An amended MaxEnt formulation for deriving Tsallis factors, and associated issues

An amended MaxEnt formulation for systems displaced from the conventional MaxEnt equilibrium is proposed. This formulation involves the minimization of the Kullback-Leibler divergence to a reference $Q$ (or maximization of Shannon $Q$-entropy), subject to a constraint that implicates a second reference distribution $P\_{1}$ and tunes the new equilibrium. In this setting, the equilibrium distribution is the generalized escort distribution associated to $P\_{1}$ and $Q$. The account of an additional constraint, an observable given by a statistical mean, leads to the maximization of R\'{e}nyi/Tsallis $Q$-entropy subject to that constraint. Two natural scenarii for this observation constraint are considered, and the classical and generalized constraint of nonextensive statistics are recovered. The solutions to the maximization of R\'{e}nyi $Q$-entropy subject to the two types of constraints are derived. These optimum distributions, that are Levy-like distributions, are self-referential. We then propose two `alternate' (but effectively computable) dual functions, whose maximizations enable to identify the optimum parameters. Finally, a duality between solutions and the underlying Legendre structure are presented.Comment: Presented at MaxEnt2006, Paris, France, july 10-13, 200

On some entropy functionals derived from R\'enyi information divergence

We consider the maximum entropy problems associated with R\'enyi $Q$-entropy, subject to two kinds of constraints on expected values. The constraints considered are a constraint on the standard expectation, and a constraint on the generalized expectation as encountered in nonextensive statistics. The optimum maximum entropy probability distributions, which can exhibit a power-law behaviour, are derived and characterized. The R\'enyi entropy of the optimum distributions can be viewed as a function of the constraint. This defines two families of entropy functionals in the space of possible expected values. General properties of these functionals, including nonnegativity, minimum, convexity, are documented. Their relationships as well as numerical aspects are also discussed. Finally, we work out some specific cases for the reference measure $Q(x)$ and recover in a limit case some well-known entropies

Fractal time random walk and subrecoil laser cooling considered as renewal processes with infinite mean waiting times

There exist important stochastic physical processes involving infinite mean waiting times. The mean divergence has dramatic consequences on the process dynamics. Fractal time random walks, a diffusion process, and subrecoil laser cooling, a concentration process, are two such processes that look qualitatively dissimilar. Yet, a unifying treatment of these two processes, which is the topic of this pedagogic paper, can be developed by combining renewal theory with the generalized central limit theorem. This approach enables to derive without technical difficulties the key physical properties and it emphasizes the role of the behaviour of sums with infinite means.Comment: 9 pages, 7 figures, to appear in the Proceedings of Cargese Summer School on "Chaotic dynamics and transport in classical and quantum systems

The Edge of Quantum Chaos

We identify a border between regular and chaotic quantum dynamics. The border is characterized by a power law decrease in the overlap between a state evolved under chaotic dynamics and the same state evolved under a slightly perturbed dynamics. For example, the overlap decay for the quantum kicked top is well fitted with $[1+(q-1) (t/\tau)^2]^{1/(1-q)}$ (with the nonextensive entropic index $q$ and $\tau$ depending on perturbation strength) in the region preceding the emergence of quantum interference effects. This region corresponds to the edge of chaos for the classical map from which the quantum chaotic dynamics is derived.Comment: 4 pages, 4 figures, revised version in press PR

Theory for atomic diffusion on fixed and deformable crystal lattices

We develop a theoretical framework for the diffusion of a single unconstrained species of atoms on a crystal lattice that provides a generalization of the classical theories of atomic diffusion and diffusion-induced phase separation to account for constitutive nonlinearities, external forces, and the deformation of the lattice. In this framework, we regard atomic diffusion as a microscopic process described by two independent kinematic variables: (i) the atomic flux, which reckons the local motion of atoms relative to the motion of the underlying lattice, and (ii) the time-rate of the atomic density, which encompasses nonlocal interactions between migrating atoms and characterizes the kinematics of phase separation. We introduce generalized forces power-conjugate to each of these rates and require that these forces satisfy ancillary microbalances distinct from the conventional balance involving the forces that expend power over the rate at which the lattice deforms. A mechanical version of the second law, which takes the form of an energy imbalance accounting for all power expenditures (including those due to the atomic diffusion and phase separation), is used to derive restrictions on the constitutive equations. With these restrictions, the microbalance involving the forces conjugate to the atomic flux provides a generalization of the usual constitutive relation between the atomic flux and the gradient of the diffusion potential, a relation that in conjunction with the atomic balance yields a generalized Cahn-Hilliard equation.Comment: To appear in Journal of Elasticity, 18 pages, requires kluwer macr