416 research outputs found
Construction of a non-standard quantum field theory through a generalized Heisenberg algebra
We construct a Heisenberg-like algebra for the one dimensional quantum free
Klein-Gordon equation defined on the interval of the real line of length .
Using the realization of the ladder operators of this type Heisenberg algebra
in terms of physical operators we build a 3+1 dimensional free quantum field
theory based on this algebra. We introduce fields written in terms of the
ladder operators of this type Heisenberg algebra and a free quantum Hamiltonian
in terms of these fields. The mass spectrum of the physical excitations of this
quantum field theory are given by , where denotes the level of the particle with mass in an infinite
square-well potential of width .Comment: Latex, 16 page
Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: rigorous nonextensive solutions
Using the Feigenbaum renormalization group (RG) transformation we work out
exactly the dynamics and the sensitivity to initial conditions for unimodal
maps of nonlinearity at both their pitchfork and tangent
bifurcations. These functions have the form of -exponentials as proposed in
Tsallis' generalization of statistical mechanics. We determine the -indices
that characterize these universality classes and perform for the first time the
calculation of the -generalized Lyapunov coefficient . The
pitchfork and the left-hand side of the tangent bifurcations display weak
insensitivity to initial conditions, while the right-hand side of the tangent
bifurcations presents a `super-strong' (faster than exponential) sensitivity to
initial conditions. We corroborate our analytical results with {\em a priori}
numerical calculations.Comment: latex, 4 figures. Updated references and some general presentation
improvements. To appear published in Europhysics Letter
Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations
A general type of nonlinear Fokker-Planck equation is derived directly from a
master equation, by introducing generalized transition rates. The H-theorem is
demonstrated for systems that follow those classes of nonlinear Fokker-Planck
equations, in the presence of an external potential. For that, a relation
involving terms of Fokker-Planck equations and general entropic forms is
proposed. It is shown that, at equilibrium, this relation is equivalent to the
maximum-entropy principle. Families of Fokker-Planck equations may be related
to a single type of entropy, and so, the correspondence between well-known
entropic forms and their associated Fokker-Planck equations is explored. It is
shown that the Boltzmann-Gibbs entropy, apart from its connection with the
standard -- linear Fokker-Planck equation -- may be also related to a family of
nonlinear Fokker-Planck equations.Comment: 19 pages, no figure
q-Moments remove the degeneracy associated with the inversion of the q-Fourier transform
It was recently proven [Hilhorst, JSTAT, P10023 (2010)] that the
q-generalization of the Fourier transform is not invertible in the full space
of probability density functions for q > 1. It has also been recently shown
that this complication disappears if we dispose of the q-Fourier transform not
only of the function itself, but also of all of its shifts [Jauregui and
Tsallis, Phys. Lett. A 375, 2085 (2011)]. Here we show that another road exists
for completely removing the degeneracy associated with the inversion of the
q-Fourier transform of a given probability density function. Indeed, it is
possible to determine this density if we dispose of some extra information
related to its q-moments.Comment: 11 pages, 12 figure
Linear instability and statistical laws of physics
We show that a meaningful statistical description is possible in conservative
and mixing systems with zero Lyapunov exponent in which the dynamical
instability is only linear in time. More specifically, (i) the sensitivity to
initial conditions is given by with
; (ii) the statistical entropy in the infinitely fine graining limit (i.e., {\it
number of cells into which the phase space has been partitioned} ),
increases linearly with time only for ; (iii) a nontrivial,
-generalized, Pesin-like identity is satisfied, namely the . These facts (which are
in analogy to the usual behaviour of strongly chaotic systems with ), seem
to open the door for a statistical description of conservative many-body
nonlinear systems whose Lyapunov spectrum vanishes.Comment: 7 pages including 2 figures. The present version is accepted for
publication in Europhysics Letter
Option Pricing Formulas based on a non-Gaussian Stock Price Model
Options are financial instruments that depend on the underlying stock. We
explain their non-Gaussian fluctuations using the nonextensive thermodynamics
parameter . A generalized form of the Black-Scholes (B-S) partial
differential equation, and some closed-form solutions are obtained. The
standard B-S equation () which is used by economists to calculate option
prices requires multiple values of the stock volatility (known as the
volatility smile). Using which well models the empirical distribution
of returns, we get a good description of option prices using a single
volatility.Comment: final version (published
- …