parameters, one of which can be identified with the heat bath temperature, while the other determines the characteristic time scale for thermalization. Exact expressions are derived for the evolutionary dynamics of the system energy, the system entropy, and the associated density operator.
Thermalization has been shown to occur in a number of closed quantum many-body systems, but the description of the actual thermalization dynamics is prohibitively complex. Here, we present a model-in one and two dimensions-for which we can analytically show that the evolution into thermal equilibrium is governed by a Fokker-Planck equation derived from the underlying quantum dynamics. Our approach does not rely on a formal distinction of weakly coupled bath and system degrees of freedom. The results show that transitions within narrow energy shells lead to a dynamics which is dominated by entropy and establishes detailed balance conditions that determine both the eventual equilibrium state and the nonequilibrium relaxation to it.
During the evolution of coupled nonlinear oscillators on a lattice, with dynamics dictated by the discrete nonlinear Schrödinger equation (DNLSE systems), two quantities are conserved: system energy (Hamiltonian) and system density (number of particles). If the number of system oscillators is large enough, a significant portion of the array can be considered to be an "open system", in intimate energy and density contact with a "bath" - the rest of the array. Thus, as indicated in previous works, the grand canonical formulation can be exploited in order to determine equilibrium statistical properties of thermalized DNLSE systems. In this work, given the values of the two conserved quantities, we have calculated the necessary values of the two Lagrange parameters (typically designated β,μ) associated with the grand canonical partition function in two different ways. One is numerical and the other is analytic, based on a published approximate entropy expression. In addition we have accessed a purposely-derived approximate PDF expression of site-densities. Applying these mathematical tools we have generated maps of temperatures, chemical potentials, and field correlations for DNLSE systems over the entire thermalization zone of the DNLSE phase diagram, subjected to all system-nonlinearity levels. The end result is a rather complete picture, characterizing equilibrated large DNLSE systems.
Thermodynamics is a phenomenological theory describing the energy conversion of work and heat. At its origins thermodynamics was developed in order to understand and improve heat engines. In conventional thermodynamics, however, only such processes are completely describable which are slow enough to keep the system of interest in an equilibrium state with its thermal environment at all times. On the contrary, all real physical processes are accompanied by non-equilibrium phenomena. These are mathematically described with the help of the irreversible entropy production, i.e. the dissipated work divided by the temperature of the bath. The second law of thermodynamics merely states that the irreversible entropy production is on average always positive. The modern trend of miniaturization, however, leads to smaller and smaller devices. On short length scales thermal noise as well as quantum fluctuations become important. Thus, usual thermodynamic quantities as work and heat acquire stochastic nature. Moreover, in the quantum regime a completely new theory had to be invented, since classical notions of work and heat are no longer valid. The present dissertation contributes to this prevailing field by the derivation of analytical expressions for the entropy production in open and closed quantum system far from thermal equilibrium. The theoretical treatment is motivated by an experimental point of view. The obtained results were derived from the reduced dynamics of the quantum mechanical system, which may be coupled to a thermal bath. To this end, it was dealt with methods and approaches of statistical physics, conventional thermodynamics, quantum information theory and the theory of open quantum systems. In this dissertation the most important physical systems are treated. First, an isolated system is considered before later on a heat bath is included. The limit of weak as well as of strong coupling are analyzed in detail. For isolated systems it turns out that a geometric approach is able to capture the thermodynamic properties, completely. To this end, the irreversible entropy production is sharply estimated from below by the angle between the real non-equilibrium state and the corresponding equilibrium one. Further, it is shown that the Heisenberg uncertainty relation for energy and time can be formulated more precisely in terms of that angle. As experimental system cold ion traps are introduced, in which the theoretical predictions can be verified. For weakly coupled systems, then, the irreversible entropy production is identified and a fluctuation theorem is derived by merely thermodynamic arguments. As experimental system we, again, propose the cold ion traps, where the heat bath can be simulated with the help of LASER light. In the limit of strong coupling a semiclassical description of the reduced quantum system is discussed. Again, an analytically exact expression for the entropy production and the corresponding fluctuation theorem is derived. A physical system, in which strong coupling to the environment can be analyzed, is given by Josephson junctions. This dissertation discusses the entropy production for isolated, weakly coupled, and overdamped quantum systems. For these physical regimes analytical relation are derived and experimental systems proposed, with which the predictions can be verified.
This thesis fathoms out the capabilities of the theory of quantum mechanics to explain thermodynamic behavior. It covers in particular equilibration and thermalization in closed quantum systems, typicality, time scales for equilibration, quantum integrability and its connection to thermalization, decoherence, and a maximum entropy principle. Together, the presented results form the body of the theory of pure state quantum statistical mechanics. With almost 300 references, ranging from the groundbreaking works of the early 20th century to the most recent discoveries (up to 2013), this work arguably constitutes the most comprehensive review of the literature on equilibration and thermalization in closed quantum systems. All results are presented in a unified notation and many are slightly strengthened or generalized.
A general thermal machine is characterized by distinct time or energy scales: temperature, coupling to the heat baths, internal free dynamics and interactions, and external driving. We address the question: what parameters determine the speed of a given engine, and how can they be used to reliably compare different types of engine? Specifically, we consider four realizations of the Otto engine, each operating with the same Otto efficiency, and aim to characterize and compare their power outputs. The main insight of this paper is that, irrespective of their very different dynamical implementations, the power of each engine can be factorized into a common characteristic work and an implementation-specific characteristic time. The characteristic work depends only on the internal frequencies of the machine and the temperatures, whereas the characteristic time depends on the coupling strength to the baths and on the implementation-specific driving. This observation allows us to compare the power of different engines through their characteristic times and thereby reduce the relevant parameter space to parameters that capture the dynamical aspects of engine performance. Despite different dynamical implementations, this framework reveals simple common features. In all cases, the maximum operating speed is limited by a common timescale given by the sum of the characteristic thermalization times of the hot and cold baths. Moreover, the four engines fall into two distinct asymptotic classes: implementations in which driving and thermalization occur simultaneously exhibit quadratic scaling, whereas those in which work extraction and thermalization alternate exhibit linear scaling. These results expose general dynamical features that determine the operational speed of quantum Otto engines.
Quantum thermalization describes how closed quantum systems can effectively reach thermal equilibrium, resolving the apparent incongruity between the reversibility of Schrödinger's equation and the second law of thermodynamics. Despite its ubiquity and conceptual significance, the precise conditions that give rise to quantum thermalization are still not well understood. After nearly a century of efforts, we have yet to find a complete mathematical proof that an effective statistical description naturally emerges the underlying quantum dynamics in generic settings. Here, we prove that quantum thermalization must occur in any qubit system with local interactions under three conditions: (i) high effective temperature, (ii) translation invariance, and (iii) no perfect resonances in the energy spectrum. Specifically, we show that a typical, low-complexity pure state drawn from any ensemble with large entropy and well-defined effective temperature becomes locally indistinguishable from a Gibbs state upon unitary evolution. In this setting, our rigorous results prove the widely anticipated notion that statistical physics should be understood as an emergent phenomenon, explicitly derived from the first principles of quantum mechanics.