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International Journal of Quantum Chemistry

Quantum mechanics of dissipative systems and noncanonical formalism

Claude D. George, Ilya Prigogine

Published Jun 18, 2009• 9 citations
Publisher

Abstract

We present a noncanonical transformation theory of large dynamical systems that leads to a formulation of dynamics in terms of processes, in which the concepts of well-defined “units” and “interactions” can be simultaneously accommodated.
DOI
10.1002/qua.560080838

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It is difficult to incorporate the concept of interacting physical particles in the frame of conventional quantum mechanics. It comes from the fact that in this formalism the Hamiltonian plays a dual role as the generator of motion and as the observable associated with the energy. As a result it is easy to define either a representation in which particles interact but are not well defined (a representation in which the Hamiltonian is not diagonal) or a representation in which the units are well defined but do not interact. We expect that in the search for a representation in which we may speak of both well defined and still interacting particles, the distinction between reversible and irreversible processes should play a basic role. Certainly we do not want to eliminate irreversible processes such as scattering, production or decay which are basically related to the mechanisms through which we observe particles. To distinguish between reversible and irreversible processes we have to work in a representation such that a Liapounov function (the entropy) may be ascribed to the time evolution of the system. The introduction of such a representation through a specific class of non-unitary transforms is described. We then propose newmore » quantum rules directly applicable to dissipative systems with a Liapounov function. For systems with a discrete spectrum these new quantum rules reduce trivially to the usual quantum rules as formulated by Born, Jordan and Heisenberg. The theory is applied to a simple field theoretical model - the so-called Friedrichs model-for which all calculations can be performed exactly.« less
Chemical reactions are generally connected with energy transfer between reactands and their environment. There are different ways of taking this environment into account. The traditional system-plus-reservoir (S+R) approach couples the relevant system to a large number of environmental degrees of freedom, represented, e.g., by harmonic oscillators. However, this has the disadvantage—from a computational point of view—that a system of many coupled differential equations has to be solved due to the large number of degrees of freedom of the reservoir. The number of degrees of freedom can be drastically reduced if an effective description of the interaction between system and reservoir is applied. On the quantum mechanical level, the effect of the environment can be included, e.g., by the use of explicitly time-dependent Hamiltonians or nonlinear additions to the Schrödinger equation (SE). An advantage of the former method is the preservation of the canonical formalism and the linearity of the theory as well as the fact that its operators can be derived directly from the S+R approach. Therefore, it seems to be even more of a paradox that this method apparently violates the uncertainty principle. This violation does not occur in the nonlinear approaches but the nonlinearity seems, at first sight, to be problematic. However, it can be shown that a particular logarithmic nonlinear SE is equivalent to the approach using the time-dependent Hamiltonian and can be transformed into this. The fact that this transformation is nonunitary provides the key for the solution of the paradox and also allows one to show the equivalence of all three of the above-mentioned approaches. In investigating the classical counterpart, a noncanonical transformation is used between the dissipative physical level and the canonical formal level, which makes it even possible to transform the physical interacting system into a formal isolated system that can be treated in the conventional way. As a result, the effect of the interaction with the environment can be taken into account classically by noncanonical transformations and quantum mechanically by nonunitary transformations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 537–547, 1999
Abstract Nonintegrable Poincaré systems with a continuous spectrum (so‐called large Poincaré systems, LPS) lead to the appearance of diffusive terms in the frame of classical or quantum dynamics. These terms break time symmetry. They lead, therefore, to limitations to classical trajectory theory and of wave‐function formalism (Schrödinger's equation). These diffusive terms correspond to well‐defined classes of dynamical processes (i.e., so‐called vacuum‐vacuum transitions). The diffusive effects are amplified in situations corresponding to persistent interactions. As a result, we have to include, already, in the fundamental dynamical description the two basic aspects, probability and irreversibility, which are so conspicuous on the macroscopic level. We have to formulate both classical and quantum mechanics on the Liouville level of probability distributions (or density matrices). For integrable systems, we recover the usual formulation of classical or quantum mechanics. Instead of being primitive concepts, which cannot be further analyzed, trajectories and wave functions appear as special solutions of the Liouville–von Neumann equations. This extension of classical and quantum dynamics permits us to unify the two concepts of nature that we inherited from the nineteenth century, based, on the one hand, on dynamical time‐reversible laws and, on the other, on an evolutionary view associated to entropy. It leads also to a unified formulation of quantum theory, avoiding the conventional dual structure based on Schrödinger's equation, on the one hand, and on the “collapse” of the wave function, on the other. A dynamical interpretation is given to processes such as decoherence or approach to equilibrium without any appeal to extra dynamic considerations (such as many‐world theory, coarse graining, or averaging over the environment). There is a striking parallelism between classical and quantum theory. For large Poincaré systems (LPS), we have, in general, both a “collapse” of trajectories and of wave functions. In both cases, we need a generalized formulation of dynamics in terms of probability distributions or density matrices. Since the beginning of this century, we have known that classical mechanics had to be generalized to take into account the existence of universal constants. We now see that classical as well as quantum mechanics also have to be extended to include unstable dynamical systems such as LPS. As a result, we achieve a new formulation of “laws of physics” dealing no more with certitudes but with probabilities. This formulation is appropriate to describe an open, evolving universe. © 1995 John Wiley & Sons, Inc.