Inverted quantum states provide a challenge to classical thermodynamics, since they appear to contradict the classical formulation of the second law of thermodynamics. Ramsey interpreted these states as stable equilibrium states of negative thermodynamic temperature, and added a provision to allow these states to the Kelvin-Planck statement of the second law [N. F. Ramsey, Phys. Rev. 103, 20 (1956)PHRVAO0031-899X10.1103/PhysRev.103.20]. Since then, Ramsey's interpretation has prevailed in the literature. Here, we present an alternative option to accommodate inverted states within thermodynamics, which strictly enforces the original Kelvin-Planck statement of the second law, and reconciles inverted states and the second law by interpreting the former as unstable states, for which no temperature-positive or negative-can be defined. Specifically, we recognize inverted quantum states as temperature-unstable states, for which all processes are in agreement with the original Kelvin-Planck statement of the second law, and positive thermodynamic temperatures in stable equilibrium states. These temperature-unstable states can only be created by work done to the system, which is stored as energy in the unstable states, and can be released as work again, just as in a battery or a spring.
Inverted quantum states provide a challenge to classical thermodynamics, since they appear to contradict the classical formulation of the second law of thermodynamics. Ramsey interpreted these states as stable equilibrium states of negative thermodynamic temperature, and added a provision to allow these states to the Kelvin-Planck statement of the second law [N. F. Ramsey, Phys. Rev. 103, 20 (1956)PHRVAO0031-899X10.1103/PhysRev.103.20]. Since then, Ramsey's interpretation has prevailed in the literature. Here, we present an alternative option to accommodate inverted states within thermodynamics, which strictly enforces the original Kelvin-Planck statement of the second law, and reconciles inverted states and the second law by interpreting the former as unstable states, for which no temperature-positive or negative-can be defined. Specifically, we recognize inverted quantum states as temperature-unstable states, for which all processes are in agreement with the original Kelvin-Planck statement of the second law, and positive thermodynamic temperatures in stable equilibrium states. These temperature-unstable states can only be created by work done to the system, which is stored as energy in the unstable states, and can be released as work again, just as in a battery or a spring.
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.
The second law of classical equilibrium thermodynamics, based on the positivity of entropy production, asserts that any process occurs only in a direction that some information may be lost (flow out of the system) due to the irreversibility inside the system. However, any thermodynamic system can exhibit fluctuations in which negative entropy production may be observed. In particular, in stochastic quantum processes due to quantum correlations and also memory effects we may see the reversal energy flow (heat flow from the cold system to the hot system) and the backflow of information into the system that leads to the negativity of the entropy production which is an apparent violation of the Second Law. In order to resolve this apparent violation, we will try to properly extend the Second Law to quantum processes by incorporating information explicitly into the Second Law. We will also provide a thermodynamic operational meaning for the flow and backflow of information. Finally, it is shown that negative and positive entropy production can be described by a quantum thermodynamic force.
In recent years we have witnessed a concentrated effort to make sense of thermodynamics for small-scale systems. One of the main difficulties is to capture a suitable notion of work that models realistically the purpose of quantum machines, in an analogous way to the role played, for macroscopic machines, by the energy stored in the idealisation of a lifted weight. Despite several attempts to resolve this issue by putting forward specific models, these are far from realistically capturing the transitions that a quantum machine is expected to perform. In this work, we adopt a novel strategy by considering arbitrary kinds of systems that one can attach to a quantum thermal machine and defining work quantifiers . These are functions that measure the value of a transition and generalise the concept of work beyond those models familiar from phenomenological thermodynamics. We do so by imposing simple operational axioms that any reasonable work quantifier must fulfil and by deriving from them stringent mathematical condition with a clear physical interpretation. Our approach allows us to derive much of the structure of the theory of thermodynamics without taking the definition of work as a primitive. We can derive, for any work quantifier , a quantitative second law in the sense of bounding the work that can be performed using some non-equilibrium resource by the work that is needed to create it. We also discuss in detail the role of reversibility and correlations in connection with the second law. Furthermore, we recover the usual identification of work with energy in degrees of freedom with vanishing entropy as a particular case of our formalism. Our mathematical results can be formulated abstractly and are general enough to carry over to other resource theories than quantum thermodynamics.
The second law of thermodynamics for adiabatic operations -- constraints on state transitions in closed systems under external control -- is one of the fundamental principles of thermodynamics. On the other hand, it is recently established that even pure quantum states can represent thermal equilibrium. However, pure quantum states do not satisfy the second law in that they are not passive, i.e., work can be extracted from them if arbitrary unitary operations are allowed. It therefore remains unresolved how quantum mechanics can be reconciled with thermodynamics. Here, based on our key quantum-mechanical notions of thermal equilibrium and adiabatic operations, we address the emergence of the second law for adiabatic operations in the thermodynamics limit. We first introduce infinite-observable macroscopic thermal equilibrium (iMATE); a quantum state, including pure states, is in iMATE if the expectation values of all additive observables agree with their equilibrium values. We also introduce a macroscopic operation as unitary evolution generated by a time-dependent additive Hamiltonian, which is regarded as corresponding to adiabatic operations. Employing these concepts, we show that no extensive work can be extracted from any quantum state in iMATE through any macroscopic operations. Furthermore, we introduce a quantum-mechanical form of entropy density such that it agrees with thermodynamic entropy density for any quantum state in iMATE. We then prove that for any initial state in iMATE, this entropy density cannot be decreased by any macroscopic operations, followed by a time-independent relaxation process. Our theory thus proves two different forms of the second law, by adopting macroscopically reasonable classes of observables, equilibrium states, and operations. We also discuss the time scales of macroscopic operations in these results.
The second law of thermodynamics places constraints on state transformations. It applies to systems composed of many particles, however, we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath. Is there a second law of thermodynamics in this regime? Here, we find that for processes which are approximately cyclic, the second law for microscopic systems takes on a different form compared to the macroscopic scale, imposing not just one constraint on state transformations, but an entire family of constraints. We find a family of free energies which generalize the traditional one, and show that they can never increase. The ordinary second law relates to one of these, with the remainder imposing additional constraints on thermodynamic transitions. We find three regimes which determine which family of second laws govern state transitions, depending on how cyclic the process is. In one regime one can cause an apparent violation of the usual second law, through a process of embezzling work from a large system which remains arbitrarily close to its original state. These second laws are relevant for small systems, and also apply to individual macroscopic systems interacting via long-range interactions. By making precise the definition of thermal operations, the laws of thermodynamics are unified in this framework, with the first law defining the class of operations, the zeroth law emerging as an equivalence relation between thermal states, and the remaining laws being monotonicity of our generalized free energies.
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.