494 research outputs found
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
Canonical equilibrium distribution derived from Helmholtz potential
Plastino and Curado [Phys. Rev. E 72, 047103 (2005)] recently determined the
equilibrium probability distribution for the canonical ensemble using only
phenomenological thermodynamical laws as an alternative to the entropy
maximization procedure of Jaynes. In the current paper we present another
alternative derivation of the canonical equilibrium probability distribution,
which is based on the definition of the Helmholtz free energy (and its being
constant at the equilibrium) and the assumption of the uniqueness of the
equilibrium probability distribution. Noting that this particular derivation is
applicable for all trace-form entropies, we also apply it to the Tsallis
entropy showing that the Tsallis entropy yields genuine inverse power laws.Comment: 7 pages. Accepted for publication in Physica
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
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
Bose-Einstein Condensation in the Framework of -Statistics
In the present work we study the main physical properties of a gas of
-deformed bosons described through the statistical distribution
function . The
deformed -exponential , recently proposed in Ref.
[G.Kaniadakis, Physica A {\bf 296}, 405, (2001)], reduces to the standard
exponential as the deformation parameter , so that
reproduces the Bose-Einstein distribution. The condensation temperature
of this gas decreases with increasing value, and
approaches the transition temperature
, improving the result obtained in the standard case
(). The heat capacity is a continuous function and
behaves as for , in
contrast with the standard case , it is always increasing.
Pacs: 05.30.Jp, 05.70.-a Keywords: Generalized entropy; Boson gas; Phase
transition.Comment: To appear in Physica B. Two fig.p
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
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