Observing how long a dynamical system takes to return to some state is one of the most simple ways to model and quantify its dynamics from data series. This work proposes two formulas to estimate the KS entropy and a lower bound of it, a sort of Shannon's entropy per unit of time, from the recurrence times of chaotic systems. One formula provides the KS entropy and is more theoretically oriented since one has to measure also the low probable very long returns. The other provides a lower bound for the KS entropy and is more experimentally oriented since one has to measure only the high probable short returns. These formulas are a consequence of the fact that the series of returns do contain the same information of the trajectory that generated it. That suggests that recurrence times might be valuable when making models of complex systems.
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。
Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy and the Largest Lyapunov Exponents for Dilute, Hard-Ball Gases and for Dilute, Random Lorentz Gases
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…
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 …
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…
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…
利用符号动力学的方法，我们证明了一大类经典混沌映射对扰动具有指数超敏性，即，描述刘维尔密度的扰动时间演化所需的信息随时间的快速增加，该信息获得的值指数地大于由扰动平均产生的熵增加。信息与熵之比的指数增长率由映射的Kolmogorov-Sinai熵给出。这些结果推广和发展了关于面包师映射[R. Schack和C. M. Caves，Phys. Rev. Lett. 69，3413（1992）]。
