In this paper, heat and Carnot's working substance are defined respectively by the equations $E=3pV$ and $E=3/2pV$, relating the energy $E$, the pressure $p$ and the volume $V$ of these two corpuscular systems. The thermodynamic-kinetic theoretical analysis of their equilibrium reveals that the constraints imposed by the second law on the working substance are the consequence of the relativistic and the quantum taut constraints, respectively, $|v|\le c$ and $|ΔpΔq|\ge h$, imposed on the motion of its molecules. It is further shown that the action of these restrictions is described by the ladder operators. By revealing the laws that rule the entropy formation mechanism, thereby explaining the time asymmetry, the present approach supersedes statistical mechanics. Furthermore, it establishes the epistemological chain connecting the laws of thermodynamics to the axioms of quantum mechanics, leading to yet another unorthodox interpretation of that theory.
In this paper, heat and Carnot's working substance are defined respectively by the equations E=3pV and E=3/2pV, relating the energy E, the pressure p and the volume V of these two corpuscular systems. The thermodynamic-kinetic theoretical analysis of their equilibrium reveals that the constraints imposed by the second law on the working substance are the consequence of the relativistic and the quantum taut constraints, respectively, ∣v∣≤c and ∣ΔpΔq∣≥h, imposed on the motion of its molecules. It is further shown that the action of these restrictions is described by the ladder operators. By revealing the laws that rule the entropy formation mechanism, thereby explaining the time asymmetry, the present approach supersedes statistical mechanics. Furthermore, it establishes the epistemological chain connecting the laws of thermodynamics to the axioms of quantum mechanics, leading to yet another unorthodox interpretation of that theory.
Abstract Part II of this two-part paper refutes the beliefs about the statistical interpretation of thermodynamics, and the association of entropy with disorder that are summarized in Part I. The refutation of the statistical approach is based on either a nonstatistical unified quantum theory of mechanics and thermodynamics, or an almost equivalent, novel, nonquantal exposition of thermodynamics. Entropy is shown to be: (i) valid for any system (both macroscopic and microscopic, including one-particle systems), and any state (both thermodynamic or stable equilibrium, and not stable equilibrium); (ii) a measure of the quantum-theoretic pliable shape of the molecules of a system; and (iii) a monotonic indicator of order. In contrast to statistics which associates a thermodynamic equilibrium macrostate with the largest number of compatible microstates, the second law avers that, for each set of values of energy, volume, and amounts of constituents of either a macroscopic or a microscopic system, there exists one and only one thermodynamic or stable equilibrium state. So, even if Boltzmann’s definition were used, a thermodynamic equilibrium state is one of perfect order.
Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)
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 one of the main pillars of theoretical physics, and it has a special appeal of having wide applicability to a large variety of different physical systems. However, many assumptions in thermodynamics apply only to systems which are bulk material, i.e. consisting a large number of microscopic classical particles. Due to the advancement of designing nanoscale engines, especially in the light of devices that are used today in the processing of quantum information, is thermodynamics still applicable? Can we refine the core principles of thermodynamics to suit such nanoscale quantum systems as well? The central aim of this thesis is to construct a theory of thermodynamics that holds for nanoscale quantum systems, even those as small and simple as a single qubit. We do this by starting out from the core basics of quantum theory: unitary dynamics on closed quantum systems. We adapt a resource theoretic approach inspired by quantum information theory, which defines the quantum states and operations allowed to be used in a thermodynamic evolution. With this framework that naturally adopts the first law as an energy preserving condition, we show the refinement of both the zeroeth and second law of thermodynamics. The zeroeth law explains the physical significance of the Gibbs thermal state. On the other hand, we show that the second law sees refinement in the quantum nanoregime: instead of having the free energy as the sole quantity dictating the possibility of a thermodynamic state transition, we derive a family of generalized free energies that also constitute necessary conditions for a transition to occur. Moreover, these conditions become sufficient for states which are block-diagonal in the energy eigenbasis. In this thesis, we also brought our approach of thermodynamics to the next step: we apply our findings on the second laws, in order to analyze the maximum achievable efficiency for quantum heat engines. In classical thermodynamics, the Carnot efficiency has been long known as the theoretical maximum which does not depend on the specific structure of the thermal baths used, but only on its temperature. With the additional free energies we discover, we show that although quantum heat engines may achieve the Carnot efficiency, such an achievability is no longer independent of the Hamiltonians of the thermal baths. In other words, we find additional restrictions that surface in the study of quantum nanoscale heat engines, which are a direct consequence of the generalized second laws. This has provided us with a deeper understanding into the fundamental limitations of how efficient devices can be made in the realm of microscopic quantum systems.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.