Coarse-Grained Probabilistic Automata Mimicking Chaotic Systems

Abstract

Discretization of phase-space usually nullifies chaos in dynamical systems. We show that if randomness is associated with discretization dynamical chaos may survive, and be indistinguishable from that of the original chaotic system, when an entropic, coarse-grained analysis is performed. Relevance of this phenomenon to the problem of quantum chaos is discussed.

Mixing and reaction efficiency in closed domains

“…On the contrary for the reacting case (1), the presence of large-scale chaos has a minor role. This result is rather close to those obtained in other subtle issues such as the classical limit of quantum mechanics [14], or metastable balance between chaos and diffusion [15]. The paper is organized as follows: In Section II we introduce two flow models for the velocity field u and we discuss the mixing efficiency (for inert particles) in closed domains at varying the chaotic properties of Eq.…”

“…The above results confirm the subtle and intriguing combined effect due to Lagrangian chaos, diffusion, and reaction. This issue is important to many different fields including the classical limit of quantum mechanics [14]. In Ref.…”

About the Concept of Quantum Chaos

“…Others, by contrast, claim that chaos does not threaten the Correspondence Principle, but rather it expresses the emergence of the logarithmic timescale proportional to logh −D (with 2D the dimension of the phase space). This marks out the non-commutativity of the limits t → ∞ andh → 0 and the region where the Kolmogorov-Sinai entropy and its quantum versions agree [35][36][37][38][39]. The key point that compatibilizes both positions consists in realizing that the distinction between continuous and discrete spectrum is unambiguous only in the asymptotic limit t → ∞.…”

Properties making a chaotic system a good pseudo random number generator

“…We propose the multi-dimensional cat map as a PRNG having both these properties. Another important example of system with both the properties is the one proposed by Knuth [36]: one iterates the Fibonacci generator (11) with M = 2 31 − 1, τ 1 = 37 and τ 2 = 100, with this choice the period is extremely large, then the output sequence is obtained taking the variable in Eq. (11) every T steps (T = 1009 or 2009).…”

“…However, there are situations where the system is, strictly speaking, non chaotic (h KS = 0) but its features appear irregular to a certain extent. Such property (denoted with the term pseudo-chaos [11,12,13]) is basically due to the presence of long transient effects [14].…”