2 research outputs found
Long memory constitutes a unified mesoscopic mechanism consistent with nonextensive statistical mechanics
We unify two paradigmatic mesoscopic mechanisms for the emergence of
nonextensive statistics, namely the multiplicative noise mechanism leading to a
{\it linear} Fokker-Planck (FP) equation with {\it inhomogenous} diffusion
coefficient, and the non-Markovian process leading to the {\it nonlinear} FP
equation with {\it homogeneous} diffusion coefficient. More precisely, we
consider the equation , where and , being the potential under which diffusion occurs. Our aim
is to find whether exists such that the inhomogeneous linear and
the homogeneous nonlinear FP equations become unified in such a way that the
(ubiquitously observed) -exponentials remain as stationary solutions. It
turns out that such solutions indeed exist for a wide class of systems, namely
when , where , , and
are (real) constants.
Our main result can be sumarized as follows: For and
arbitrary confining potential , , where . The present approach unifies into a single mechanism,
essentially {\it long memory}, results currently discussed and applied in the
literature.Comment: 5 pages including 1 figur
Preferential attachment growth model and nonextensive statistical mechanics
We introduce a two-dimensional growth model where every new site is located,
at a distance from the barycenter of the pre-existing graph, according to
the probability law , and is attached to
(only) one pre-existing site with a probability ; is the number of links of the site of the
pre-existing graph, and its distance to the new site). Then we
numerically determine that the probability distribution for a site to have
links is asymptotically given, for all values of , by , where is the function
naturally emerging within nonextensive statistical mechanics. The entropic
index is numerically given (at least for not too large) by , and the characteristic number of links by . The particular case belongs to the same
universality class to which the Barabasi-Albert model belongs. In addition to
this, we have numerically studied the rate at which the average number of links
increases with the scaled time ; asymptotically, , the exponent being close to for , and zero otherwise.
The present results reinforce the conjecture that the microscopic dynamics of
nonextensive systems typically build (for instance, in Gibbs -space for
Hamiltonian systems) a scale-free network.Comment: 5 pages including 5 figures (the original colored figures 1 and 5a
can be asked directly to the authors