Abstract The 2nd law of thermodynamics is derived from the principle of least action, positing that the quantum of action is the indivisible and indestructible basic building block of everything. On their least-time paths to balance, the quanta move from the system to its surroundings, or vice versa, so that the kinetic, potential, and dissipated energy tally. When re-expressed in logarithmic terms, this current toward more probable states with decreasing free energy equates to the principle of increasing entropy, the 2nd law of thermodynamics, including path-independent dynamic and path dependent geometric phase shifts. Despite being exact, the equation of evolution to entropy maximum, equivalent to free energy minimum, cannot be solved because evolution, consuming its own driving forces, becomes path dependent. Thus, the future remains open within free energy bounds. As discussed, the entropy derived from the statistical physics of open quantum systems sums states distinguishable in energy; whereas, Boltzmann’s entropy enumerates microstates indistinguishable in energy. Consequently, the statistical physics of open systems differs from that of closed systems: The irreversible evolution in the state space toward thermodynamic balance contrasts with the steady-state revolution in phase space between conceivable configurations. This concrete comprehension explains, among other things, that increasing disorder is not a law of nature itself but a consequence of the law to attain balance with incoherent surroundings in the least time.
Abstract The 2nd law of thermodynamics is derived from the principle of least action, positing that the quantum of action is the indivisible and indestructible basic building block of everything. On their least-time paths to balance, the quanta move from the system to its surroundings, or vice versa, so that the kinetic, potential, and dissipated energy tally. When re-expressed in logarithmic terms, this current toward more probable states with decreasing free energy equates to the principle of increasing entropy, the 2nd law of thermodynamics, including path-independent dynamic and path dependent geometric phase shifts. Despite being exact, the equation of evolution to entropy maximum, equivalent to free energy minimum, cannot be solved because evolution, consuming its own driving forces, becomes path dependent. Thus, the future remains open within free energy bounds. As discussed, the entropy derived from the statistical physics of open quantum systems sums states distinguishable in energy; whereas, Boltzmann’s entropy enumerates microstates indistinguishable in energy. Consequently, the statistical physics of open systems differs from that of closed systems: The irreversible evolution in the state space toward thermodynamic balance contrasts with the steady-state revolution in phase space between conceivable configurations. This concrete comprehension explains, among other things, that increasing disorder is not a law of nature itself but a consequence of the law to attain balance with incoherent surroundings in the least time.
Basing his work on a new formulation of thermodynamics called the Unified Quantum Theory of Mechanics and Thermodynamics first published in a series of four ground breaking papers in 1976 (Hatsopoulos and Gyftopoulos, 1976a, b, c, d), Beretta develops a dynamical postulate (Beretta et al. 1984; Beretta, Gyftopoulos, and Park, 1985) consistent both with the non-dynamical quantum mechanical postulates of the Unified Theory as well as with its thermodynamic ones (the 2nd Law in particular). The theory itself simply and elegantly extends in a unified fashion the concepts of thermodynamics to quantum mechanics and the concepts of quantum mechanics to thermodynamics. It does so without the bridge traditionally used, i.e. statistical mechanics, eliminating a number of the ambiguities, tautologies, and inconsistencies (including a built-in violation of the 2nd Law) inherent in the presentations of both Classical and Statistical Thermodynamics. This new formulation generalizes thermodynamics so that it applies to all systems large or small (including one particle systems) either in a state of thermodynamic (i.e. stable) equilibrium or not in a state of thermodynamic equilibrium. The Beretta equation of motion describes the time evolution of the state of a system via a density operator which is uniquely based on an unambiguous preparation of an ensemble of identical systems, i.e. the so-called homogenous or irreducible ensemble, and does so both for unitary and non-unitary reversible as well as irreversible processes. In this paper, we present a simple application of this general equation of motion to the time evolution of the entropy of a closed system comprised of a Boltzmann type gas consisting of one or of many particles undergoing an irreversible process. A number of different energy eigenlevels and initial states and their effects on entropy generation and the final state of maximum entropy, i.e. stable equilibrium, are examined. A simple time-dependent work interaction is introduced into the formulation to show how this in turn affects the evolution of the state of the system.
Abstract The quantum thermodynamics of the ideal gas is treated by means of the particle entropies σ=ka. The non-dimensional entropy numbers a are calculated by the extended, temperature-dependent Schrödinger equation applying periodic boundary conditions. The Boltzmann constant k represents the atomic entropy unit. The quantized particle entropies a are used to calculate the Boltzmann distribution f=exp(α−a), the density of entropy levels D(a), the dimensionless chemical potential α=μ/kT, etc. – The thermodynamics of the ideal gas is treated on the basis of entropy quanta by means of the density matrix formalism. In this way we obtain the total particle entropy A, the thermodynamic entropy S (equation of Sackur-Tetrode), the internal and the Helmholtz free energy E and F, respectively, etc. Moreover, it can also be shown on the basis of entropy quanta that the entropy of a closed system in thermal equilibrium has a maximum value, S=S max. Against this, the entropy S ′ of an arbitrary non-equilibrium state is always smaller, S′ < S max (second law of thermodynamics). – Application of ordinary density matrix theory to a closed system of N particles leads to the result that the entropy S does not change in time, dS/dt=0, regardless whether we consider an equilibrium state, S=S max, or a non-equilibrium state, S=S′. Consequently, S cannot irreversibly change from S′ to S max. However, any irreversible process is accompanied by a positive entropy production P=dS/dt>0. In order to overcome this obvious contradiction, we discuss the entropy evolution in time by means of the commutator equation G =i[ P , t ], which is deduced from very general assumptions. Here P and t are the operators of the entropy production P and the time t. Accordingly, P and t do not commute in general, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties ΔP and Δt, which are expressed by the uncertainty relation of the N-particle system, ΔPΔt≥(1/2)|〈 G 〉|=(γ/2)k. This P − t uncertainty relation easily allows a discussion of the evolution of the entropy S in time t from S′ to S max. Now the irreversible steps are correctly described by the entropy production P=dS/dt > 0, and the thermal equilibrium by P=0, ∆P=0, and thus the lifetime ∆t=∞.
The principle of least action, which has so successfully been applied to diverse fields of physics looks back at three centuries of philosophical and mathematical discussions and controversies. They could not explain why nature is applying the principle and why scalar energy quantities succeed in describing dynamic motion. When the least action integral is subdivided into infinitesimal small sections each one has to maintain the ability to minimize. This however has the mathematical consequence that the Lagrange function at a given point of the trajectory, the dynamic, available energy generating motion, must itself have a fundamental property to minimize. Since a scalar quantity, a pure number, cannot do that, energy must fundamentally be dynamic and time oriented for a consistent understanding. It must have vectorial properties in aiming at a decrease of free energy per state (which would also allow derivation of the second law of thermodynamics). Present physics is ignoring that and applying variation calculus as a formal mathematical tool to impose a minimization of scalar assumed energy quantities for obtaining dynamic motion. When, however, the dynamic property of energy is taken seriously it is fundamental and has also to be applied to quantum processes. A consequence is that particle and wave are not equivalent, but the wave (distributed energy) follows from the first (concentrated energy). Information, provided from the beginning, an information self-image of matter, is additionally needed to recreate the particle from the wave, shaping a “dynamic” particle-wave duality. It is shown that this new concept of a “dynamic” quantum state rationally explains quantization, the double slit experiment and quantum correlation, which has not been possible before. Some more general considerations on the link between quantum processes, gravitation and cosmological phenomena are also advanced.
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.
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 central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.