224 research outputs found
Fluctuations induce transitions in frustrated sparse networks
We analyze, by means of statistical mechanics, a sparse network with random
competitive interactions among dichotomic variables pasted on the nodes, namely
a Viana-Bray model. The model is described by an infinite series of order
parameters (the multi-overlaps) and has two tunable degrees of freedom: the
noise level and the connectivity (the averaged number of links). We show that
there are no multiple transition lines, one for every order parameter, as a
naive approach would suggest, but just one corresponding to ergodicity
breaking. We explain this scenario within a novel and simple mathematical
technique via a driving mechanism such that, as the first order parameter (the
two replica overlap) becomes different from zero due to a real second order
phase transition (with properly associated diverging rescaled fluctuations), it
enforces all the other multi-overlaps toward positive values thanks to the
strong correlations which develop among themselves and the two replica overlap
at the critical line
A numerical investigation of the jamming transition in traffic flow on diluted planar networks
In order to develop a toy model for car's traffic in cities, in this paper we
analyze, by means of numerical simulations, the transition among fluid regimes
and a congested jammed phase of the flow of "kinetically constrained" hard
spheres in planar random networks similar to urban roads. In order to explore
as timescales as possible, at a microscopic level we implement an event driven
dynamics as the infinite time limit of a class of already existing model (e.g.
"Follow the Leader") on an Erdos-Renyi two dimensional graph, the crossroads
being accounted by standard Kirchoff density conservations. We define a
dynamical order parameter as the ratio among the moving spheres versus the
total number and by varying two control parameters (density of the spheres and
coordination number of the network) we study the phase transition. At a
mesoscopic level it respects an, again suitable adapted, version of the
Lighthill-Whitham model, which belongs to the fluid-dynamical approach to the
problem. At a macroscopic level the model seems to display a continuous
transition from a fluid phase to a jammed phase when varying the density of the
spheres (the amount of cars in a city-like scenario) and a discontinuous jump
when varying the connectivity of the underlying network.Comment: accepted in Int.J.Mod.Phys.
Exact solution of the van der Waals model in the critical region
The celebrated van der Waals model describes simple fluids in the
thermodynamic limit and predicts the existence of a critical point associated
to the gas-liquid phase transition. However the behaviour of critical isotherms
according to the equation of state, where a gas-liquid phase transition occurs,
significantly departs from experimental observations. The correct critical
isotherms are heuristically re-established via the Maxwell equal areas rule. A
long standing open problem in mean field theory is concerned with the analytic
description of van der Waals isotherms for a finite size system that is
consistent, in the thermodynamic limit, with the Maxwell prescription. Inspired
by the theory of nonlinear conservation laws, we propose a novel mean field
approach, based on statistical mechanics, that allows to calculate the van der
Waals partition function for a system of large but finite number of particles
. Our partition function naturally extends to the whole space of
thermodynamic variables, reproduces, in the thermodynamic limit ,
the classical results outside the critical region and automatically encodes
Maxwell's prescription. We show that isothermal curves evolve in the space of
thermodynamic variables like nonlinear breaking waves and the criticality is
explained as the mechanism of formation of a classical hydrodynamic shock
Overlap Fluctuations from Random Overlap Structures
We investigate overlap fluctuations of the Sherrington-Kirkpatrick mean field
spin glass model in the framework of the Random Over- lap Structure (ROSt). The
concept of ROSt has been introduced recently by Aizenman and coworkers, who
developed a variational approach to the Sherrington-Kirkpatrick model. We
propose here an iterative procedure to show that, in the so-called Boltzmann
ROSt, Aizenman-Contucci (AC) polynomials naturally arise for almost all values
of the inverse temperature (not in average over some interval only). The same
results can be obtained in any ROSt, including therefore the Parisi structure.
The AC polynomials impose restric- tions on the overlap fluctuations in
agreement with Parisi theory.Comment: 18 page
Dreaming neural networks: forgetting spurious memories and reinforcing pure ones
The standard Hopfield model for associative neural networks accounts for
biological Hebbian learning and acts as the harmonic oscillator for pattern
recognition, however its maximal storage capacity is , far
from the theoretical bound for symmetric networks, i.e. . Inspired
by sleeping and dreaming mechanisms in mammal brains, we propose an extension
of this model displaying the standard on-line (awake) learning mechanism (that
allows the storage of external information in terms of patterns) and an
off-line (sleep) unlearningconsolidating mechanism (that allows
spurious-pattern removal and pure-pattern reinforcement): this obtained daily
prescription is able to saturate the theoretical bound , remaining
also extremely robust against thermal noise. Both neural and synaptic features
are analyzed both analytically and numerically. In particular, beyond obtaining
a phase diagram for neural dynamics, we focus on synaptic plasticity and we
give explicit prescriptions on the temporal evolution of the synaptic matrix.
We analytically prove that our algorithm makes the Hebbian kernel converge with
high probability to the projection matrix built over the pure stored patterns.
Furthermore, we obtain a sharp and explicit estimate for the "sleep rate" in
order to ensure such a convergence. Finally, we run extensive numerical
simulations (mainly Monte Carlo sampling) to check the approximations
underlying the analytical investigations (e.g., we developed the whole theory
at the so called replica-symmetric level, as standard in the
Amit-Gutfreund-Sompolinsky reference framework) and possible finite-size
effects, finding overall full agreement with the theory.Comment: 31 pages, 12 figure
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