Kolmogorov-Sinai entropy

SbS = Smea.s B) (compare to the Kolmogorov-Sinai entropy, defined in equation 4.101). 

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... 

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

The Lyapunov exponents and the Kolmogorov-Sinai entropy per unit time concern the short time scale of the kinetics of collisions taking place in the fluid. The longer time scales of the hydrodynamics are instead characterized by the decay of the statistical averages or the time correlation functions of the... 

On the other hand, the dynamical randomness is characterized by the Kolmogorov-Sinai entropy per unit time ... 

In systems with two degrees of freedom such as the two-dimensional Lorentz gases, there is a single positive Lyapunov exponent X and the partial Hausdorff dimension of the set of nonescaping trajectories can be estimated by the ratio of the Kolmogorov-Sinai entropy to the Lyapunov exponent [ 1, 38]... 

Figure 13. Diagram showing how dynamical instability characterized by the sum of positive Lyapunov exponents > o contributes to dynamical randomness characterized by the Kolmogorov-Sinai entropy per unit time h s and to the escape y due to transport according to the chaos-transport formula (95).

The Kolmogorov-Sinai entropy per unit time is defined in Eq. (89) as the supremum of h over all the possible partitions V. Since we expect that the probability of the nonequilibrium steady state is not time-reversal symmetric, the probability of the time-reversed paths should decay at a different rate, which can be called a time-reversed entropy per unit time [3]... 

The phase space is partitioned into cells co of diameter 5. In the limit of an arbitrarily fine partition, the entropy per unit time tends to the Kolmogorov-Sinai entropy per unit time which is equal to the sum of positive Lyapunov exponents by Pesin theorem [16] ... 

According to dynamical systems theory, the escape rate is given by the difference (92) between the sum of positive Lyapunov exponents and the Kolmogorov-Sinai entropy. Since the dynamics is Hamiltonian and satisfies Liouville s theorem, the sum of positive Lyapunov exponents is equal to minus the sum of negative ones ... 

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.

P. Gaspard Concerning multiple-pulse echo experiments, I would like to know if there are results on the decay of the amplitude of the echo as the number of pulses increases with equal-time spacing between pulses. If the decay is exponential, the rate of decay may characterize dynamical randomness since it is closely related to the so-called Kolmogorov-Sinai entropy per unit time [see P. Gaspard, Prog. Theor. Phys. Suppl. 116, 369 (1994)]. 

The problem of a kinematic dynamo in a steady velocity field can be treated mathematically as a problem of the effect of a small diffusion or round-off error on the Kolmogorov-Sinai entropy (or Lyapunov exponent) of a dynamical system which is specified by the velocity field v. This problem, on which Ya.B. worked actively, therefore has a general mathematical nature as well, and each step toward its solution is simultaneously a step forward in several seemingly distant areas of modern mathematics. 

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 dynamical randomness is characterized by the Kolmogorov-Sinai entropy per unit length. The Kolmogorov-Sinai (KS) entropy is the rate of accumulation of data necessary and sufficient to follow unambiguously an orbit on the repellor. is given by the relation ... 

---