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Progress of Theoretical Physics

Variation Principle of Linear and Nonlinear Irreversible Processes

Masakazu Ichiyanagi

Published Jul 1, 1986• 5 citations
Publisher

Abstract

A method is proposed for investigating nonequilibrium processes in an arbitrary system under an action of external perturbations. We formulate a new quantum mechanical variation principle, which is valid beyond linear irreversible processes. The definition of our variation functional is made on the basis of our recognition of the importance of a change in the internal energy of the system. In this article, we go beyond the theory of Nakano by developing our new variation principle in the framework of the closed-time-path method by Schwinger. The relationship of the thermodynamical stability criterion to the exact solution of transport problems is discussed.

DOI
10.1143/ptp.76.37

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A method is proposed for investigating nonequilibrium processes in an arbitrary system under an action of external perturbations. We formulate a new quantum mechanical variation principle, which is valid beyond linear irreversible processes. The definition of our variation functional is made on the basis of our recognition of the importance of a change in the internal energy of the system. In this article, we go beyond the theory of Nakano by developing our new variation principle in the framework of the closed-time-path method by Schwinger. The relationship of the thermodynamical stability criterion to the exact solution of transport problems is discussed.
Principles of development of the quantum theory of nonlinear processes on the basis of Lagrangian formulation are discussed. It is shown that in the framework of this formulation it is possible to preserve succession from the classical theory and, in particular, use these methods for studies of quantum systems. The quantum dispersion of a nonlinear oscillator excited by an external source and of a parametric generator is calculated. Its role is established in the solution of the problem of stability of oscillations.
Throughout our discussion of the general principles of quantum mechanics we have emphasized the importance of the eigenvalues and eigenfunctions of the operators representing physical quantities. However, we have seen that solving the basic eigenvalue equations determining these is often not straightforward. In the early chapters discussing the energy eigenvalue equation (or time-independent Schro¨dinger equation) for example, we found that this could often not be solved exactly and that, even when a solution was possible, it frequently required considerable mathematical analysis. Because of this, methods have been developed to obtain approximate solutions to eigenvalue equations. One of the most important of such techniques is known as time-independent perturbation theory and will be the first method to be discussed in the present chapter. Another, known as the variational principle, will be discussed later. We shall confine our discussion to the timeindependent Schro¨dinger equation, but generalization to other eigenvalue equations is quite straightforward. We shall mainly use the wave function representation, but we shall restate some of the important results in Dirac notation so that they can be readily transformed to a matrix representation if desired.
I propose a new and direct connection between classical mechanics and quantum mechanics where I derive the quantum mechanical propagator from a variational principle. This variational principle is Hamilton's modified principle generalized to allow many paths due to the non-local existence of particles in phase space. This principle allows a physical system to evolve non-locally in phase space while still allowing a representation that uses many classical paths. Whereas a point in phase space represents a classical system's state, I represent the state of a non-local system by a mixed trajectory. This formulation naturally leads to the transactional interpretation for resolving the paradoxes of the measurement problem. This principle also suggests a more flexible framework for formulating theories based on invariant actions and provides a single conceptual framework for discussing many areas of science.
Abstract The genuine variational principle of macroscopic dissipative continua is generalized to complex scalar fields in the universal form of GAUSSian type. The generalization makes it possible to extend the validity of the principle to the PS field of the non‐relativistic SCHRÖDINGER equation, and to certain types of non‐linear problems. Following MORSE's and FESHBACH's sequence of thoughtful ideas in reverse direction the fundamentals of an “abstract” transport theory, isomorphic with one‐particle quantum theory, are developed and the possibility of interpreting the quantum theory by classical models is discussed. On the basis of new arguments, the impossibility of interpreting quantum mechanics in terms of classical transport theories (hydrodynamical model, diffusional and stochastic models) is established.
Quantum theory is developed from a q-number (operator) action principle with a representation-invariant technique for limiting the number of independent system variables. It is shown that in a q-number theory such a limitation on the number of variations is necessary, since a completely arbitrary q-number variation implies an infinite number of conditions to be satisfied. This implication arises because, in a q-number theory, the system variables are matrices which, in most cases, have an infinite number of elements, each of which is independently varied in a completely arbitrary q-number variation. A review of Schwinger's quantum field theory formalism shows that the limitation to c- number variations, while accomplishing the desired limitation on the number of independent variations, is not representation-invariant, i.e., under a change of representation, c-number variations applied to the original system variables transform to q-number variations of the new system variables. In the Hamiltonian formulation of classical mechanics, invariant infinitesimal canonical transformations are generated by constants of the motion. It is also possible to generate infinitesimal invariant transformations from constants of the motion in the classical Lagrangian formulation. This latter possibility is exploited to obtain a q-number theory. It is demanded that the variations of the system variablesmore » in the action principle be the invariant infinitesimal transformations generated by constants of the motion. This requirement accomplishes the desired limitation of the number of independent variations in a representation-invariant manner. The resulting theory is developed in detail for a certain class of Lagrangian functions. A set of commutation relations is derived and a dynamical principle for determining the temporal development of the system is postulated. A discussion of the results, including consideration of the self-consistency of the formalism developed, is presented. Finally, a specific problem is worked through and it is shown that the results obtained correspond with those of the canonical method of quantization.« less
We describe the large applicability of the Nonequilibrium Statistical Operator Method (NSOM) for the study of dissipative dynamic systems far from equilibrium. It is shown that the NSOM can be encompassed by a unifying variational principle, which produces a large family of NSO that contains existing examples as particular cases. Further, we review the application of the NSOM for the construction of a nonlinear quantum theory of large scope, and for the generation of a response function theory, for far-from-equilibrium Hamiltonian systems. An accompanying non-equilibrium thermodynamic Green's function theory is briefly described. Also it is shown that the NSOM provides mechano-statistical foundations for phenomenological irreversible thermodynamics, and for the important question of stability of far-from-equilibrium steady states and the emergence of self-organized dissipative structures in condensed matter.