Shocks and Universal Statistics in (1+1)-Dimensional Relativistic Turbulence

We propose that statistical averages in relativistic turbulence exhibit universal properties. We consider analytically the velocity and temperature differences structure functions in the (1+1)-dimensional relativistic turbulence in which shock waves provide the main contribution to the structure functions in the inertial range. We study shock scattering, demonstrate the stability of the shock waves, and calculate the anomalous exponents. We comment on the possibility of finite time blowup singularities.Comment: 37 pages, 7 figure

Lyapunov exponent of the random frequency oscillator: cumulant expansion approach

We consider a one-dimensional harmonic oscillator with a random frequency, focusing on both the standard and the generalized Lyapunov exponents, $\lambda$ and $\lambda^\star$ respectively. We discuss the numerical difficulties that arise in the numerical calculation of $\lambda^\star$ in the case of strong intermittency. When the frequency corresponds to a Ornstein-Uhlenbeck process, we compute analytically $\lambda^\star$ by using a cumulant expansion including up to the fourth order. Connections with the problem of finding an analytical estimate for the largest Lyapunov exponent of a many-body system with smooth interactions are discussed.Comment: 6 pages, 4 figures, to appear in J. Phys. Conf. Series - LAWNP0

Response to arXiv:0811.3876 "Comment on a recent conjectured solution of the three dimensional Ising model" by Wu et al

This is a Response to a recent Comment [F.Y. Wu et al., Phil. Mag. 88, 3093 (2008), arXiv:0811.3876] on the conjectured solution of the three-dimensional (3D) Ising model [Z.D. Zhang, Phil. Mag. 87, 5309 (2007), arXiv:0705.1045]. Several points are made: 1) Conjecture 1, regarding the additional rotation, is understood as performing a transformation for smoothing all the crossings of the knots; 2) The weight factors in Conjecture 2 are interpreted as a novel topologic phase; 3) The conjectured solution and its low- and high-temperature expansions are supported by the mathematical theorems for the analytical behavior of the Ising model. The physics behind the extra dimension is also discussed briefly.Comment: 11 pages, 0 figure

Uniqueness of Gibbs Measure for Models With Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geq 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.Comment: 13 page

Mutual information rate and bounds for it

The amount of information exchanged per unit of time between two nodes in a dynamical network or between two data sets is a powerful concept for analysing complex systems. This quantity, known as the mutual information rate (MIR), is calculated from the mutual information, which is rigorously defined only for random systems. Moreover, the definition of mutual information is based on probabilities of significant events. This work offers a simple alternative way to calculate the MIR in dynamical (deterministic) networks or between two data sets (not fully deterministic), and to calculate its upper and lower bounds without having to calculate probabilities, but rather in terms of well known and well defined quantities in dynamical systems. As possible applications of our bounds, we study the relationship between synchronisation and the exchange of information in a system of two coupled maps and in experimental networks of coupled oscillators