Escort mean values ͑or q-moments͒ constitute useful theoretical tools for describing basic features of some probability densities such as those which asymptotically decay like power laws. They naturally appear in the study of many complex dynamical systems, particularly those obeying nonextensive statistical mechanics, a current generalization of the Boltzmann-Gibbs theory. They recover standard mean values ͑or moments͒ for q = 1. Here we discuss the characterization of a ͑non-negative͒ probability density by a suitable set of all its escort mean values together with the set of all associated normalizing quantities, provided that all of them converge. This opens the door to a natural extension of the well-known characterization, for the q = 1 instance, of a distribution in terms of the standard moments, provided that all of them have finite values. This question would be specially relevant in connection with probability densities having divergent values for all nonvanishing standard moments higher than a given one ͑e.g., probability densities asymptotically decaying as power laws͒, for which the standard approach is not applicable. The Cauchy-Lorentz distribution, whose second and higher even order moments diverge, constitutes a simple illustration of the interest of this investigation. In this context, we also address some mathematical subtleties with the aim of clarifying some aspects of an interesting nonlinear generalization of the Fourier transform, namely, the so-called q-Fourier transform.

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Thomson scattering (TS) of electromagnetic radiation by relativistic electrons is of great interest as an electron distribution diagnostics in high temperature plasma devices. A Monte Carlo approach to TS simulation has been implemented. It is based on first principles, i.e. on the integration of electron trajectories and on the computation and averaging of radiated Liénard–Wiechert fields. The treatment is relativistic. Nonlinear effects, arising from ultraintense lasers (both linear and circular polarizations having been considered) and/or from highly energetic electrons, are included from the outset. TS spectra from isotropic monoenergetic electron distribution functions and from the relativistic Maxwellian distribution have been extensively studied, both for low-intensity and for ultraintense incoming laser. A particularly simple quadratic dependence of S 2(ω) on ω has been found in the case of monoenergetic distributions. Numerical results are checked against known analytical results for low-intensity lasers, and found to be in agreement with them. For high-intensity lasers, new effects on scattered spectra arise, like an overall red-shift or the appearance of higher harmonics. The numerical methods developed in this work are general enough to deal with anisotropic and/or non-Maxwellian distribution functions.

Motivated by the melting transition of DNA, we study genuinely three-dimensional models for two interacting open, flexible and homogeneous macromolecular chains, bound or unbound to each other, at thermal equilibrium from about room temperature up to about the denaturation temperature (Tun). In each chain, angular constraints on bond angles (due to covalent bonding) determine monomers: each monomer contains ne nucleotides and has an effective length de. These monomers could remain practically unaltered for temperatures in a range above and below Tun, down to 300 K. Estimates for ne and de are provided and justified. Upon proceeding from Quantum Mechanics to the classical limit and using suitable large-distance approximations (partly, due to those monomer configurations), we get a generalization of Edwards’ model, which includes effective potentials between monomers. The classical partition function for the two-chain system is reduced to an integral of a generalized and discretized two-chain Green’s function. We analyse conditions for the denaturing transition. The fact that each single chain is an extended one-dimensional system modifies their mutual global interaction, in comparison with typical potentials between nucleotides: this is simply illustrated by computing a global effective potential between the two chains. Applications for Morse potentials are presented. Our models seem to be physically compatible with some previous one-dimensional ones and could allow us to efficiently extend the latter to three spatial dimensions.