A recent proposal for a quantum entropy S univ Q for a pure state of a system-environment “universe” is developed to encompass a much more realistic temperature bath. Microcanonical entropy is formulated in the context of the idea of a quantum microcanonical shell. The fundamental relation that holds for the classical microcanonical ensemble − T Δ S univ = Δ F sys is tested for the quantum entropy Δ S univ Q in numerical simulations. It is found that there is “excess entropy production” Δ S x due to quantum time-energy uncertainty and spreading of states in the zero-order basis. The excess entropy production is found numerically to become small as the magnitude of the system-environment coupling nears zero, as one would hope for in the limit of the classical microcanonical ensemble. The quantum microcanonical ensemble and the new “universe entropy” thereby appear as well-founded concepts poised to serve as a point of departure for time-dependent processes in which excess entropy production is physically significant.
A recent proposal for a quantum entropy S univ Q for a pure state of a system-environment “universe” is developed to encompass a much more realistic temperature bath. Microcanonical entropy is formulated in the context of the idea of a quantum microcanonical shell. The fundamental relation that holds for the classical microcanonical ensemble − T Δ S univ = Δ F sys is tested for the quantum entropy Δ S univ Q in numerical simulations. It is found that there is “excess entropy production” Δ S x due to quantum time-energy uncertainty and spreading of states in the zero-order basis. The excess entropy production is found numerically to become small as the magnitude of the system-environment coupling nears zero, as one would hope for in the limit of the classical microcanonical ensemble. The quantum microcanonical ensemble and the new “universe entropy” thereby appear as well-founded concepts poised to serve as a point of departure for time-dependent processes in which excess entropy production is physically significant.
We explore a recently introduced quantum thermodynamic entropy SunivQ of a pure state of a composite system-environment computational "universe" with a simple system S coupled to a constant temperature bath E. The principal focus is "excess entropy production" in which the quantum entropy change is greater than expected from the classical entropy-free energy relationship. We analyze this in terms of quantum spreading of time dependent states, and its interplay with the idea of a microcanonical shell. The entropy takes a basis-dependent Shannon information definition. We argue for the zero-order SE energy basis as the unique choice that gives classical thermodynamic relations in the limit of weak coupling and high density of states, including an exact division into system and environment components. Entropy production takes place due to two kinds of processes. The first is classical "ergodization" that fills the full density of states within the microcanonical shell. The second is excess entropy production related to quantum spreading or "quantum ergodization" of the wavepacket that effectively increases the width of the energy shell. Lorentzian superpositions with finite microcanonical shell width lead to classical results as the limiting case, with no excess entropy. We then consider a single SE zero-order initial state, as the examplar of extreme excess entropy production. Systematic formal results are obtained for a unified treatment of excess entropy production for time-dependent Lorentzian superpositions, and verified computationally. It is speculated that the idea of free energy might be extended to a notion of "available energy" corresponding to the excess entropy production. A unified perspective on quantum thermodynamic entropy is thereby attained from the classical limit to extreme quantum conditions.
We take the view that the standard von Neumann definition, in which the entropy SvN of a pure state is zero, is in evident conflict with the statement of the second law that the entropy of the universe Suniv increases in spontaneous processes, $ ΔS_{univ} > 0$. Here we seek an alternative entropy of the universe Suniv that is in accord with the second law, in a spirit not dissimilar to von Neumann himself in lesser-known work. We perform simulations of time dependent dynamics for a previously developed \cite{polyadbath} model quantum system becoming entangled with a quantum environment. We test the new definition of the entropy of the system-environment "universe" against the standard thermodynamic relation $ ΔF_{sys} = - T ΔS_{univ} $, calculating system properties using the reduced density matrix and standard von Neumann entropy. Good agreement is obtained, showing the compatibility of an entropy for a pure state of a universe with the statement of the second law and the concept of free energy. Interesting deviation from microcanonical behavior within the zero order energy shell is observed in a context of effectively microcanonical behavior within the much larger total basis of the time dependent universe.
Recent theoretical and experimental work on the foundations of statistical mechanics and thermodynamics has shown that quantum pure states typically evolve to thermal equilibrium. We focus on formulating the second law of thermodynamics for these states. The standard quantum von Neumann quantum entropy is constant during equilibration, S^{vN}=0, in apparent conflict with the entropy increase of the second law, ΔS_{univ} > 0. We explore a recently developed entropy S_{univ}^Q for a pure state and test its behavior in simulations of a model system and environment evolving in time with heat flow to equilibrium. We find that the entropy approaches the correct classical value in a type of classical limit with weak coupling. With stronger coupling, we find a new source of quantum "excess entropy production," which has its origin in the quantum spreading of the wavepacket. Are there quantum thermodynamic effects related to this new source of entropy? To test this, we developed a model for a small variable temperature quantum oscillator bath. We performed simulations where two small baths are linked by a system, with unequal couplings to the baths. The model evolves to a novel type of equilibrium state with unequal temperatures in the baths, with heat flow from cold to hot along the path to equilibrium. We give an account of this behavior in terms of the second law with the quantum entropy S_{univ}^Q. The new formulation of the second law thus appears well-founded and fruitful, with surprising new quantum thermodynamic effects that may still be awaiting discovery, for example in molecules or systems far from equilibrium. \n\nThis dissertation includes previously published and unpublished co-authored material.
Bridging the second law of thermodynamics and microscopic reversible dynamics has been a longstanding problem in statistical physics. Here, we address this problem on the basis of quantum many-body physics, and discuss how the entropy production saturates in isolated quantum systems under unitary dynamics. First, we rigorously prove that the entropy production does indeed saturate in the long time regime, even when the total system is in a pure state. Second, we discuss the non-negativity of the entropy production at saturation, implying the second law of thermodynamics. This is based on the eigenstate thermalization hypothesis, which states that even a single energy eigenstate is thermal. We also numerically demonstrate that the entropy production saturates at a non-negative value even when the initial state of a heat bath is a single energy eigenstate. Our results reveal fundamental properties of the entropy production in isolated quantum systems at late times.
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.
A consistent theory of non-equilibrium thermodynamics for Markovian open quantum systems has been developed in the late seventies in analogy with Classical Irreversible Thermodynamics. The time-evolution of these open systems is usually described by means of effective master equations in Lindblad form, that turn out to be reliable when there is a separation of time-scales between system and environment, such that memory effects are negligible and the so-called Markovian approximation is justified. In this framework, the variations of energy and entropy in the system are consistently described, distinguishing between heat and work contributions and providing a statement of the second law of thermodynamics as positivity of the entropy production.\nHowever, there is empirical evidence that many physical systems like photosynthetic complexes,\nopto-mechanical resonators and superconducting qubits, just to mention a few, experience more general non-Markovian dynamics. The formulation of the laws of thermodynamics in a non-Markovian setting is matter of current research and represents the main topic of the present work. As a first step, we show that the entropy production defined as in the Markovian case can be negative for a class of non-Markovian dynamics. We argue that this outcome should not be interpreted as a violation of the second law of thermodynamics because the environment must be explicitly taken into account in the balance of entropy in the non-Markovian setting. In order to justify this claim we adopt a more general point of view, studying a closed bipartite quantum system, such that each of the two subsystems plays the role of a finite environment out of equilibrium for the other one. We concentrate on the balance of energy first and construct an effective Hamiltonian for each subsystem using physically reasonable requirements; then we define heat and work as in the standard Markovian treatment, with the effective Hamiltonian replacing the free Hamiltonian. It turns out that, in our framework, the work power is perfectly balanced between subsystems, while the correlations can store a part of energy locally inaccessible and exchange it with both subsystems in the form of heat.\nConcerning the balance of entropy, a quite general formulation of the second law of thermodynamics can be given as follows: under the assumption of a factorized initial state for the compound system, the sum of the total variations of the entropies in the two subsystems is always nonnegative. We show with an explicit example that this general formulation does not correspond to the statement presented in the framework of Markovian master equations, which should not be considered a priori the second law of thermodynamics.\nIn the last part of the thesis we concentrate of the so-called fluctuation relations, that are results extending the thermodynamic formalism beyond the behavior of average quantities. After reviewing the main theoretical outcomes, such as the Jarzynski equality and the Crooks fluctuation theorem, we describe a proposal to access experimentally the work performed on an ensemble of diatomic molecules by a time-dependent electric field coupled with their vibrational degree of freedom. This procedure could then be used to test the quantum Jarzynski equality. With respect to the results so far appeared in the literature, in which the left-hand side of the equality is inferred from an experiment and the right-hand side is computed according to a model, in our proposed setting we should be able to estimate from the experiment both the left-hand side and the right-hand side of the equality, independently.