Chaotic Dynamics

We present an efficient method for fast, complete, and accurate detection of unstable periodic orbits in chaotic systems. Our method consists of a new iterative scheme and an effective technique for selecting initial points. The iterative scheme is based on the semi-implicit Euler method, which has both fast and global convergence, and only a small number of initial points is sufficient to detect all unstable periodic orbits of a given period. The power of our method is illustrated by numerical examples of both two- and four-dimensional maps.

The eigenvalue statistics of quantum ideal gases with single particle energies e n = n α are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. For integer α>1 one expects and finds number theoretic degeneracies and deviations from the Poissonian spacing distribution. By semiclassical arguments, the length spectrum of the classical system is shown to be related to sums of integers to the power α/(α−1) . In particular, for α=3/2 , the periodic orbits are related to sums of cubes, for which one again expects number theoretic degeneracies, with consequences for the two point correlation function.

To understand the mechanism allowing for long-term storage of excess energy in proteins, we study a Hamiltonian system consisting of several coupled pendula in partial contact with a heat bath. It is found that energy absorption and storage are possible when the motion of each pendulum switches between oscillatory (vibrational) and rotational modes. The relevance of our mechanism to protein motors is discussed.

The Reynolds number dependence of the statistics of energy dissipation is investigated in a shell model of fully developed turbulence. The results are in agreement with a model which accounts for fluctuations of the dissipative scale with the intensity of energy dissipation. It is shown that the assumption of a fixed dissipative scale leads to a different scaling with Reynolds which is not compatible with numerical results.

Landauer discussed the minimum energy necessary for computation and stated that erasure of information is accompanied by heat generation to the amount of kT ln2/bit. Modifying the above statement, we claim that erasure of information is accompanied by entropy generation k ln2/bit. Some new concepts will be introduced in the field of thermodynamics that are implicitly included in our statement. The new concepts that we will introduce are ``partitioned state'', which corresponds to frozen state such as in ice, ``partitioning process'' and ``unifying process''. Developing our statement, i.e., our thermodynamics of computation, we will point out that the so-called ``residual entropy'' does not exist in the partitioned state. We then argue that a partioning process is an entropy decreasing process. Finally we reconsider the second law of thermodynamics especially when computational processes are involved.

To make clear several issues relating with the thermodynamics of computations, we perform a simulation of a binary device using a Langevin equation. Based on our numerical results, we consider how to estimate thermodynamic entropy of computational devices. We then argue against the existence of the so-called residual entropy in frozen systems such as ice.

The ergodic hypothesis outgrew from the ancient conception of motion as periodic or quasi periodic. It did cause a revision of our views of motion, particularly through Boltzmann and Poincaré: we discuss how Boltmann's conception of motion is still very modern and how it can provide ideas and methods to study the problem of nonequilibrium in mechanics and in fluids. This leads to the chaotic hypothesis, a recent interpretation of a very ambitious principle conceived by D. Ruelle: it is a possible extension of the ergodic hypothesis and it implies general parameterless relations. Together with further ideas, it appears to be consistent with some recent experiments as we discuss here.

We investigate fluid transport in random velocity fields with unsteady drift. First, we propose to quantify fluid transport between flow regimes of different characteristic motion, by escape probability and mean residence time. We then develop numerical algorithms to solve for escape probability and mean residence time, which are described by backward Fokker-Planck type partial differential equations. A few computational issues are also discussed. Finally, we apply these ideas and numerical algorithms to a tidal flow model.

By computing 254 unstable stationary solutions of the Kuramoto-Sivashinsky equation in the extensive chaos regime (Lyapunov fractal dimension D=8.8), we find that 30% satisfy the symmetry of the time-average pattern of the spatiotemporal chaos. Using a symmetry pruning of unstable stationary solutions, the escape-time weighting average converges to the time-average pattern of the chaotic attractor as O(1/N), where N is the total number of unstable stationary solutions and unstable periodic orbits in the average.

We apply periodic orbit theory to study the asymptotic distribution of escape times from an intermittent map. The dynamical zeta function exhibits a branch point which is associated with an asymptotic power law escape. By an analytic continuation technique we compute a zero of the zeta function beyond its radius of convergence leading to a pre-asymptotic exponential decay. The time of crossover from an exponential to a power law is also predicted. The theoretical predictions are confirmed by numerical simulation. Applications to conductance fluctuations in quantum dots are discussed.