284 research outputs found
Beyond the Mean Field Approximation for Spin Glasses
We study the d-dimensional random Ising model using a Bethe-Peierls
approximation in the framework of the replica method. We take into account the
correct interaction only inside replicated clusters of spins. Our ansatz is
that the interaction of the borders of the clusters with the external world can
be described via an effective interaction among replicas. The Bethe-Peierls
model is mapped into a single Ising model with a random gaussian field, whose
strength (related to the effective coupling between two replicas) is determined
via a self-consistency equation. This allows us to obtain analytic estimates of
the internal energy and of the critical temperature in d dimensions.Comment: plane TeX file,19 pages. 3 figures may be requested to Paladin at
axscaq.aquila.infn.i
On the concept of complexity in random dynamical systems
We introduce a measure of complexity in terms of the average number of bits
per time unit necessary to specify the sequence generated by the system. In
random dynamical system, this indicator coincides with the rate K of divergence
of nearby trajectories evolving under two different noise realizations.
The meaning of K is discussed in the context of the information theory, and
it is shown that it can be determined from real experimental data. In presence
of strong dynamical intermittency, the value of K is very different from the
standard Lyapunov exponent computed considering two nearby trajectories
evolving under the same randomness. However, the former is much more relevant
than the latter from a physical point of view as illustrated by some numerical
computations for noisy maps and sandpile models.Comment: 35 pages, LaTe
Constrained annealing for spin glasses
The quenched free energy of spin glasses is estimated by means of annealed
averages where the frustration is constrained to its average value. We discuss
the case of d-dimensional Ising models with random nearest neighbour coupling,
and we introduce a new method to obtain constrained annealed averages without
recurring to Lagrange multipliers. It requires to perform quenched averages
either on small volumes in an analytic way, or on finite size strips via
products of random transfer matrices. We thus give a sequence of converging
lower bounds for the quenched free energy of 2d spin glasses.Comment: plane TeX file,24 pages. 6 figures may be requested to Paladin at
vxscaq.aquila.infn.i
Stochastic Resonance in Deterministic Chaotic Systems
We propose a mechanism which produces periodic variations of the degree of
predictability in dynamical systems. It is shown that even in the absence of
noise when the control parameter changes periodically in time, below and above
the threshold for the onset of chaos, stochastic resonance effects appears. As
a result one has an alternation of chaotic and regular, i.e. predictable,
evolutions in an almost periodic way, so that the Lyapunov exponent is positive
but some time correlations do not decay.Comment: 9 Pages + 3 Figures, RevTeX 3.0, sub. J. Phys.
Lack of self-average in weakly disordered one dimensional systems
We introduce a one dimensional disordered Ising model which at zero
temperature is characterized by a non-trivial, non-self-averaging, overlap
probability distribution when the impurity concentration vanishes in the
thermodynamic limit. The form of the distribution can be calculated
analytically for any realization of disorder. For non-zero impurity
concentration the distribution becomes a self-averaging delta function centered
on a value which can be estimated by the product of appropriate transfer
matrices.Comment: 17 pages + 5 figures, TeX dialect: Plain TeX + IOP macros (included
Statistical Mechanics of Shell Models for 2D-Turbulence
We study shell models that conserve the analogues of energy and enstrophy,
hence designed to mimic fluid turbulence in 2D. The main result is that the
observed state is well described as a formal statistical equilibrium, closely
analogous to the approach to two-dimensional ideal hydrodynamics of Onsager,
Hopf and Lee. In the presence of forcing and dissipation we observe a forward
flux of enstrophy and a backward flux of energy. These fluxes can be understood
as mean diffusive drifts from a source to two sinks in a system which is close
to local equilibrium with Lagrange multipliers (``shell temperatures'')
changing slowly with scale. The dimensional predictions on the power spectra
from a supposed forward cascade of enstrophy, and from one branch of the formal
statistical equilibrium, coincide in these shell models at difference to the
corresponding predictions for the Navier-Stokes and Euler equations in 2D. This
coincidence have previously led to the mistaken conclusion that shell models
exhibit a forward cascade of enstrophy.Comment: 25 pages + 9 figures, TeX dialect: RevTeX 3.
Predictability in Systems with Many Characteristic Times: The Case of Turbulence
In chaotic dynamical systems, an infinitesimal perturbation is exponentially
amplified at a time-rate given by the inverse of the maximum Lyapunov exponent
. In fully developed turbulence, grows as a power of the
Reynolds number. This result could seem in contrast with phenomenological
arguments suggesting that, as a consequence of `physical' perturbations, the
predictability time is roughly given by the characteristic life-time of the
large scale structures, and hence independent of the Reynolds number. We show
that such a situation is present in generic systems with many degrees of
freedom, since the growth of a non-infinitesimal perturbation is determined by
cumulative effects of many different characteristic times and is unrelated to
the maximum Lyapunov exponent. Our results are illustrated in a chain of
coupled maps and in a shell model for the energy cascade in turbulence.Comment: 24 pages, 10 Postscript figures (included), RevTeX 3.0, files packed
with uufile
2d frustrated Ising model with four phases
In this paper we consider a 2d random Ising system on a square lattice with
nearest neighbour interactions. The disorder is short range correlated and
asymmetry between the vertical and the horizontal direction is admitted. More
precisely, the vertical bonds are supposed to be non random while the
horizontal bonds alternate: one row of all non random horizontal bonds is
followed by one row where they are independent dichotomic random variables. We
solve the model using an approximate approach that replace the quenched average
with an annealed average under the constraint that the number of frustrated
plaquettes is keep fixed and equals that of the true system. The surprising
fact is that for some choices of the parameters of the model there are three
second order phase transitions separating four different phases:
antiferromagnetic, glassy-like, ferromagnetic and paramagnetic.Comment: 17 pages, Plain TeX, uses Harvmac.tex, 4 ps figures, submitted to
Physical Review
Bethe-Peierls Approximation for the 2D Random Ising Model
The partition function of the 2d Ising model with random nearest neighbor
coupling is expressed in the dual lattice made of square plaquettes. The dual
model is solved in the the mean field and in different types of Bethe-Peierls
approximations, using the replica method.Comment: Plane TeX file, 21 pages, 5 figures available under request to
[email protected]
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