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p. 222-237
Received: 30 December 2008 / Accepted: 31 March 2009 / Published: 1 April 2009
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| Download PDF Full-text (563 KB) Abstract: In a compact group the Haar probability measure plays the role of uniform distribution. The entropy and rate distortion theory for this uniform distribution is studied. New results and simplified proofs on convergence of convolutions on compact groups are presented and they can be formulated as entropy increases to its maximum. Information theoretic techniques and Markov chains play a crucial role. The convergence results are also formulated via rate distortion functions. The rate of convergence is shown to be exponential.
p. 238-248
Received: 30 December 2008 / Accepted: 20 April 2009 / Published: 21 April 2009
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| Download PDF Full-text (181 KB) Abstract: In this paper, we briefly discuss a field theory approach of classical statistical mechanics. We show how an essentially entropic functional accounts for fundamental symmetries related to quantum mechanical properties which hold out in the classical limit of the quantum description. Within this framework, energetic and entropic properties are treated at equal level. Based on a series of examples on electrolytes, we illustrate how this framework gives simple interpretations where entropic fluctuations of anions and cations compete with the energetic properties related to the interaction potential.
p. 249-270
Received: 12 January 2009 / Accepted: 24 April 2009 / Published: 29 April 2009
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| Download PDF Full-text (1740 KB) Abstract: This paper provides a new approach for the analysis and eventually the classification of dynamical systems. The objective is pursued by extending the concept of the entropy of plane curves, first introduced within the theory of the thermodynamics of plane curves, to Rn space. Such a generalised entropy of a curve is used to evaluate curves that are obtained by connecting several points in the phase space. As the points change their coordinates according to the equations of a dynamical system, the entropy of the curve connecting them is used to infer the behaviour of the underlying dynamics. According to the proposed method all linear dynamical systems evolve at constant zero entropy, while higher asymptotic values characterise nonlinear systems. The approach proves to be particularly efficient when applied to chaotic systems, in which case it has common features with other classic approaches. Performances of the proposed method are tested over several benchmark problems.
p. 271-284
Received: 13 April 2009 / Accepted: 6 May 2009 / Published: 11 May 2009
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| Download PDF Full-text (241 KB) Abstract: This elucidation studies ergodicity and equilibrium measures for additive cellular automata with prime states. Additive cellular automata are ergodic with respect to Bernoulli measure unless it is either an identity map or constant. The formulae of measure-theoretic and topological entropies can be expressed in closed forms and the topological pressure is demonstrated explicitly for potential functions that depend on finitely many coordinates. According to these results, Parry measure is inferred to be an equilibrium measure.
p. 285-294
Received: 7 May 2009 / Accepted: 9 June 2009 / Published: 11 June 2009
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| Download PDF Full-text (183 KB) Abstract: We show that the configurational probability distribution of a classical gas always belongs to the q-exponential family. One of the consequences of this observation is that the thermodynamics of the configurational subsystem is uniquely determined up to a scaling function. As an example we consider a system of non-interacting harmonic oscillators. In this example, the scaling function can be determined from the requirement that in the limit of large systems the microcanonical temperature of the configurational subsystem should coincide with that of the canonical ensemble. The result suggests that R´enyi’s entropy function is the relevant one rather than that of Tsallis.
p. 295-325
Received: 24 March 2009 / Accepted: 19 June 2009 / Published: 22 June 2009
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| Download PDF Full-text (710 KB) Abstract: In this paper, we propose definitions for a general, domain-independent concept of replicability and specifically focus on the notion of self-replication. We argue that selfreplication should not be viewed as a binary property of a system, but rather as a continuously valued property of the interaction between a system and its environment. This property, selfreplicability, represents the effect of the presence of a system upon the future presence of similar systems. We demonstrate both analytical and computational analysis of self-replicability for four distinct systems involving both discrete and continuous, formal and physical behaviors.
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