Computation of the Kolmogorov-Sinai entropy using statis- titical mechanics: Application of an exchange Monte Carlo method

– We propose a method for computing the Kolmogorov-Sinai (KS) entropy of chaotic systems. In this method, the KS entropy is expressed as a statistical average over the canonical ensemble for a Hamiltonian with many ground states. This Hamiltonian is constructed directly from an evolution equation that exhibits chaotic dynamics. As an example, we compute the KS entropy for a chaotic repeller by evaluating the thermodynamic entropy of a system with many ground states. Chaotic systems are characterized by properties of O(e ) distinguishable trajectories over a time interval N in a phase space in which the limit of observational accuracy is given by some finite value ǫ [1–3]. In particular, if these trajectories are coded as sequences of symbols, the optimal compression ratio of such sequences is an important quantity from the point of view of information theory [4, 5]. This quantity corresponds to the Kolmogorov-Sinai (KS) entropy, hks. Although the KS entropy provides a simple characterization of chaotic systems, its direct numerical evaluation is not simple. One reason for this is that a precise mathematical definition of distinguishable trajectories relies on an (ǫ,N) separated set that is difficult to investigate numerically [1]. However, this difficulty can be overcome when a set of periodic orbits alone can provide information that is sufficient to obtain a useful characterization of the set of all distinguishable trajectories [6–8]. However, even in such cases, the difficulty involved in numerical computations remains, because the number of periodic orbits of period N increases exponentially as a function of N . Here, let us recall that such an exponential dependence of the number of explored states appears in the statistical mechanics of glassy systems [9] and decision problems [10]. Recently, efficient methods for computing thermodynamic quantities of glassy systems have been developed [11, 12] and applied to several problems [13–15]. Inspired by this success, in this (∗) E-mail:[email protected] (∗∗) E-mail:[email protected]