16 research outputs found
Multiplicative noise: A mechanism leading to nonextensive statistical mechanics
A large variety of microscopic or mesoscopic models lead to generic results
that accommodate naturally within Boltzmann-Gibbs statistical mechanics (based
on ). Similarly, other classes of models
point toward nonextensive statistical mechanics (based on , where the value of the entropic index depends on
the specific model). We show here a family of models, with multiplicative
noise, which belongs to the nonextensive class. More specifically, we consider
Langevin equations of the type , where
and are independent zero-mean Gaussian white noises with
respective amplitudes and . This leads to the Fokker-Planck equation
. Whenever the
deterministic drift is proportional to the noise induced one, i.e., , the stationary solution is shown to be (with and ). This distribution is
precisely the one optimizing with the constraint constant. We also
introduce and discuss various characterizations of the width of the
distributions.Comment: 3 PS figure
Social games in a social network
We study an evolutionary version of the Prisoner's Dilemma game, played by
agents placed in a small-world network. Agents are able to change their
strategy, imitating that of the most successful neighbor. We observe that
different topologies, ranging from regular lattices to random graphs, produce a
variety of emergent behaviors. This is a contribution towards the study of
social phenomena and transitions governed by the topology of the community
Exact results and scaling properties of small-world networks
We study the distribution function for minimal paths in small-world networks.
Using properties of this distribution function, we derive analytic results
which greatly simplify the numerical calculation of the average minimal
distance, , and its variance, . We also discuss the
scaling properties of the distribution function. Finally, we study the limit of
large system sizes and obtain some analytic results.Comment: RevTeX, 4 pages, 5 figures included. Minor corrections and addition
Fragmentation Experiment and Model for Falling Mercury Drops
The experiment consists of counting and measuring the size of the many
fragments observed after the fall of a mercury drop on the floor. The size
distribution follows a power-law for large enough fragments. We address the
question of a possible crossover to a second, different power-law for small
enough fragments. Two series of experiments were performed. The first uses a
traditional film photographic camera, and the picture is later treated on a
computer in order to count the fragments and classify them according to their
sizes. The second uses a modern digital camera. The first approach has the
advantage of a better resolution for small fragment sizes. The second, although
with a poorer size resolution, is more reliable concerning the counting of all
fragments up to its resolution limit. Both together clearly indicate the real
existence of the quoted crossover.
The model treats the system microscopically during the tiny time interval
when the initial drop collides with the floor. The drop is modelled by a
connected cluster of Ising spins pointing up (mercury) surrounded by Ising
spins pointing down (air). The Ising coupling which tends to keep the spins
segregated represents the surface tension. Initially the cluster carries an
extra energy equally shared among all its spins, corresponding to the coherent
kinetic energy due to the fall. Each spin which touches the floor loses its
extra energy transformed into a thermal, incoherent energy represented by a
temperature used then to follow the dynamics through Monte Carlo simulations.
Whenever a small piece becomes disconnected from the big cluster, it is
considered a fragment, and counted. The results also indicate the existence of
the quoted crossover in the fragment-size distribution.Comment: 6 pages, 3 figure
Geometric origin of mechanical properties of granular materials
Some remarkable generic properties, related to isostaticity and potential
energy minimization, of equilibrium configurations of assemblies of rigid,
frictionless grains are studied. Isostaticity -the uniqueness of the forces,
once the list of contacts is known- is established in a quite general context,
and the important distinction between isostatic problems under given external
loads and isostatic (rigid) structures is presented. Complete rigidity is only
guaranteed, on stability grounds, in the case of spherical cohesionless grains.
Otherwise, the network of contacts might deform elastically in response to load
increments, even though grains are rigid. This sets an uuper bound on the
contact coordination number. The approximation of small displacements (ASD)
allows to draw analogies with other model systems studied in statistical
mechanics, such as minimum paths on a lattice. It also entails the uniqueness
of the equilibrium state (the list of contacts itself is geometrically
determined) for cohesionless grains, and thus the absence of plastic
dissipation. Plasticity and hysteresis are due to the lack of such uniqueness
and may stem, apart from intergranular friction, from small, but finite,
rearrangements, in which the system jumps between two distinct potential energy
minima, or from bounded tensile contact forces. The response to load increments
is discussed. On the basis of past numerical studies, we argue that, if the ASD
is valid, the macroscopic displacement field is the solution to an elliptic
boundary value problem (akin to the Stokes problem).Comment: RevTex, 40 pages, 26 figures. Close to published paper. Misprints and
minor errors correcte
Shortest paths on systems with power-law distributed long-range connections
We discuss shortest-path lengths on periodic rings of size L
supplemented with an average of pL randomly located long-range links whose
lengths are distributed according to P_l \sim l^{-\xpn}. Using rescaling
arguments and numerical simulation on systems of up to sites, we show
that a characteristic length exists such that for
. For small p we find
that the shortest-path length satisfies the scaling relation
\ell(r,\xpn,p)/\xi = f(\xpn,r/\xi). Three regions with different asymptotic
behaviors are found, respectively: a) \xpn>2 where , b)
1<\xpn<2 where 0<\theta_s(\xpn)<1/2 and, c) \xpn<1 where
behaves logarithmically, i.e. . The characteristic length is
of the form with \nu=1/(2-\xpn) in region b), but depends
on L as well in region c). A directed model of shortest-paths is solved and
compared with numerical results.Comment: 10 pages, 10 figures, revtex4. Submitted to PR
Modulated Scale-free Network in the Euclidean Space
A random network is grown by introducing at unit rate randomly selected nodes
on the Euclidean space. A node is randomly connected to its -th predecessor
of degree with a directed link of length using a probability
proportional to . Our numerical study indicates that the
network is Scale-free for all values of and the degree
distribution decays stretched exponentially for the other values of .
The link length distribution follows a power law:
where is calculated exactly for the whole range of values of .Comment: 4 pages, 4 figures. To be published in Physical Review
Relaxation Properties of Small-World Networks
Recently, Watts and Strogatz introduced the so-called small-world networks in
order to describe systems which combine simultaneously properties of regular
and of random lattices. In this work we study diffusion processes defined on
such structures by considering explicitly the probability for a random walker
to be present at the origin. The results are intermediate between the
corresponding ones for fractals and for Cayley trees.Comment: 16 pages, 6 figure
Effective dimensions and percolation in hierarchically structured scale-free networks
We introduce appropriate definitions of dimensions in order to characterize
the fractal properties of complex networks. We compute these dimensions in a
hierarchically structured network of particular interest. In spite of the
nontrivial character of this network that displays scale-free connectivity
among other features, it turns out to be approximately one-dimensional. The
dimensional characterization is in agreement with the results on statistics of
site percolation and other dynamical processes implemented on such a network.Comment: 5 pages, 5 figure
Stress response inside perturbed particle assemblies
The effect of structural disorder on the stress response inside three
dimensional particle assemblies is studied using computer simulations of
frictionless sphere packings. Upon applying a localised, perturbative force
within the packings, the resulting {\it Green's} function response is mapped
inside the different assemblies, thus providing an explicit view as to how the
imposed perturbation is transmitted through the packing. In weakly disordered
arrays, the resulting transmission of forces is of the double-peak variety, but
with peak widths scaling linearly with distance from the source of the
perturbation. This behaviour is consistent with an anisotropic elasticity
response profile. Increasing the disorder distorts the response function until
a single-peak response is obtained for fully disordered packings consistent
with an isotropic description.Comment: 8 pages, 7 figure captions To appear in Granular Matte