Dean J. Driebe

Transport as a dynamical property of a simple map

Abstract We study a simple chaotic system given by a map which displays diffusion. The decaying eigenstates of the Frobenius-Perron operator of the map are constructed. These eigenstates are associated with the Ruelle resonances. The eigenstate associated with diffusion is considered in particular. The result implies a deep connection between the thermodynamical properties of a system and its underlying exact dynamics.

Spectral determination and physical conditions for a class of chaotic piecewise-linear maps

Abstract We study the determination of the spectrum of the Frobenius-Perron operatorfor a class of piecewise-linear maps that satisfy an intertwining relation between the derivative operator and the Frobenius-Perron operator. The multi-Bernoulli map is used for illustration by employing an Euler-Maclaurin expansion of the resolvent of its Frobenius-Perron operator. We find that the spectrum is determined by the smoothness of the functions in the domain of the Frobenius-Perron operator. The decaying left eigenstates are Schwartz distributions which we express in terms of a class of fractal functions.

Using symmetries of the Frobenius-Perron operator to determine spectral decompositions

Abstract The spectral decomposition of the Frobenius-Perron operator for a class of piecewise-linear maps is determined from symmetry transformations of the dyadic Bernoulli map. This approach enables one to construct explicit compact expressions for the eigenstates.

The Bernoulli Map

In this chapter we study in detail the simplest example of a chaotic transformation for which we may construct a spectral decomposition of the Frobenius—Perron operator that explicitly incorporates the discrete, physical decay modes of the system. As we will see, this decomposition has elements belonging to a generalized functional space, which implies a restriction of the domain of the Frobenius—Perron operator to smooth densities. We give full details of the construction for this system since the main aspects of the analysis will be applicable to all the systems we will consider.

Spectral decomposition of the tent map with varying height

The generalized spectral decomposition of the Frobenius-Perron operator of the tent map with varying height is determined at the band-splitting points. The decomposition includes both decay onto the attracting set and the approach to the asymptotically periodic state on the attractor. Explicit compact expressions for the polynomial eigenstates are obtained using algebraic techniques. (c) 1998 American Institute of Physics.

Spectral decomposition of tent maps using symmetry considerations

The spectral decomposition of the Frobenius-Perron operator of maps composed of many tents is determined from symmetry considerations. The eigenstates involve Euler as well as Bernoulli polynomials.

Generalized second law for a simple chaotic system

The generalized second law (nonequilibrium maximum work formulation) is derived for a simple chaotic system. We consider a probability density, prepared in the far past, which weakly converges to an invariant density due to the mixing property. The generalized second law is then rewritten for an initial invariant density. Gibbs-Shannon entropy is constant in time, but the invariant density has a greater entropy than the prepared density. The maximum work is reduced due to the greater entropy of the invariant density. If and only if the invariant density is a canonical distribution, work is not extractable by any cyclic operation. This gives us the unique equilibrium state. Our argument is extended for a power invariant density such as the Tsallis distribution. On the basis of the Tsallis entropy, the maximum q-work formulation is derived. If and only if the invariant density is a Tsallis distribution, the q-work is no longer extractable by any cyclic operation.

Application of the Generalized Work Relation for an N-level Quantum System

An efficient periodic operation to obtain the maximum work from a nonequilibrium initial state in an N–level quantum system is shown. Each cycle consists of a stabilization process followed by an isentropic restoration process. The instantaneous time limit can be taken in the stabilization process from the nonequilibrium initial state to a stable passive state. In the restoration process that preserves the passive state a minimum period is needed to satisfy the uncertainty relation between energy and time. An efficient quantum feedback control in a symmetric two–level quantum system connected to an energy source is proposed.

Ensemble Dynamics of Intermittency and Power-Law Decay

A spectral decomposition of the Frobenius–Perron operator is constructed for one-dimensional maps with intermittent chaos, using the method of coherent states. A technique using the spectral density function is applied to the the well-known cusp map, which generates weak type-II intermittency. Higher-order corrections are obtained to the leading 1/t long-time behavior of the x−x autocorrelation.

Statistical Mechanics of Chaotic Maps

We give now a more precise formulation of some of the ideas mentioned in the first chapter. The basic tools needed for a statistical mechanics approach to chaotic dynamics are given. Resonances are discussed by considering the various types of correlation decay in mixing systems.