Kolmogorov-Sinai entropy production

Very recently, a new concept of time-reversed entropy per unit time was introduced as the complement of the Kolmogorov-Sinai entropy per unit time in order to make the connection with nonequilibrium thermodynamics and its entropy production [3]. This connection shows that the origin of entropy production can be... 

The number of typical paths generated by the random process increases as exp(/ifl). In this regard, the Kolmogorov-Sinai entropy per unit time is the rate of production of information by the random process. On the other hand, the time-reversed entropy per unit time is the rate of production of information by the time reversals of the typical paths. The thermodynamic entropy production is the difference between these two rates of information production. With the formula (101), we can recover a result by Landauer [50] and Bennett [51] that erasing information in the memory of a computer is an irreversible process of... 

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 = 2j3. The process is perfectly cyclic at a = 0 where the path is. .. 123123123123. .. and the Kolmogorov-Sinai entropy h vanishes as a consequence.

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. 

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