366 research outputs found
Structural Stability and Renormalization Group for Propagating Fronts
A solution to a given equation is structurally stable if it suffers only an
infinitesimal change when the equation (not the solution) is perturbed
infinitesimally. We have found that structural stability can be used as a
velocity selection principle for propagating fronts. We give examples, using
numerical and renormalization group methods.Comment: 14 pages, uiucmac.tex, no figure
The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory
Perturbative renormalization group theory is developed as a unified tool for
global asymptotic analysis. With numerous examples, we illustrate its
application to ordinary differential equation problems involving multiple
scales, boundary layers with technically difficult asymptotic matching, and WKB
analysis. In contrast to conventional methods, the renormalization group
approach requires neither {\it ad hoc\/} assumptions about the structure of
perturbation series nor the use of asymptotic matching. Our renormalization
group approach provides approximate solutions which are practically superior to
those obtained conventionally, although the latter can be reproduced, if
desired, by appropriate expansion of the renormalization group approximant. We
show that the renormalization group equation may be interpreted as an amplitude
equation, and from this point of view develop reductive perturbation theory for
partial differential equations describing spatially-extended systems near
bifurcation points, deriving both amplitude equations and the center manifold.Comment: 44 pages, 2 Postscript figures, macro \uiucmac.tex available at macro
archives or at ftp://gijoe.mrl.uiuc.edu/pu
Geometric origin of scaling in large traffic networks
Large scale traffic networks are an indispensable part of contemporary human
mobility and international trade. Networks of airport travel or cargo ships
movements are invaluable for the understanding of human mobility
patterns\cite{Guimera2005}, epidemic spreading\cite{Colizza2006}, global
trade\cite{Imo2006} and spread of invasive species\cite{Ruiz2000}. Universal
features of such networks are necessary ingredients of their description and
can point to important mechanisms of their formation. Different
studies\cite{Barthelemy2010} point to the universal character of some of the
exponents measured in such networks. Here we show that exponents which relate
i) the strength of nodes to their degree and ii) weights of links to degrees of
nodes that they connect have a geometric origin. We present a simple robust
model which exhibits the observed power laws and relates exponents to the
dimensionality of 2D space in which traffic networks are embedded. The model is
studied both analytically and in simulations and the conditions which result
with previously reported exponents are clearly explained. We show that the
relation between weight strength and degree is , the relation
between distance strength and degree is and the relation
between weight of link and degrees of linked nodes is
on the plane 2D surface. We further analyse the
influence of spherical geometry, relevant for the whole planet, on exact values
of these exponents. Our model predicts that these exponents should be found in
future studies of port networks and impose constraints on more refined models
of port networks.Comment: 17 pages, 5 figures, 1 tabl
Renormalization Group Theory for Global Asymptotic Analysis
We show with several examples that renormalization group (RG) theory can be
used to understand singular and reductive perturbation methods in a unified
fashion. Amplitude equations describing slow motion dynamics in nonequilibrium
phenomena are RG equations. The renormalized perturbation approach may be
simpler to use than other approaches, because it does not require the use of
asymptotic matching, and yields practically superior approximations.Comment: 13 pages, plain tex + uiucmac.tex (available from babbage.sissa.it),
one PostScript figure appended at end. Or (easier) get compressed postscript
file by anon ftp from gijoe.mrl.uiuc.edu (128.174.119.153), file
/pub/rg_sing_prl.ps.
Solution of reduced equations derived with singular perturbation methods
For singular perturbation problems in dynamical systems, various appropriate
singular perturbation methods have been proposed to eliminate secular terms
appearing in the naive expansion. For example, the method of multiple time
scales, the normal form method, center manifold theory, the renormalization
group method are well known. In this paper, it is shown that all of the
solutions of the reduced equations constructed with those methods are exactly
equal to sum of the most divergent secular terms appearing in the naive
expansion. For the proof, a method to construct a perturbation solution which
differs from the conventional one is presented, where we make use of the theory
of Lie symmetry group.Comment: To be published in Phys. Rev.
Robust ecological pattern formation induced by demographic noise
We demonstrate that demographic noise can induce persistent spatial pattern
formation and temporal oscillations in the Levin-Segel predator-prey model for
plankton-herbivore population dynamics. Although the model exhibits a Turing
instability in mean field theory, demographic noise greatly enlarges the region
of parameter space where pattern formation occurs. To distinguish between
patterns generated by fluctuations and those present at the mean field level in
real ecosystems, we calculate the power spectrum in the noise-driven case and
predict the presence of fat tails not present in the mean field case. These
results may account for the prevalence of large-scale ecological patterns,
beyond that expected from traditional non-stochastic approaches.Comment: Revised version. Supporting simulation at:
http://guava.physics.uiuc.edu/~tom/Netlogo
Stiffness of the Edwards-Anderson Model in all Dimensions
A comprehensive description in all dimensions is provided for the scaling
exponent of low-energy excitations in the Ising spin glass introduced by
Edwards and Anderson. A combination of extensive numerical as well as
theoretical results suggest that its lower critical dimension is {\it exactly}
. Such a result would be an essential feature of any complete model of
low-temperature spin glass order and imposes a constraint that may help to
distinguish between theories.Comment: 4 RevTex pages, 2 eps Figures included; related information available
at http://www.physics.emory.edu/faculty/boettcher/publications.html#EO, as to
appear in PR
Origin of the approximate universality of distributions in equilibrium correlated systems
We propose an interpretation of previous experimental and numerical
experiments, showing that for a large class of systems, distributions of global
quantities are similar to a distribution originally obtained for the
magnetization in the 2D-XY model . This approach, developed for the Ising
model, is based on previous numerical observations. We obtain an effective
action using a perturbative method, which successfully describes the order
parameter fluctuations near the phase transition. This leads to a direct link
between the D-dimensional Ising model and the XY model in the same dimension,
which appears to be a generic feature of many equilibrium critical systems and
which is at the heart of the above observations.Comment: To appear in Europhysics Letter
Thermodynamics of Born-Infeld-anti-de Sitter black holes in the grand canonical ensemble
The main objective of this paper is to study thermodynamics and stability of
static electrically charged Born-Infeld black holes in AdS space in D=4. The
Euclidean action for the grand canonical ensemble is computed with the
appropriate boundary terms. The thermodynamical quantities such as the Gibbs
free energy, entropy and specific heat of the black holes are derived from it.
The global stability of black holes are studied in detail by studying the free
energy for various potentials. For small values of the potential, we find that
there is a Hawking-Page phase transition between a BIAdS black hole and the
thermal-AdS space. For large potentials, the black hole phase is dominant and
are preferred over the thermal-AdS space. Local stability is studied by
computing the specific heat for constant potentials. The non-extreme black
holes have two branches: small black holes are unstable and the large black
holes are stable. The extreme black holes are shown to be stable both globally
as well as locally. In addition to the thermodynamics, we also show that the
phase structure relating the mass and the charge of the black holes is
similar to the liquid-gas-solid phase diagram.Comment: Accepted to be published in Physical Review D. Minor change
Universal Scaling in Non-equilibrium Transport Through a Single-Channel Kondo Dot
Scaling laws and universality play an important role in our understanding of
critical phenomena and the Kondo effect. Here we present measurements of
non-equilibrium transport through a single-channel Kondo quantum dot at low
temperature and bias. We find that the low-energy Kondo conductance is
consistent with universality between temperature and bias and characterized by
a quadratic scaling exponent, as expected for the spin-1/2 Kondo effect. The
non-equilibrium Kondo transport measurements are well-described by a universal
scaling function with two scaling parameters.Comment: v2: improved introduction and theory-experiment comparsio
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