Application of the maximum information entropy principle to selforganizing systems

Self-organizing systems are systems which can acquire macroscopic spatial, temporal, or spatio-temporal structures by means of internal processes. Hitherto the distribution functions of the order parameters governing the macroscopic structures could be calculated by microscopic theories only. In the present paper we derive them from macroscopic quantities, where we demonstrate the procedure explicitly by means of the single and multimode laser close to the lasing threshold.     

Atom-field correlation,conservation laws and the phase transition of the laser

The Fokker-Planck equation of a single-mode laser is solved by a novel method, which shows that the correlation between photon-number and atomic inversion is a consequence of conservation laws. Thus the behaviour of the atomic system at the phase transition threshold of the laser is intimately connected with the characteristic change of the photon statistics.     

Unbiased determination of forces causing observed processes

Using measured short time correlation functions of a stochastic process as constraints in the maximum calibre principle of Jaynes, we formulate the joint probability distribution function of the process. The Lagrange multipliers which hereby occur are determined by minimizing a time-dependent form of the (Kullback) information gain. This step can alternatively be interpreted as if our system builds a neural network which learns the Lagrange multipliers. Next, we proceed to determine explicit formulas-expressed in terms of the Lagrange multipliers-for the drift and diffusion coefficients appearing in the corresponding Ito-Langevin equation, which describe the forces underlying the process. Computer-simulations of two processes are presented, showing good confirmation of the theory.     

H-theorem for a mean field model describing coupled oscillator systems under external forces

This article studies the asymptotic behavior of solutions of Fokker–Planck equations describing mean field approximations of weakly coupled oscillator systems subjected to external forces. Using an H-theorem we show that transient probability densities converge to stationary ones. Furthermore, stability criteria are derived for the stationary solutions of these Fokker–Planck equations. The obtained results are applied to a model that combines the Haken–Kelso–Bunz model and the models of weakly coupled oscillators proposed by Winfree and Kuramoto. The stability criteria based on the H-theorem agree with those derived in our earlier analyses.     

An invariance property of the geometrical phase and its consequence in detuned lasers

We show that the geometrical phase defined for the dissipative systems [1] is invariant under a unitary transformation. This implies for lasers an invariance of the geometrical phase for different choices of the reference frequency.     

The mechanisms of thermal degradation of polymethyl acrylate using pyrolysis gas chromatography mass spectrometry

The pyrolysis of polymethyl acrylate is reported with identification of the major pyrolysis products. The mechanism involving random homolytic scission of the chain followed by a series of intermolecular and intramolecular transfer reactions as proposed by Cameron and Kane and extended by Haken, Ho, and Houghton is further developed. Reaction mechanisms are postulated for all of the products produced and ion fragmentation mechanisms for the various spectra produced are shown. The paper also forms the basis for a study of the homologous alkyl polyacrylates, both straight and branched chain esters being considered these having not been previously extensively studied.     

An attempt to understand the four color problem

Is the recently obtained, computer-aided proof of the Four Color Theorem an isolated phenomenon or is its combinatorial complexity typical for a significantly large class of mathematical problems? While it is too early to give a definite answer to this question, an informal discussion is undertaken in this article.