Aspects of the modern dynamical systems approach to thermodynamics of stationary states out of equilibrium with attention to the original conceptions which arose at the beginnings of Statistical Mechanics

Nonequilibrium thermodynamics

The information-theoretic maximal-entropy procedure for the analysis of collision processes is derived as a consequence of the dynamics, be they quantal or classical. The method centers attention on the minimal number of operators (the "dynamic constraints") whose expectation values are both necessary and sufficient to completely characterize the collision dynamics. For a given Hamiltonian and initial state, the constraints required to obtain an exact solution of the equations of motion are determined by a purely algebraic procedure. It is furthermore found possible to derive equations of motion for the conjugate Lagrange parameters. Immediate applications are noted, e.g., a family of similar reactions is shown to have a common set of dynamic constraints and simple illustrative applications are provided. The determination of the scattering matrix is discussed, with examples. The general formalism consists in solving the scattering problem in two stages. The first is purely algebraic. At the end of this stage one obtains the functional form of, say, the scattering matrix or of the density matrix after the collision expressed in terms of parameters whose number equals the number of dynamic constraints. The end result of this algebraic stage suffices to analyze the scattering pattern for any initial state. The second stage is the predictive procedure. Explicit coupled first-order nonlinear differential equations are obtained for the parameters.

Connection between the maximal entropy and the scattering theoretic analyses of collision processes

In a previous paper, we attempted to lay a conceptual framework for an ultimate physics of small semiconductor devices and concentrated on the medium small device. Here we treat the very small device (VSD), characterized by an effective channel length of 250 A. We demonstrate how such a device could conceivably be fabricated using two side processing of the wafer. In treating the transport, however, it is found that the time and distance scales are such that the Boltzmann transport equation is completely invalidated. Here we develop the appropriate quantum transport equations based upon the density matrix for the entire system, device plus boundaries plus environment. It is found that the boundaries and environment can lead to renormalization of the energy spectrum as well as long range dissipative interactions. Two special cases of the transport equations are treated. If the transport is dominantly stochastic, an exact Langevin equation is found for the various transport parameters. In a second case, a parameterized density matrix is used in analogy to the displaced Maxwellian. In this latter case, a hierarchy of moment equations can be developed to yield, e.g. energy and momentum balance equations.

On the physics and modeling of small semiconductor devices—II: The very small device

The aim of this paper is to provide better understanding of a few approaches that have been proposed for treating nonequilibrium (time-dependent) processes in statistical mechanics with the emphasis on the interrelation between theories. The ensemble method, as it was formulated by Gibbs, has great generality and broad applicability to equilibrium statistical mechanics. Different macroscopic environmental constraints lead to different types of ensembles, with particular statistical characteristics. In the present work, the statistical theory of nonequilibrium processes which is based on nonequilibrium ensemble formalism is discussed. We also outline the reasoning leading to some other useful approaches to the description of the irreversible processes. The kinetic approach to dynamic many-body problems, which is important from the point of view of the fundamental theory of irreversibility, is alluded to. Appropriate references are made to papers dealing with similar problems arising in other fields. The em...

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Theory of transport processes and the method of the nonequilibrium statistical operator

We describe a particular approach for the construction of a nonequilibrium statistical ensemble formalism for the treatment of dissipative many-body systems. This is the so-called Nonequilibrium Statistical Operator Method, based on the seminal and fundamental ideas set forward by Boltzmann and Gibbs. The existing approaches can be unified under a unique variational principle, namely, MaxEnt, which we consider here. The main six basic steps that are at the foundations of the formalism are presented and the fundamental concepts are discussed. The associated nonlinear quantum kinetic theory and the accompanying Statistical Thermodynamics (the Informational Statistical Thermodynamics) are very briefly described. The corresponding response function theory for systems away from equilibrium allows to connected the theory with experiments, and some examples are summarized; there follows a good agreement between theory and experimental data in the cases in which the latter are presently available. We also present an overview of some conceptual questions and associated criticisms.

V.P. Kalashnikov

Papers

Derivation of the nonequilibrium statistical operator from the extremum of the information entropy