Kolmogorov-Sinai entropy of the dilute wet granular gas

We present an analytical expression for the Kolmogorov-Sinai entropy of a wet granular gas. The influence of the liquid is modeled by a hysteretic interaction force. For the dilute limit (two-particle collisions only), we find a simple expression accounting for the contribution of both the scattering states and the bound states in arbitrary dimensions. It is shown that the system is significantly more chaotic than a gas of (dry) hard spheres, as reflected by a pronounced increase of the Kolmogorov-Sinai entropy.     

An approach to comparing Kolmogorov-Sinai and permutation entropy

In this paper we discuss the relationship between permutation entropy and Kolmogorov-Sinai entropy in the one-dimensional case. For this, we consider partitions of the state space of a dynamical system using ordinal patterns of order (d + n? 1) on the one hand, and using n-letter words of ordinal patterns of order d on the other hand. The answer to the question of how different these partitions are provides an approach to comparing the entropies.     

Hamilton-Jacobi formulation of Kolmogorov-Sinai entropy for classical and quantum dynamics

A Hamilton-Jacobi formulation of the Lyapunov spectrum and Kolmogorov-Sinai (KS) entropy is developed. It is numerically efficient and reveals a close relation between the KS invariant and the classical action. This formulation is extended to the quantum domain using the Madelung-Bohm orbits associated with the Schroedinger equation. The resulting quantum KS invariant for a given orbit equals the mean decay rate of the probability density along the orbit, while its ensemble average measures the mean growth rate of configuration-space information for the quantum system.     

On the relation between the divergence of trajectories and the Kolmogorov-Sinai entropy

Automorphisms of the torus are considered and the Kolmogorov-Sinai entropy explicitly computed via rate of growth of volumes. The usual rate of growth of exponentiating trajectories is shown to give only a lower bound to K-S entropy.     

New universal scaling laws of diffusion and Kolmogorov-Sinai entropy in simple liquids

A new universal scaling law relating the self-diffusivities of the components of a binary fluid mixture to their excess entropies is derived using mode coupling theory. These scaling laws yield numerical results, for a hard sphere as well as Lennard-Jones fluid mixtures, in excellent agreement with simulation results even at a low density region, where the empirical scaling laws of Dzugutov [Nature (London) 381, 137 (1996)]] and Hoyt, Asta, and Sadigh [Phys. Rev. Lett. 85, 594 (2001)]] fail completely. A new scaling law relating the Kolmogorov-Sinai entropy to the excess entropy is also obtained.     

Network structure entropy and its dynamical evolution for recurrence networks from earthquake magnitude time series

Based on the theory of complex network, we construct a recurrence network for earthquakemagnitude time series from California. Network structure entropy and its dynamicalevolution of the network is studied. It is found that the network structure entropy of therecurrence network exhibits a peculiar behavior: it stays at a small value before mainshock, jumps to a great value at the main shock, and then recovers to normal valuesgradually. The network structure entropy therefore provides us an approach to characterizemain shocks quantitatively.     

On the distribution of recurrence times in nonlinear systems

A recent result on the recurrence properties of the sequence m=min_pZ |Mm–p| for irrational , together with a rather old, but little-known result by Florek and Slater on the recurrence properties of the sequence m mod 1 with respect to connected intervals in the interval [0, 1] show that integrable Hamiltonian systems with two degrees of freedom have at most three different recurrence times with respect to an arbitrary Poincaré section of the invariant tori in phase space. We discuss a possible extension of this result to arbitrary integrable systems with any number of degrees of freedom and propose the recurrence time spectrum as a new quantity for characterising simple and complex behaviour of general nonlinear systems.     

Estimating the Shannon entropy: recurrence plots versus symbolic dynamics

Recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics. A few years later, the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis. Among the different measures introduced, a Shannon entropy was found to be correlated with the inverse of the largest Lyapunov exponent. The discrepancy between this and the usual interpretation of a Shannon entropy is solved here by using a new definition--still based on the recurrence plots--and it is verified that this new definition is correlated with the largest Lyapunov exponent, as expected from the Pesin conjecture. A comparison with a Shannon entropy computed from symbolic dynamics is also provided.     

Entropy functionals of Kolmogorov-Sinai type and their limit theorems

Two functionals and are introduced forC ^*-dynamical systems with invariant states and stationary channels. It is shown that the Kolmogorov-Sinai-type theorems hold for these functionals and . Our functionals and are set within the framework of quantum information theory and generalize a quantum KS entropy by CNT and the mutual entropy by Ohya.     

Maximum Kolmogorov-Sinai Entropy Versus Minimum Mixing Time in Markov Chains

We establish a link between the maximization of Kolmogorov Sinai entropy (KSE) and the minimization of the mixing time for general Markov chains. Since the maximisation of KSE is analytical and easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time dynamics. It could be interesting in computer sciences and statistical physics, for computations that use random walks on graphs that can be represented as Markov chains.     

Intermittency from maximum entropy distribution

A maximum entropy distribution has been formulated in which the imposed constraint contains a stochastic (rather than a deterministic) variable. The distribution depends on the observational bin size through the smoothing of population by intrabin averaging. Moments of fluctuations calculated with this distribution give bin-size dependences (intermittency exponents) that agree reasonably with those obtained from the size dependence in nuclear multifragmentation. The exponents depend on the spread of the stochastic mechanism (supposed to be a cascading, multiplicative process) and on the magnitude of the constraint imposed. An information-theoretic interpretation is provided for the relation between statistical and mechanism-induced (dynamic) fluctuations.     

Maximum entropy image reconstruction from projections

The maximum entropy method is applied to image reconstruction from projections, of which angular view is restricted. The relaxation parameters are introduced to the maximum entropy reconstruction and after iteration the median filtering is implemented. These procedures improve the quality of the reconstructed image from noisy projections.     

Calculation of entropy from data of motion

We discuss the question of determining the entropy given the phase space trajectory which describes the detailed history of a many-body system over a period of observation. Our viewpoint is that the determination of entropy, as well as all other thermodynamic properties, should require no concepts or information other than those given and defined by the trajectory. The counting of coincidence (or repetition) of states along the trajectory is presented as a way to determine entropy given the trajectory. An illustrative program based on the kinetic Ising model is described in detail.     

Black ring entropy from the Weyl tensor

A black ring is an asymptotically flat vacuum solution of the n-dimensional Einstein equations with an event horizon of topology S¹×Sⁿ⁻³. In this study, a connection between the black ring entropy and the Weyl tensor C_μνλρ is explored by interpreting the Weyl scalar invariant C_μνλρC^μνλρ as the entropy density in five-dimensional space-time. It is shown that the proper volume integral of C_μνλρC^μνλρ for a neutral black ring is proportional to the black ring entropy in the thin-ring limit. Similar calculations are extended to more general cases: a black string, a black ring with two angular momenta, and a black ring with a cosmological constant. The proportionality is also found to be valid for these complex black objects at the leading order.