Canonical nonequilibrium ensembles and subdynamics

A method for constructing a canonical nonequilibrium ensemble for systems in which correlations decay exponentially has recently been proposed by Coveney and Penrose. In this paper, we show that the method is equivalent to the subdynamics formalism, developed by Prigogine and others, when the dimension of the subdynamic kinetic subspace is finite. The comparison between the two approaches helps to clarify the nature of the various operators used in the Brussels formalism. We discuss further the relationship between these two approaches, with particular reference to a simple discrete-time dynamical system, based on the baker's transformation, which we call the baker's urn.     

Quantum computation in triangular decoherence-free subdynamic space

A formalism of quantum computing with 2000 qubits or more in decoherence-free subspaces is presented. The subspace is triangular with respect to the index related to the environment. The quantum states in the subspaces are projected states ruled by a subdynamic kinetic equation. These projected states can be used to perform general, large-scale decoherence-free quantum computing.     

Vacuum subdynamics in large quantum systems

We present a constructive procedure to deal with large quantum systems in the thermodynamic limit. Starting with a discrete spectrum, we perform a complete decomposition of the evolution into one-dimensional subdynamics. We then go to the limit of a continuous spectrum after collecting them into global subdynamics of given degrees of correlation. Previously obtained results for the vacuum subdynamics are justified. The procedure is applied to the problem of potential scattering.It is a privilege for the authors to dedicate this paper to Prof. I. Prigogine. They are both genuine offspring of the Brussels school. As such, most of the ideas they profess owe their inspiration to I. P., their mentor, colleague, and friend through fascinating teaching, stimulating discussions, and illuminating speculations, but they take full responsability for their obvious mistakes, pernicious misconceptions, and malignant deviations from the orthodoxy they have modestly helped to create.     

Expansive Subdynamics

This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let be a continuous action of on an infinite compact metric space. For each subspace of we introduce a notion of expansiveness for along , and show that there are nonexpansive subspaces in every dimension . For each the set of expansive -dimensional subspaces is open in the Grassmann manifold of all -dimensional subspaces of . Various dynamical properties of are constant, or vary nicely, within a connected component of , but change abruptly when passing from one expansive component to another. We give several examples of this sort of ``phase transition,' including the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an . The unresolved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our analysis.

     

Order and disorder in the approach to equilibrium of a classical gas

We study the approach to equilibrium of a classical gas. The initial condition corresponds to a Maxwell velocity distribution, but to a nonequilibrium binary correlation. We consider two cases. In the first, there are initially no spatial correlations, while in the second, initial correlations correspond to long-range spatial order. We show that the gas leaves the Maxwell velocity distribution function in the process of building up equilibrium correlations. The spatial correlations in the equilibrium state are seen to emerge from a self-organization process in the gas. Non-Markovian effects play an essential role in this process by coupling the velocity distribution and the binary correlations. For the case of initial long-range correlations we obtain anti-Boltzmann behavior in the evolution of the velocity distribution as the Boltzmann entropy decreases from the nonequilibrium to the equilibrium state. For this case we also have nontrivial behavior on a short time scale due to the non-Markovian effects. The approach used here is based on the theory of subdynamics as developed in previous publications. The results obtained show the interplay between irreversible processes leading to disorder and to order in a classical gas.     

A Framework for Non-Equilibrium Statistical Ensemble Theory

Since Gibbs synthesized a general equilibrium statistical ensemble theory, many theorists have attempted to generalized the Gibbsian theory to
non-equilibrium phenomena domain, however the status of the theory of non-equilibrium phenomena can not be said as firm as well established as the
Gibbsian ensemble theory. In this work, we present a framework for the non-equilibrium statistical ensemble formalism based on a subdynamic kinetic
equation (SKE) rooted from the Brussels-Austin school and followed by some up-to-date works. The constructed key is to use a similarity transformation between Gibbsian ensembles formalism based on Liouville equation and the subdynamic ensemble formalism based on the SKE. Using this formalism, we study the spin-Boson system, as cases of weak coupling or strongly coupling, and obtain the reduced density operators for the Canonical ensembles easily.