10 research outputs found
Three-dimensional Ising model confined in low-porosity aerogels: a Monte Carlo study
The influence of correlated impurities on the critical behaviour of the 3D
Ising model is studied using Monte Carlo simulations. Spins are confined into
the pores of simulated aerogels (diffusion limited cluster-cluster aggregation)
in order to study the effect of quenched disorder on the critical behaviour of
this magnetic system. Finite size scaling is used to estimate critical
couplings and exponents. Long-range correlated disorder does not affect
critical behavior. Asymptotic exponents differ from those of the pure 3D Ising
model (3DIS), but it is impossible, with our precision, to distinguish them
from the randomly diluted Ising model (RDIS).Comment: 10 pages, 10 figures. Submitted to Physical Review
Weak quenched disorder and criticality: resummation of asymptotic(?) series
In these lectures, we discuss the influence of weak quenched disorder on the
critical behavior in condensed matter and give a brief review of available
experimental and theoretical results as well as results of MC simulations of
these phenomena. We concentrate on three cases: (i) uncorrelated random-site
disorder, (ii) long-range-correlated random-site disorder, and (iii) random
anisotropy.
Today, the standard analytical description of critical behavior is given by
renormalization group results refined by resummation of the perturbation theory
series. The convergence properties of the series are unknown for most
disordered models. The main object of these lectures is to discuss the
peculiarities of the application of resummation techniques to perturbation
theory series of disordered models.Comment: Lectures given at the Second International Pamporovo Workshop on
Cooperative Phenomena in Condensed Matter (28th July - 7th August 2001,
Pamporovo, Bulgaria). 51 pages, 12 figures, 1 style files include
Critical dynamics and effective exponents of magnets with extended impurities
We investigate the asymptotic and effective static and dynamic critical
behavior of (d=3)-dimensional magnets with quenched extended defects,
correlated in dimensions (which can be considered as the
dimensionality of the defects) and randomly distributed in the remaining
dimensions. The field-theoretical renormalization group
perturbative expansions being evaluated naively do not allow for the reliable
numerical data. We apply the Chisholm-Borel resummation technique to restore
convergence of the two-loop expansions and report the numerical values of the
asymptotic critical exponents for the model A dynamics. We discuss different
scenarios for static and dynamic effective critical behavior and give values
for corresponding non-universal exponents.Comment: 12 pages, 6 figure
Where two fractals meet: the scaling of a self-avoiding walk on a percolation cluster
The scaling properties of self-avoiding walks on a d-dimensional diluted
lattice at the percolation threshold are analyzed by a field-theoretical
renormalization group approach. To this end we reconsider the model of Y. Meir
and A. B. Harris (Phys. Rev. Lett. 63:2819 (1989)) and argue that via
renormalization its multifractal properties are directly accessible. While the
former first order perturbation did not agree with the results of other
methods, we find that the asymptotic behavior of a self-avoiding walk on the
percolation cluster is governed by the exponent nu_p=1/2 + epsilon/42 +
110epsilon^2/21^3, epsilon=6-d. This analytic result gives an accurate numeric
description of the available MC and exact enumeration data in a wide range of
dimensions 2<=d<=6.Comment: 4 pages, 2 figure
Critical behavior of magnetic systems with extended impurities in general dimensions
We investigate the critical properties of d-dimensional magnetic systems with
quenched extended defects, correlated in
dimensions (which can be considered as the dimensionality of the
defects) and randomly distributed in the remaining dimensions;
both in the case of fixed dimension d=3 and when the space dimension
continuously changes from the lower critical dimension to the upper one. The
renormalization group calculations are performed in the minimal subtraction
scheme. We analyze the two-loop renormalization group functions for different
fixed values of the parameters . To this end, we apply the
Chisholm-Borel resummation technique and report the numerical values of the
critical exponents for the universality class of this system.Comment: 8 figures. To appear in Phys. Rev.
Polymers in long-range-correlated disorder
We study the scaling properties of polymers in a d-dimensional medium with
quenched defects that have power law correlations ~r^{-a} for large separations
r. This type of disorder is known to be relevant for magnetic phase
transitions. We find strong evidence that this is true also for the polymer
case. Applying the field-theoretical renormalization group approach we perform
calculations both in a double expansion in epsilon=4-d and delta=4-a up to the
1-loop order and secondly in a fixed dimension (d=3) approach up to the 2-loop
approximation for different fixed values of the correlation parameter, 2=<a=<3.
In the latter case the numerical results need appropriate resummation. We find
that the asymptotic behavior of self-avoiding walks in three dimensions and
long-range-correlated disorder is governed by a set of separate exponents. In
particular, we give estimates for the 'nu' and 'gamma' exponents as well as for
the correction-to-scaling exponent 'omega'. The latter exponent is also
calculated for the general m-vector model with m=1,2,3.Comment: 13 pages, 5 figure
Network Harness: Metropolis Public Transport
We analyze the public transport networks (PTNs) of a number of major cities
of the world. While the primary network topology is defined by a set of routes
each servicing an ordered series of given stations, a number of different
neighborhood relations may be defined both for the routes and the stations. The
networks defined in this way display distinguishing properties, the most
striking being that often several routes proceed in parallel for a sequence of
stations. Other networks with real-world links like cables or neurons embedded
in two or three dimensions often show the same feature - we use the car
engineering term "harness" for such networks. Geographical data for the routes
reveal surprising self-avoiding walk (SAW) properties. We propose and simulate
an evolutionary model of PTNs based on effectively interacting SAWs that
reproduces the key features.Comment: 5 pages, 4 figure
Entropy-induced separation of star polymers in porous media
We present a quantitative picture of the separation of star polymers in a
solution where part of the volume is influenced by a porous medium. To this
end, we study the impact of long-range-correlated quenched disorder on the
entropy and scaling properties of -arm star polymers in a good solvent. We
assume that the disorder is correlated on the polymer length scale with a
power-law decay of the pair correlation function . Applying
the field-theoretical renormalization group approach we show in a double
expansion in and that there is a range of
correlation strengths for which the disorder changes the scaling
behavior of star polymers. In a second approach we calculate for fixed space
dimension and different values of the correlation parameter the
corresponding scaling exponents that govern entropic effects. We
find that , the deviation of from its mean field value
is amplified by the disorder once we increase beyond a threshold. The
consequences for a solution of diluted chain and star polymers of equal
molecular weight inside a porous medium are: star polymers exert a higher
osmotic pressure than chain polymers and in general higher branched star
polymers are expelled more strongly from the correlated porous medium.
Surprisingly, polymer chains will prefer a stronger correlated medium to a less
or uncorrelated medium of the same density while the opposite is the case for
star polymers.Comment: 14 pages, 7 figure
The three-dimensional randomly dilute Ising model: Monte Carlo results
We perform a high-statistics simulation of the three-dimensional randomly
dilute Ising model on cubic lattices with . We choose a
particular value of the density, x=0.8, for which the leading scaling
corrections are suppressed. We determine the critical exponents, obtaining , , , and ,
in agreement with previous numerical simulations. We also estimate numerically
the fixed-point values of the four-point zero-momentum couplings that are used
in field-theoretical fixed-dimension studies. Although these results somewhat
differ from those obtained using perturbative field theory, the
field-theoretical estimates of the critical exponents do not change
significantly if the Monte Carlo result for the fixed point is used. Finally,
we determine the six-point zero-momentum couplings, relevant for the
small-magnetization expansion of the equation of state, and the invariant
amplitude ratio that expresses the universality of the free-energy
density per correlation volume. We find .Comment: 34 pages, 7 figs, few correction