Computation of the Kolmogorov-Sinai entropy using statis- titical mechanics: Application of an exchange Monte Carlo method
نویسندگان
چکیده
– 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]
منابع مشابه
Observational Modeling of the Kolmogorov-Sinai Entropy
In this paper, Kolmogorov-Sinai entropy is studied using mathematical modeling of an observer $ Theta $. The relative entropy of a sub-$ sigma_Theta $-algebra having finite atoms is defined and then the ergodic properties of relative semi-dynamical systems are investigated. Also, a relative version of Kolmogorov-Sinai theorem is given. Finally, it is proved that the relative entropy of a...
متن کاملEntropy of infinite systems and transformations
The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with ...
متن کاملApplication of Monte Carlo Simulation in the Assessment of European Call Options
In this paper, the pricing of a European call option on the underlying asset is performed by using a Monte Carlo method, one of the powerful simulation methods, where the price development of the asset is simulated and value of the claim is computed in terms of an expected value. The proposed approach, applied in Monte Carlo simulation, is based on the Black-Scholes equation which generally def...
متن کاملThe concept of logic entropy on D-posets
In this paper, a new invariant called {it logic entropy} for dynamical systems on a D-poset is introduced. Also, the {it conditional logical entropy} is defined and then some of its properties are studied. The invariance of the {it logic entropy} of a system under isomorphism is proved. At the end, the notion of an $ m $-generator of a dynamical system is introduced and a version of the Kolm...
متن کاملAn Algebraic Approach to the Kolmogorov-sinai Entropy
We revisit the notion of Kolmogorov-Sinai entropy for classical dynamical systems in terms of an algebraic formalism. This is the starting point for deening the entropy for general non-commutative systems. Hereby typical quantum tools are introduced in the statistical description of classical dynamical systems. We illustrate the power of these techniques by providing a simple, self-contained pr...
متن کامل