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.
The Kolmogorov-Sinai entropy for dilute systems of hard particles in equilibrium
在平衡系统中，柯尔莫哥洛夫-西奈熵（$h_{\mathrm{KS}}$）等于正的李雅普诺夫指数（无穷小扰动的指数发散率）之和。动力学理论可以用来计算在低数密度$n$下处于平衡的许多硬盘或球体的稀释气体的Kolmogorov-Sinai熵。$h_{\mathrm{KS}}$的密度展开式是$N \barνA [\ln n + B + O(n)]$，其中$\barν$是单粒子碰撞频率。以前对$A$的计算是成功的。然而，$B$的计算结果并不令人满意。在本文中，我展示了如何拉伸因子的概率分布可以确定从一个非线性微分方程的迭代方法。由此得出的柯尔莫哥洛夫-西奈熵是单位时间拉伸因子的对数的平均值。我计算近似值的$B$和比较这些现有的模拟结果。协议是好的。
Radius of curvature approach to the Kolmogorov-Sinai entropy of dilute hard particles in equilibrium
本文研究了$N$硬盘或球体中稀气体的Kolmogorov-Sinai熵。这可以在密度上扩展为$h_{\mathrm{KS}} \propto n N [\ln n a^d+ B + O(n a^d)+O(1/N)]$，其中$a$是球体或圆盘的直径，$n$是密度，而$d$是系统的维数。我们估计常数$B$通过求解线性微分方程的近似分布的特征值的反曲率半径张量。我们比较结果值的$B$以前的估计和现有的模拟结果，发现非常好的协议与后者。此外，我们比较了新的模拟结果的反曲率曲率半径张量的特征值的分布从我们的计算。对于大多数光谱，我们的计算和模拟之间的一致性非常好。
We study the dynamics of a Bose-Einstein condensate in a Sinai-oscillator trap under a monochromatic driving force. Such a trap is formed by a harmonic potential and a repulsive disk located in the center vicinity corresponding to the first experiments of condensate formation by Ketterle group in 1995. We argue that the external driving allows to model the regime of weak wave turbulence with the Kolmogorov energy flow from low to high energies. We show that in a certain regime of weak driving and weak nonlinearity such a turbulent energy flow is defeated by the Anderson localization that leads…
Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases
Yang-Mills fields are an important part of the non-Abelian space theory describing the properties of quark-gluon plasma. The dynamics of the classical fields are given by the Hamiltonian equations of motion, which contain the member of the field strength tensor SU(2) \cite{1.}. This system exhibits chaotic behavior. The homogeneous Yang-Mills equation includes the quadratic part of the field strength tensor $F_{μν}^a$ expressed in Minkowski space, which was determined by the fields $A_μ^a $. The dynamics of the classical Yang-Mills field equations arise from the Hamiltonian SU(2) field tensor …
In classical chaotic systems the entropy, averaged over initial phase space distributions, follows an universal behavior. While approaching thermal equilibrium it passes through a stage where it grows linearly, while the growth rate, the Kolmogorov-Sinai entropy, is given by the sum over all positive Lyapunov exponents. A natural question is whether a similar relation is valid for quantum systems. We argue that the Maldacena-Shenker-Stanford bound on quantum Lyapunov exponents $λ$ implies that the upper bound on the growth rate of the entropy, averaged over states in Hilbert space that evolve …
By performing a large number of fully resolved simulations of incompressible homogeneous and isotropic two dimensional turbulence, we study the scaling behavior of the maximal Lyapunov exponent, the Kolmogorov-Sinai entropy and attractor dimension. The scaling of the maximal Lyapunov exponent is found to be in good agreement with the dimensional predictions. For the cases of the Kolmogorov-Sinai entropy and attractor dimension the simple dimensional predictions are found to be insufficient. A dependence on the system size and the forcing length scale is found, suggesting non-universal behavior…
A powerful tool is developed for the characterization of chaotic signals. The approach is based on the symbolic encoding of time series (according to their ordinal patterns) combined with the ensuing characterization of the corresponding cylinder sets. Quantitative estimates of the Kolmogorov-Sinai entropy are obtained by introducing a modified permutation entropy which takes into account the average width of the cylinder sets. The method works also in hyperchaotic systems and allows estimating the fractal dimension of the underlying attractors.
Time-series analysis in terms of ordinal patterns is revisited by introducing a generalized permutation entropy $H_p(w,L)$, which depends on two different window lengths: $w$, implicitly defining the resolution of the underlying partition; $L$, playing the role of an embedding dimension, analogously to standard nonlinear time-series analysis. The $w$-dependence provides information on the structure of the corresponding invariant measure, while the $L$-dependence helps determining the Kolmogorov-Sinai entropy. We finally investigate the structure of the partition with the help of principal comp…
We study the Reynolds number scaling of the Kolmogorov-Sinai entropy and attractor dimension for three dimensional homogeneous isotropic turbulence through the use of direct numerical simulation. To do so, we obtain Lyapunov spectra for a range of different Reynolds numbers by following the divergence of a large number of orthogonal fluid trajectories. We find that the attractor dimension grows with the Reynolds number as Re$^{2.35}$ with this exponent being larger than predicted by either dimensional arguments or intermittency models. The distribution of Lyapunov exponents is found to be fini…
We analyse the infinite-dimensional limit of the maximally chaotic dynamical systems that are defined on N-dimensional tori. These hyperbolic systems found successful application in computer algorithms that generate high-quality pseudorandom numbers for advanced Monte Carlo simulations. The chaotic properties of these systems are increasing with $N$ because the corresponding Kolmogorov-Sinai entropy grows linearly with $N$. We calculated the spectrum and the entropy of the system that appears in the infinite dimensional limit. We demonstrated that the limiting system has exponentially expandin…
