Dynamical entropy

The discretized symbol-sequence defined in equation 8.39 suggests that we might use two other familiar measures of complexity to characterize the various dynamical regimes of behavior namely, the pattern entropy and dynamical entropy. 

Dynamical Entropy In order to capture the dynamics of a CML pattern, Kaneko has constructed what amounts to it mutual information between two successive patterns at a given time interval [kaneko93]. It is defined by first obtaining an estimate, through spatio-temporal samplings, of the probability transition matrix Td,d = transition horn domain of size D to a domain of size D. The dynamical entropy, Sd, is then given by... 

Regime Spatial Power Spectra S(k) Pattern Distribution Q(.D) Pattern Entropy Sp Dynamical Entropy Sd... 

GEN.29.1. Prigogine, Entropie et Dynamique (Entropy and dynamics), Entropie 57, 5—11 (1974). GEN.30.1. Prigogine, Leon Rosenfeld et les fondements de la physique moderne (Leon Rosenfeld and the foundations of modern physics), Bull. Cl. Sci. Acad. Roy. Belg. 60, 841-854 (1974). 

We notice that the only difference between both dynamical entropies is the exchange of oa and oa in the transition probabilities appearing in the logarithm. According to Eq. (101), the thermodynamic entropy production of this process would be equal to... 

Figure 18. The dynamical entropies (126) and (127) as well as the entropy production (128) for the three-state Markov chain defined by the matrix (125) of transition probabilities versus the parameter a. The equilibrium corresponds to the value a — Ij i. The process is perfectly cyclic at a = 0 where the path is. .. 123123123123. .. and the Kohnogorov-Sinai entropy h vanishes as a...

In addition, it is possible to derive an expression for the time variation of the local entropy density by writing a dynamic entropy balance ... 

We have obtained several interesting results from the theorem If the period of the external transformation is much longer than the relaxation time, then thermodynamic entropy production is proportional to the ratio of the period and relaxation time. The relaxation time is proportional to the inverse of the Kolmogorov-Sinai entropy for small strongly chaotic systems. Thermodynamic entropy production is proportional to the inverse of the dynamical entropy [11]. On the other hand, thermodynamic entropy production is proportional to the dynamical entropy when the period of the external transformation is much shorter than the relaxation time. Furthermore, we found fractional scaling of the excess heat for long-period external transformations, when the system has longtime correlation such as 1 /fa noise. Since excess heat is measured as the area of a hysteresis loop [12], these properties can be confirmed in experiments. 

The area of the hysteresis loop is proportional to jT, so it increases for greater T. The thermodynamic entropy production is proportional to the dynamical entropy production in this case [22, 23]. In a large chaotic system, we can expect less thermodynamic entropy production for weak diffusion. 

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