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Multiscale measures of equilibrium on finite dynamic systems. (English) Zbl 1075.37530

Summary: This article presents a new method for studying the evolution of dynamical systems based on the notion of quantity of information. The system is divided into elementary cells and the quantity of information is studied with respect to the cell size. We have introduced an analogy between quantity of information and entropy, and defined the intrinsic entropy as the entropy of the whole system independent of the size of the cells. It is shown that the intrinsic entropy follows a Gaussian probability density function and thereafter, the time needed by the system to reach the equilibrium is a random variable. For a finite system, statistical analysis shows that this entropy converges to a state of equilibrium and an algorithmic method is proposed to quantify the time needed to reach the equilibrium for a given confidence interval level. A Monte-Carlo simulation of diffusion of A\(^{\ast}\) atoms in A is then provided to illustrate the proposed simulation. It follows that the time to reach the equilibrium for a constant error probability, \(t_{\text e}\), depends on the number, \(n\), of elementary cells as: \(t_{\text e} \propto n^{2.22_{\pm 0.06}}\). For an infinite system size (\(n\) infinite), the intrinsic entropy obtained by statistical modelling is a pertinent characteristic number of the system at the equilibrium.

MSC:

37M05 Simulation of dynamical systems
28A80 Fractals
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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[1] Brillouin, L., Science and information theory (1956), Academic Press: Academic Press New York · Zbl 0071.13104
[2] Zurek, W. H., Complexity, entropy and the physics of information (1990), Addison-Wesley · Zbl 0800.94140
[3] Shannon, C. E., Bell System Technical Journal, 27, 379-423 (1948), 623-56
[4] Bunde, A.; Havlin, S., Fractals and disordered systems (1995), Springer
[5] Peitgen, H. O.; Jrgens, H.; Saupe, D., Chaos and fractals new frontiers of science (1992), Springer-Verlag · Zbl 0779.58004
[6] El Naschie, M. S., On the uncertaintly of Cantorian geometry and the two-slit experiment, Chaos, Solitons & Fractals, 9, 3, 517-529 (1998) · Zbl 0935.81009
[7] El Naschie, M. S., Cobe satellite measurement, hyperspheres, superstrings and the dimension of space-time, Chaos, Solitons & Fractals, 9, 8, 1445-1471 (1998) · Zbl 1038.83508
[8] El Naschie, M. S., Branching polymers and the fractal Cantorian space-time, Chaos, Solitons & Fractals, 9, 1/2, 131-141 (1998) · Zbl 1085.82513
[9] Rief, F., Fundamentals of statistical and thermal physics (1965), McGrawHill: McGrawHill New York
[10] Conrad, M., Anti-entropy and the origin of initial condition, Chaos, Solitons & Fractals, 7, 5, 725-745 (1996) · Zbl 1080.80503
[11] Conrad, M., Across-scale information processing in evolution, development and intelligence, BioSystems, 38, 97-109 (1996)
[12] Conrad, M., Quantum gravity and life, BioSystems, 46, 29-39 (1998)
[13] Conrad, M., Origin of life and the underlying physics of the universe, BioSystems, 42, 177-190 (1998)
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