A. Le Méhauté

Optical wave diffraction on fractal objects

In the frame of a generalized functions analysis, it is shown that the light scattered by fractal objects has fractal properties. The difference between mass diffraction and surface diffraction is presented and illustrated through clear and easy experiments based on the fundamental difference between the concepts of dimension and codimension.

Measuring information growth in fractal phase space

Abstract We look at chaotic systems evolving in fractal phase space. The entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement. Due to the incompleteness, at any scale, of the information calculation in fractal support, the incomplete normalization ∑ipiq=1 is applied throughout the paper. It is shown that the information growth is nonadditive and is proportional to the trace-form ∑ipi−∑ipiq so that it can be connected to several nonadditive entropies. This information growth can be extremized to give, for nonequilibrium systems, power law distributions of evolving stationary state which may be called “maximum entropic evolution”.

Reformulation of a stochastic action principle for irregular dynamics

A stochastic action principle for stochastic dynamics is revisited. We present first numerical diffusion experiments showing that the diffusion path probability depend exponentially on average Lagrangian action. This result is then used to derive an uncertainty measure defined in a way mimicking the heat or entropy in the first law of thermodynamics. It is shown that the path uncertainty (or path entropy) can be measured by the Shannon information and that the maximum entropy principle and the least action principle of classical mechanics can be unified into a concise form. It is argued that this action principle, hence the maximum entropy principle, is simply a consequence of the mechanical equilibrium condition extended to the case of stochastic dynamics.

Introduction to wave phenomena and uncertainty in a fractal space—I

Abstract The purpose of this note is to introduce the principle of dynamics in a fractal space. Although a lot of work has been devoted to fractal geometry, the introduction of time with regard to a fractal space must not be undertaken without some restrictions. In a first step, our analysis restricts itself to the case of electromagnetic theory which will be described in terms of differential geometry and homologic algebra. By an appropriate extension, some efficient mathematical tools, like the fractional derivative operator, will be introduced to enlarge dynamics to ‘fractal dynamics’ or, more generally speaking, to ‘scale dynamics’. This analysis enlightens the opportunities opened by the fractional derivative in quantum mechanics and, at the same time, restrains the generalisation. It is shown that one of the authorised generalisations is related to the ‘uncertainty principle’ which can be considered as a ‘universal principle’ concerning the defect of accuracy of the analysis with respect to a physical phenomenon. In the frame of quantum mechanics, the approach puts the emphasis on some conclusions that deal with the use of such operators and some adequate extension.

About "On certain incomplete statistics" by Lima et al

Abstract Lima et al. recently claim that (Chaos, Solitons & Fractals 19 (2004) 1005) the entropy for the incomplete statistics based on the normalization ∑ipiq=1 should be S=−∑ipi2q−1lnqpi instead of S=−∑ipiqlnqpi initially proposed by Wang. We indicate here that this conclusion is a result of erroneous use of temperature definition for the incomplete statistics.

GENERALIZED MEASURE OF UNCERTAINTY AND THE MAXIMIZABLE ENTROPY

Huaqiao University, China [07BS105]; region des Pays de la Loire of France [2007-6088]; National Natural Science Foundation, China [10947114, 10875100]

Magnetometric analysis of a new class of copolymer magnets based on aminonaphthalenesulfonic acids

Abstract Recently, a new class of copolymer based on aminonaphthalenesulfonic acid and aniline were synthesized. By means of a VSM, an AGFM and a SQUID, it was shown that these materials are ferromagnetic, above room temperature. These results, consistent with temperature dependent EPR tests and magneto-optic experiments, support the achievement of high temperature magnets ( T