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Maximum Kolmogorov-Sinai entropy vs minimum mixing time in Markov chains  [PDF]
Martin Mihelich,Berengere Dubrulle,Didier Paillard,Davide Faranda,Quentin Kral
Physics , 2015,
Abstract: Many modern techniques employed in physics, such a computation of path integrals, rely on random walks on graphs that can be represented as Markov chains. Traditionally, estimates of running times of such sampling algorithms are computed using the number of steps in the chain needed to reach the stationary distribution. This quantity is generally defined as mixing time and is often difficult to compute. In this paper, we suggest an alternative estimate based on the Kolmogorov-Sinai entropy, by establishing a link between the maximization of KSE and the minimization of the mixing time. Since KSE are easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time that could be interesting in computer sciences and statistical physics. Beyond this, our finding will also be of interest to the out-of-equilibrium community, by providing a new rational to select stationary states in out-of-equilibrium physics: it seems reasonable that in a physical system with two simultaneous equiprobable possible dynamics, the final stationary state will be closer to the stationary state corresponding to the fastest dynamics (smallest mixing time).Through the empirical link found in this letter, this state will correspond to a state of maximal Kolmogorov-Sinai entropy. If this is true, this would provide a more satisfying rule for selecting stationary states in complex systems such as climate than the maximization of the entropy production.
Kolmogorov-Sinai entropy and black holes  [PDF]
Kostyantyn Ropotenko
Physics , 2008, DOI: 10.1088/0264-9381/25/19/195005
Abstract: It is shown that stringy matter near the event horizon of a Schwarzschild black hole exhibits chaotic behavior (the spreading effect) which can be characterized by the Kolmogorov-Sinai entropy. It is found that the Kolmogorov-Sinai entropy of a spreading string equals to the half of the inverse gravitational radius of the black hole. But the KS entropy is the same for all objects collapsing into the black hole. The nature of this universality is that the KS entropy possesses the main property of temperature: it is the same for all bodies in thermal equilibrium with the black hole. The Kolmogorov-Sinai entropy measures the rate at which information about the string is lost as it spreads over the horizon. It is argued that it is the maximum rate allowed by quantum theory. A possible relation between the Kolmogorov-Sinai and Bekenstein-Hawking entropies is discussed.
An approach to comparing Kolmogorov-Sinai and permutation entropy  [PDF]
Valentina A. Unakafova,Anton M. Unakafov,Karsten Keller
Mathematics , 2015, DOI: 10.1140/epjst/e2013-01846-7
Abstract: In this paper we investigate a quantity called conditional entropy of ordinal patterns, akin to the permutation entropy. The conditional entropy of ordinal patterns describes the average diversity of the ordinal patterns succeeding a given ordinal pattern. We observe that this quantity provides a good estimation of the Kolmogorov-Sinai entropy in many cases. In particular, the conditional entropy of ordinal patterns of a finite order coincides with the Kolmogorov-Sinai entropy for periodic dynamics and for Markov shifts over a binary alphabet. Finally, the conditional entropy of ordinal patterns is computationally simple and thus can be well applied to real-world data.
Kolmogorov-Sinai entropy from the ordinal viewpoint  [PDF]
Karsten Keller,Mathieu Sinn
Physics , 2009, DOI: 10.1016/j.physd.2010.02.006
Abstract: In the case of ergodicity much of the structure of a one-dimensional time-discrete dynamical system is already determined by its ordinal structure. We generally discuss this phenomenon by considering the distribution of ordinal patterns, which describe the up and down in the orbits of a Borel measurable map on a subset of the real numbers. In particular, we give a natural ordinal description of Kolmogorov-Sinai entropy of a large class of one-dimensional dynamical systems and relate Kolmogorov-Sinai entropy to the permutation entropy recently introduced by Bandt and Pompe.
Eigenvalue Estimates Using the Kolmogorov-Sinai Entropy  [PDF]
Shih-Feng Shieh
Entropy , 2011, DOI: 10.3390/e13122036
Abstract: The scope of this paper is twofold. First, we use the Kolmogorov-Sinai Entropy to estimate lower bounds for dominant eigenvalues of nonnegative matrices. The lower bound is better than the Rayleigh quotient. Second, we use this estimate to give a nontrivial lower bound for the gaps of dominant eigenvalues of A and A + V.
The Kolmogorov-Sinai Entropy for Dilute Gases in Equilibrium  [PDF]
H. van Beijeren,J. R. Dorfman,H. A. Posch,Ch. Dellago
Physics , 1997, DOI: 10.1103/PhysRevE.56.5272
Abstract: We use the kinetic theory of gases to compute the Kolmogorov-Sinai entropy per particle for a dilute gas in equilibrium. For an equilibrium system, the KS entropy, h_KS is the sum of all of the positive Lyapunov exponents characterizing the chaotic behavior of the gas. We compute h_KS/N, where N is the number of particles in the gas. This quantity has a density expansion of the form h_KS/N = a\nu[-\ln{\tilde{n}} + b + O(\tilde{n})], where \nu is the single-particle collision frequency and \tilde{n} is the reduced number density of the gas. The theoretical values for the coefficients a and b are compared with the results of computer simulations, with excellent agreement for a, and less than satisfactory agreement for b. Possible reasons for this difference in b are discussed.
Renormalization Group versus Kolmogorov-Sinai entropy: a very simple remark  [PDF]
Gavriel Segre
Physics , 2005,
Abstract: A very simple remark concerning a link between the notions of Kolmogorov-Sinai entropy and of Renormalization Group is performed.
The Kolmogorov-Sinai entropy for dilute systems of hard particles in equilibrium  [PDF]
Astrid S. de Wijn
Physics , 2004, DOI: 10.1103/PhysRevE.71.046211
Abstract: In an equilibrium system, the Kolmogorov-Sinai entropy, $h_{\mathrm{KS}}$, equals the sum of the positive Lyapunov exponents, the exponential rates of divergence of infinitesimal perturbations. Kinetic theory may be used to calculate the Kolmogorov-Sinai entropy for dilute gases of many hard disks or spheres in equilibrium at low number density $n$. The density expansion of $h_{\mathrm{KS}}$ is $N \bar\nu A [\ln n + B + O(n)]$, where $\bar\nu$ is the single-particle collision frequency. Previous calculations of $A$ were succesful. Calculations of $B$, however, were unsatisfactory. In this paper, I show how the probability distribution of the stretching factor can be determined from a nonlinear differential equation by an iterative method. From this the Kolmogorov-Sinai entropy follows as the average of the logarithm of the stretching factor per unit time. I calculate approximate values of $B$ and compare these to results from existing simulations. The agreement is good.
Comment on `Universal relation between the Kolmogorov-Sinai entropy and the thermodynamic entropy in simple liquids'  [PDF]
E. G. D. Cohen,L. Rondoni
Physics , 1999, DOI: 10.1103/PhysRevLett.84.394
Abstract: The intriguing relations between Kolmogorov-Sinai entropy and self diffusion coefficients and the excess (thermodynamic) entropy found by Dzugutov and collaborators do not appear to hold for hard sphere and hard disks systems.
The Problem of Positive Kolmogorov-Sinai entropy for the Standard map  [PDF]
Oliver Knill
Mathematics , 1999,
Abstract: The problem of positive Kolmogorov-Sinai entropy of the Chirikov-Standard map with respect to the invariant Lebesgue measure on the two-dimensional is open. In 1999, we believed to have a proof that the entropy can be bounded below. This approach was based on an idea of Herman to do subharmonic estimates.This document replaces an announcement I had circulated in 1999. In the present document, the incorrect parts have been deleted. The entropy conjecture is open. The references given in the text might still be helpful for people trying an operator theoretical or analytic approach to this problem in ergodic theory.
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