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Transport Theory and Statistical Physics

Application of the statistical mechanics transformation theory to a soluble model

R. W. Gibberd, Claude D. George

Published Jan 1, 1972• 3 citations
Publisher

Abstract

One of us (C.G) has developed a transformation theory in nonequilibrium statistical mechanics that generalizes the usual transformation theory of quantum mechanics in the framework of the resolvent formalism of the Liouville-von Neumann operator extensively used by the Brussels school. Prigogine, George, and Henin have shown that the evolution of a dissipative system can be decomposed into separate subdynamics in orthogonal subspaces. The π subspace, containing all thermodynamics, can be seen as an extension of the subspace of the invariants in the absence of dissipation. In this subspace a suitable contracted description of the system can be given through a dressing operator χ, which has the property that HR = χ−1H0 can be considered as the renormalized energy. A differential equation that determines χ has been proposed by Mandel. It is useful to see, even for a nondissipative system, what relation the transformation theory in the Liouville-von Neumann formalism has to the standard transformations used in quantum mechanics. In this paper we consider a system that can be completely diagonalized by the Bogoliubov transformation. It is shown that in this case the Mandel equation can be solved and that χ−1H0 = HD, where HD is the diagonalized Hamiltonian, obtained by the Bogoliubov transformation.

DOI
10.1080/00411457208232531

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International Journal of Quantum Chemistry•Jan 1, 2004•1 citation
One of us (C.G) has developed a transformation theory in nonequilibrium statistical mechanics that generalizes the usual transformation theory of quantum mechanics in the framework of the resolvent formalism of the Liouville-von Neumann operator extensively used by the Brussels school. Prigogine, George, and Henin have shown that the evolution of a dissipative system can be decomposed into separate subdynamics in orthogonal subspaces. The π subspace, containing all thermodynamics, can be seen as an extension of the subspace of the invariants in the absence of dissipation. In this subspace a suitable contracted description of the system can be given through a dressing operator χ, which has the property that HR = χ−1H0 can be considered as the renormalized energy. A differential equation that determines χ has been proposed by Mandel. It is useful to see, even for a nondissipative system, what relation the transformation theory in the Liouville-von Neumann formalism has to the standard transformations used in quantum mechanics. In this paper we consider a system that can be completely diagonalized by the Bogoliubov transformation. It is shown that in this case the Mandel equation can be solved and that χ−1H0 = HD, where HD is the diagonalized Hamiltonian, obtained by the Bogoliubov transformation.
We study some mathematical problems posed in nonequilibrium statistical mechanics and subdynamics theory developed by Prigogine and coworkers. We study in the superspace of the Hilbert–Schmidt operators the solution of the Liouville–von Neumann equation. The application in this frame of the spectral deformation methods yields expression of analytic continuations of a class of matrix elements of the resolvent of the Liouville–von Neumann operator, and this allows the analytic continuation of the collision and destruction operators. However, it is impossible to obtain simultaneously analytic continuation of the creation operator in this frame. The above results will be used in the second part of the article in order to study the pseudo-Markovian equation.
Recently, a number of authors have begun to study the evolution of quantum fields in the early Universe characterized by a time-dependent density matrix ${\ensuremath{\rho}}_{S}$(t). All of this work is predicated on the assumption that one's ``subsystem'' of interest is in some sense ``decoupled'' from the rest of the Universe, so that ${\ensuremath{\rho}}_{S}$ satisfies a Liouville--von Neumann equation which implies, e.g., an isentropic evolution. Starting from ``first principles,'' i.e., the Schr\"odinger equation for the totality of ``subsystem'' plus surroundings (``bath''), it is shown here how such a picture can be derived as the limiting case of a more complete statistical description. Quite generally, one finds that ${\ensuremath{\rho}}_{S}$ and the ``bath'' density matrix ${\ensuremath{\rho}}_{B}$ satisfy coupled nonlinear generalizations of the Liouville--von Neumann equation and evidence a nonisentropic evolution. However, in a Vlasov-type approximation, ${\ensuremath{\rho}}_{S}$ and ${\ensuremath{\rho}}_{B}$ satisfy instead much simpler bilinear equations which imply an isentropic evolution. And finally, in the limit that the ``back reaction'' of ${\ensuremath{\rho}}_{S}$ on ${\ensuremath{\rho}}_{B}$ can be neglected in computing the evolution of ${\ensuremath{\rho}}_{B}$, one recovers a true Liouville--von Neumann equation for the evolution of ${\ensuremath{\rho}}_{S}$ in an external field.
The investigation of the recently described generalized transformation theory which leads to a non-Hamiltonian description of dynamics is pursued. The concept of generalized unitary transformations of superoperators is introduced and a specific class of transformations studied. For nondissipative systems it is equivalent to the usual unitary transformations that diagonalize the Hamiltonian. The important point is that this class of transformations remains meaningful for dissipative systems, hence a new representation of dynamics that we shall call the "physical particle" representation. It has the following properties: (a) The energy (or an arbitrary function of the energy) is represented by a diagonal matrix. (b) In the ((0))II space (see these Proceedings, 65, 789 (1970)) corresponding to the coherent processes, the evolution can be described in terms of the changes in population of the physical particles. (c) At thermodynamic equilibrium, the physical particles are uncorrelated and behave as independent entities; the entropy has a purely combinatorial meaning.A full description of dynamics in this representation will be given in a forth-coming paper.
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.
Abstract A challenging problem facing physicists is the compatibility of irreversibility, as expressed by the second law of thermodynamics, with ordinary quantum mechanics. We present a solution through an extension of the latter. The presence of self‐energy processes (already found in quantum electrodynamics) is fully exploited to get an extension of a reversible formalism. They enable the widening of the dynamics by the introduction of specific auxiliary variables, leading to the recognition of such processes. A projection technique modeled on the one introduced long ago by the Brussels group defines a subdynamics that is shown to contain entirely the starting formulation (it corresponds to a restricted class of initial conditions, namely, the reversible one). A new kinetic description is obtained in which, for example, collision processes for privileged components arise from the original interaction. A very simple model and its generalizations are considered. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004