Hiroshi H. Hasegawa

Decaying eigenstates for simple chaotic systems

Abstract We introduce a new approach to the study of chaotic classical dynamical systems. The method, based on the resolvent formalism of kinetic theory, expresses the time evolution of a probability distribution in terms of exponentially decaying eigenstates of the Perron-Frobenius operator. We illustrate the method by applying it to some simple chaotic maps.

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.

Non-equilibrium statistical mechanics of the baker map : Ruelle resonances and subdynamics

Abstract We examine the time evolution of a probability distribution under the baker map using the subdynamics approach of the Brussels group. Our method decomposes the time evolution into independent decaying modes associated with generalized complex eigenvalues of the generator of time evolution. These are the resonances introduced by Ruelle and others to describe the power spectrum of correlation functions in chaotic dynamical systems.

Spectral representations of the Bernoulli map

Abstract We discuss spectral representations of the Perron-Frobenius operator, U , associated with a highly chaotic map. The continuous spectrum of U does not contain (except coincidentally) information about physically accessible quantities such as decay rates of correlation functions. We show constructively that decay rates can be incorporated into a generalized spectral decomposition of U if its domain is restricted to smooth functions.

Exact power spectrum for a system of intermittent chaos

Abstract We introduce a systematic method to construct the spectral resolution of the Frobenius-Perron operator of a class of chaotic maps. We exactly obtain a power spectrum through the spectral resolution for a piecewise linear version of the Manneville-Pomeau map and compare it with data of numerical experiment.

Fractional power scaling of excess heat production

Abstract We derive the thermodynamics of a system with long-time correlations. We demonstrate fractional power scaling of excess heat production with respect to the period of an external transformation. There is a simple relation between the scaling power and the power α of 1/fα noise. We construct thermodynamics for a one-dimensional intermittent chaotic system and numerically confirm the fractional power scaling of excess heat production. We comment on the scaling of the area of the hysteresis loop in magnetic inversion experiments.

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.