111 research outputs found
Geometrical universality in vibrational dynamics
A good generalization of the Euclidean dimension to disordered systems and
non crystalline structures is commonly required to be related to large scale
geometry and it is expected to be independent of local geometrical
modifications. The spectral dimension, defined according to the low frequency
density of vibrational states, appears to be the best candidate as far as
dynamical and thermodynamical properties are concerned. In this letter we give
the rigorous analytical proof of its independence of finite scale geometry. We
show that the spectral dimension is invariant under local rescaling of
couplings and under addition of finite range couplings, or infinite range
couplings decaying faster then a characteristic power law. We also prove that
it is left unchanged by coarse graining transformations, which are the
generalization to graphs and networks of the usual decimation on regular
structures. A quite important consequence of all these properties is the
possibility of dealing with simplified geometrical models with
nearest-neighbors interactions to study the critical behavior of systems with
geometrical disorder.Comment: Latex file, 1 figure (ps file) include
A Diffusive Strategic Dynamics for Social Systems
We propose a model for the dynamics of a social system, which includes
diffusive effects and a biased rule for spin-flips, reproducing the effect of
strategic choices. This model is able to mimic some phenomena taking place
during marketing or political campaigns. Using a cost function based on the
Ising model defined on the typical quenched interaction environments for social
systems (Erdos-Renyi graph, small-world and scale-free networks), we find, by
numerical simulations, that a stable stationary state is reached, and we
compare the final state to the one obtained with standard dynamics, by means of
total magnetization and magnetic susceptibility. Our results show that the
diffusive strategic dynamics features a critical interaction parameter strictly
lower than the standard one. We discuss the relevance of our findings in social
systems.Comment: Major revisions; to appear on the Journal of Statistical Physic
Superdiffusion and Transport in 2d-systems with L\'evy Like Quenched Disorder
We present an extensive analysis of transport properties in superdiffusive
two dimensional quenched random media, obtained by packing disks with radii
distributed according to a L\'evy law. We consider transport and scaling
properties in samples packed with two different procedures, at fixed filling
fraction and at self-similar packing, and we clarify the role of the two
procedures in the superdiffusive effects. Using the behavior of the filling
fraction in finite size systems as the main geometrical parameter, we define an
effective L\'evy exponents that correctly estimate the finite size effects. The
effective L\'evy exponent rules the dynamical scaling of the main transport
properties and identify the region where superdiffusive effects can be
detected.Comment: 12 pages, 19 figure
Diffusion on non exactly decimable tree-like fractals
We calculate the spectral dimension of a wide class of tree-like fractals by
solving the random walk problem through a new analytical technique, based on
invariance under generalized cutting-decimation transformations. These fractals
are generalizations of the NTD lattices and they are characterized by non
integer spectral dimension equal or greater then 2, non anomalous diffusion
laws, dynamical dimension splitting and absence of phase transitions for spin
models.Comment: 5 pages Latex, 3 figures (figures are poscript files
Topological regulation of activation barriers on fractal substrates
We study phase-ordering dynamics of a ferromagnetic system with a scalar
order-parameter on fractal graphs. We propose a scaling approach, inspired by
renormalization group ideas, where a crossover between distinct dynamical
behaviors is induced by the presence of a length associated to the
topological properties of the graph. The transition between the early and the
asymptotic stage is observed when the typical size of the growing
ordered domains reaches the crossover length . We consider two
classes of inhomogeneous substrates, with different activated processes, where
the effects of the free energy barriers can be analytically controlled during
the evolution. On finitely ramified graphs the free energy barriers encountered
by domains walls grow logarithmically with while they increase as a
power-law on all the other structures. This produces different asymptotic
growth laws (power-laws vs logarithmic) and different dependence of the
crossover length on the model parameters. Our theoretical picture
agrees very well with extensive numerical simulations.Comment: 13 pages, 4 figure
Anomalous diffusion and response in branched systems: a simple analysis
We revisit the diffusion properties and the mean drift induced by an external
field of a random walk process in a class of branched structures, as the comb
lattice and the linear chains of plaquettes. A simple treatment based on
scaling arguments is able to predict the correct anomalous regime for different
topologies. In addition, we show that even in the presence of anomalous
diffusion, Einstein's relation still holds, implying a proportionality between
the mean square displacement of the unperturbed systems and the drift induced
by an external forcing.Comment: revtex.4-1, 16 pages, 7 figure
Fast rare events in exit times distributions of jump processes
Rare events in the first-passage distributions of jump processes are capable
of triggering anomalous reactions or series of events. Estimating their
probability is particularly important when the jump probabilities have
broad-tailed distributions, and rare events are therefore not so rare. We study
three jump processes that are used to model a wide class of stochastic
processes ranging from biology to transport in disordered systems, ecology and
finance. We consider discrete time random-walks, continuous time random-walks
and the L\'evy-Lorentz gas and determine the exact form of the scaling function
for the probability distribution of fast rare events, in which the jump process
exits in a very short time at a large distance opposite to the starting point.
For this estimation we use the so called big jump principle, and we show that
in the regime of fast rare events the exit time distributions are not
exponentially suppressed, even in the case of normal diffusion. This implies
that fast rare events are actually much less rare than predicted by the usual
estimates of large deviations and can occur on timescales orders of magnitude
shorter than expected. Our results are confirmed by extensive numerical
simulations.Comment: 6 pages, 4 figure
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