We review our approach to the second law of thermodynamics as a theorem asserting the growth of the mean (Gibbs–von Neumann) entropy of quantum spin systems undergoing automorphic (unitary) adiabatic transformations. Non-automorphic interactions with the environment, although known to produce on the average a strict reduction of the entropy of systems with finite number of degrees of freedom, are proved to conserve the mean entropy on the average. The results depend crucially on two properties of the mean entropy, proved by Robinson and Ruelle for classical systems and Lanford and Robinson for quantum lattice systems: upper semicontinuity and affinity.
We review our approach to the second law of thermodynamics as a theorem asserting the growth of the mean (Gibbs–von Neumann) entropy of quantum spin systems undergoing automorphic (unitary) adiabatic transformations. Non-automorphic interactions with the environment, although known to produce on the average a strict reduction of the entropy of systems with finite number of degrees of freedom, are proved to conserve the mean entropy on the average. The results depend crucially on two properties of the mean entropy, proved by Robinson and Ruelle for classical systems and Lanford and Robinson for quantum lattice systems: upper semicontinuity and affinity.
We improve on our version of the second law of thermodynamics as a deterministic theorem for quantum spin systems in two basic aspects. The first concerns the general statement of the second law: spontaneous changes in an adiabatically closed system will always be in the direction of increasing mean entropy, which rises to a maximal value. Two specific examples concern the transition from pure to mixed states in two different universality classes of dynamics in one dimension, one being the exponential model, the other the Dyson model, the dynamics of the latter exhibiting strong graphical evidence of quantum chaos, as a consequence of the results of Albert and Kiessling on the Cloitre function.
We improve on our version of the second law of thermodynamics as a deterministic theorem for quantum spin systems [ 1 ] in two basic aspects. The first concerns the general statement of the second law: spontaneous changes in an adiabatically closed system will always be in the direction of increasing (mean) entropy, which rises to a maximal value. In order to arrive at this statement, we have to extend the previous definition of adiabatic transformation, in order to include sudden interactions, and thereby generalize the “barrier model”. The cornerstone of the second law is then seen to be the fact that the dynamics may induce a basic structural transition between states of infinite systems in the limit of large times, which, physically, represent times much larger than specific relaxation times. Two specific examples concern the transition from pure to mixed states in two different universality classes of dynamics in one dimension, one being the exponential model, which has exactly soluble dynamics and does not display a phase transition, the other the Dyson model(s), for which neither the dynamics nor the statistical mechanics are exactly soluble, and which do exhibit a (ferromagnetic) phase transition, first demonstrated by Dyson. It is also shown, as a consequence of results of Albert and Kiessling on the Cloitre function, that there is strong graphical evidence that the dynamics of the Dyson models are chaotic for large times, and, thereby, that the mechanisms of approach to equilibrium differ for these two universality classes in a profound way, which depend exclusively on the dynamics, and not on the states and observables.
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.
We study dynamical conservation laws and the propagation of thermodynamic structure in translation-invariant lattice quantum spin systems. Using Lieb-Robinson bounds [CMP 28 (1972), 251-257] in arbitrary dimension, and Araki's bounds [CMP 14 (1969), 120-157] in dimension one, we prove conservation of the specific entropy, specific energy, and specific relative entropy under suitable assumptions on the interactions. The transported local Hamiltonians admit surface energies that vanish per unit volume in the van Hove limit, and local observables remain a core for the transported generators at every finite time. Weak Gibbsianity is preserved for sufficiently small times in arbitrary dimension and for all times in dimension one. These results provide the analytical foundation for the structural theory of approach to equilibrium developed in a companion paper, and establish partial results toward the conjectures formulated there.
We construct a macroscopic thermodynamical theory that satisfies the second law of thermodynamics, based on quantum mechanics. More concretely, our macroscopic theory satisfies the following two features: (1) A macroscopic equilibrium state is expressed as a set of macroscopic variables a=(a1,⋯,aL), which represents coarse-grained values of a set of physical quantities, and (2) An adiabatic transformation from an equilibrium state a to another state a′ is possible if and only if sa≤sa′ holds. Here, sa is a real-valued function corresponding to the thermodynamical entropy, which depends only on macroscopic variables a. We show that such a macroscopic thermodynamical theory can be consistently constructed if the von-Neumann entropy of a generalized microcanonical state satisfies a generalized central limit theorem. We also show that even when the von-Neumann entropies do not satisfy the condition, a macroscopic thermodynamic-like theory with a similar structure exists, which could be at the foundations of nonequilibrium thermodynamics. Finally, we make a replica of (U,V,N) expression of thermodynamics satisfying (1) and (2), using a toy model of free particles in a box.
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.