58 research outputs found
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
Rigorous results in non-extensive thermodynamics
This paper studies quantum systems with a finite number of degrees of freedom
in the context of non-extensive thermodynamics. A trial density matrix,
obtained by heuristic methods, is proved to be the equilibrium density matrix.
If the entropic parameter q is larger than 1 then existence of the trial
equilibrium density matrix requires that q is less than some critical value q_c
which depends on the rate by which the eigenvalues of the hamiltonian diverge.
Existence of a unique equilibrium density matrix is proved if in addition q<2
holds. For q between 0 and 1, such that 2<q+q_c, the free energy has at least
one minimum in the set of trial density matrices. If a unique equilibrium
density matrix exists then it is necessarily one of the trial density matrices.
Note that this is a finite rank operator, which means that in equilibrium high
energy levels have zero probability of occupancy.Comment: 21 page
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 (with the nonextensive entropic
index and 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
Generalized entropy arising from a distribution of q-indices
It is by now well known that the Boltzmann-Gibbs (BG) entropy
can be usefully generalized into the
entropy (). Microscopic dynamics determines, given classes of initial
conditions, the occupation of the accessible phase space (or of a
symmetry-determined nonzero-measure part of it), which in turn appears to
determine the entropic form to be used. This occupation might be a uniform one
(the usual {\it equal probability hypothesis} of BG statistical mechanics),
which corresponds to ; it might be a free-scale occupancy, which appears
to correspond to . Since occupancies of phase space more complex than
these are surely possible in both natural and artificial systems, the task of
further generalizing the entropy appears as a desirable one, and has in fact
been already undertaken in the literature. To illustrate the approach, we
introduce here a quite general entropy based on a distribution of -indices
thus generalizing . We establish some general mathematical properties for
the new entropic functional and explore some examples. We also exhibit a
procedure for finding, given any entropic functional, the -indices
distribution that produces it. Finally, on the road to establishing a quite
general statistical mechanics, we briefly address possible generalized
constraints under which the present entropy could be extremized, in order to
produce canonical-ensemble-like stationary-state distributions for Hamiltonian
systems.Comment: 14 pages including 3 figure
Nonequilibrium Probabilistic Dynamics of the Logistic Map at the Edge of Chaos
We consider nonequilibrium probabilistic dynamics in logistic-like maps
, at their chaos threshold: We first introduce many
initial conditions within one among intervals partitioning the phase
space and focus on the unique value for which the entropic form
{\it linearly} increases with
time. We then verify that vanishes like
[]. We finally exhibit a new finite-size
scaling, . This
establishes quantitatively, for the first time, a long pursued relation between
sensitivity to the initial conditions and relaxation, concepts which play
central roles in nonextensive statistical mechanics.Comment: Final version with new Title and small modifications. REVTeX, 8 pages
and 4 eps figure
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
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
BEC in Nonextensive Statistical Mechanics
We discuss the Bose-Einstein condensation (BEC) for an ideal gas of bosons in
the framework of Tsallis's nonextensive statistical mechanics. We study the
corrections to the standard BEC formulas due to a weak nonextensivity of the
system. In particular, we consider three cases in the D-dimensional space: the
homogeneous gas, the gas in a harmonic trap and the relativistic homogenous
gas. The results show that small deviations from the extensive Bose statistics
produce remarkably large changes in the BEC transition temperature.Comment: LaTex, 7 pages, no figures, to be published in Mod. Phys. Lett. B;
corrected a typo in Eq. (2
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
Chaos edges of -logistic maps: Connection between the relaxation and sensitivity entropic indices
Chaos thresholds of the -logistic maps are numerically analysed at accumulation points of cycles 2, 3
and 5. We verify that the nonextensive -generalization of a Pesin-like
identity is preserved through averaging over the entire phase space. More
precisely, we computationally verify , where the entropy (), the sensitivity to the initial
conditions , and
(). The entropic index
depend on
both and the cycle. We also study the relaxation that occurs if we start
with an ensemble of initial conditions homogeneously occupying the entire phase
space. The associated Lebesgue measure asymptotically decreases as
(). These results led to (i) the first
illustration of the connection (conjectured by one of us) between sensitivity
and relaxation entropic indices, namely , where the positive numbers depend on the
cycle; (ii) an unexpected and new scaling, namely ( for , and for ).Comment: 5 pages, 5 figure
- …