Starting from the quantum-Boltzmann equation derived in a previous paper, we study the irreversible evolution of an electron gas in the one-particle phase space. The connection with phase space is established by expressing one-electron states in terms of the overcomplete and nonorthogonal generating system of coherent states. By using the generalized closure relation for coherent states, as well as the fact that a one-particle operator is completely determined by the ensemble of expectation values for all coherent states, we obtain the master equations in a form that allows us to follow the evolution in phase space. This form of the master equations provides a direct link between the quantum-statistical approach and the semi-classical Boltzmann equation. The latter is obtained after a coarse-graining procedure in the one-particle phase space and by using the fact that the electron-electron interaction, as well as the interactions between the electron gas and the bath subsystems provided by phonons or photons, are local in real space.
Starting from the quantum-Boltzmann equation derived in a previous paper, we study the irreversible evolution of an electron gas in the one-particle phase space. The connection with phase space is established by expressing one-electron states in terms of the overcomplete and nonorthogonal generating system of coherent states. By using the generalized closure relation for coherent states, as well as the fact that a one-particle operator is completely determined by the ensemble of expectation values for all coherent states, we obtain the master equations in a form that allows us to follow the evolution in phase space. This form of the master equations provides a direct link between the quantum-statistical approach and the semi-classical Boltzmann equation. The latter is obtained after a coarse-graining procedure in the one-particle phase space and by using the fact that the electron-electron interaction, as well as the interactions between the electron gas and the bath subsystems provided by phonons or photons, are local in real space.
Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.
We study the evolution of a many-electron system that is confined in a finite spatial region and coupled to a statistical environment. The latter may be composed of several independent bath subsystems, which are held at some statistical equilibrium. From the master equations describing the evolution of the coarse-grained N-particle density matrix, we obtain the equations, which describe the evolution of the n-particle density operators D(n) for n<N. These equations are hierarchically coupled through the electron-electron interaction. We show that the hierarchy can be truncated under the assumption that the residual interaction of the electrons in the considered system with the environment introduces a memory loss, which hinders the electronic system to build up more than two-particle correlations. We first consider a weakly excited electronic system, which looses memory but where energy exchange with the statistical environment can be neglected. This is the quantum analog of the classical Boltzmann gas in a box. We derive the master equations, which describe the irreversible evolution of the coarse-grained one-particle density matrix. Based on this result, we show that, in accord with the second law of thermodynamics, the corresponding von Neumann entropy either increases with time or it remains constant. Finally, allowing also for energy exchange with one or more bath subsystems provided, e.g., by phonons or photons, we obtain the corresponding general master equations that describe the evolution of a spatially confined interacting electron gas of metallic density.
The present paper derives conditions on which the Boltzmann equations for electron-phonon systems (two coupled equations) can be derived without the use of the objectionable random phase approximation. The treatment uses von Neumann's density matrix based on the Hamiltonian introduced some time ago by one of us in connection with superconductivity. Formally exact expressions for the rate of change of coarse-grained one-particle distribution functions are derived. They become identical with the collision terms in the Boltzmann equations if (i) the influence of an external field on them is negligible, (ii a ) particle correlation is absent at an initial time, and (ii b ) the relevant series in terms of the interaction parameter converges and is at all times dominated by a certain term. (ii b ) excludes the possibility of extensive correlations as they occur in superconductivity.
Using a newly introduced connection between the local and non-local description of open quantum system dynamics, we investigate the relationship between these two characterisations in the case of quantum semi-Markov processes. This class of quantum evolutions, which is a direct generalisation of the corresponding classical concept, guarantees mathematically well-defined master equations, while accounting for a wide range of phenomena, possibly in the non-Markovian regime. In particular, we analyse the emergence of a dephasing term when moving from one type of master equation to the other, by means of several examples. We also investigate the corresponding Redfield-like approximated dynamics, which are obtained after a coarse graining in time. Relying on general properties of the associated classical random process, we conclude that such an approximation always leads to a Markovian evolution for the considered class of dynamics.
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
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.