29,057 research outputs found
Higher order corrections for anisotropic bootstrap percolation
We study the critical probability for the metastable phase transition of the
two-dimensional anisotropic bootstrap percolation model with
-neighbourhood and threshold . The first order asymptotics for
the critical probability were recently determined by the first and second
authors. Here we determine the following sharp second and third order
asymptotics:
We note that the second and third order terms are so large that the first order
asymptotics fail to approximate even for lattices of size well beyond
.Comment: 46 page
Slow nonequilibrium dynamics: parallels between classical and quantum glasses and gently driven systems
We review an scenario for the non-equilibrium dynamics of glassy systems that
has been motivated by the exact solution of simple models. This approach allows
one to set on firmer grounds well-known phenomenological theories. The old
ideas of entropy crisis, fictive temperatures, free-volume... have clear
definitions within these models. Aging effects in the glass phase are also
captured. One of the salient features of the analytic solution, the breakdown
of the fluctuation-dissipation relations, provides a definition of a bonafide
{\it effective temperature} that is measurable by a thermometer, controls heat
flows, partial equilibrations, and the reaction to the external injection of
heat. The effective temperature is an extremely robust concept that appears in
non-equilibrium systems in the limit of small entropy production as, for
instance, sheared fluids, glasses at low temperatures when quantum fluctuations
are relevant, tapped or vibrated granular matter, etc. The emerging scenario is
one of partial equilibrations, in which glassy systems arrange their internal
degrees of freedom so that the slow ones select their own effective
temperatures. It has been proven to be consistent within any perturbative
resummation scheme (mode coupling, etc) and it can be challenged by
experimental and numerical tests, some of which it has already passed.Comment: 15 pages, 8 figure
Surface Charge Density Wave Transition in NbSe
The two charge-density wave (CDW) transitions in NbSe %at wave numbers at
and , occurring at the surface were investigated by
scanning tunneling microscopy (STM) on \emph{in situ} cleaved
plane. The temperature dependence of first-order CDW satellite spots, obtained
from the Fourier transform of the STM images, was measured between 5-140 K to
extract the surface critical temperatures (T). The low T CDW transition
occurs at T=70-75 K, more than 15 K above the bulk TK while at
exactly the same wave number. %determined by x-ray diffraction experiments.
Plausible mechanism for such an unusually high surface enhancement is a
softening of transverse phonon modes involved in the CDW formation.% The large
interval of the 2D regime allows to speculate on % %the special
Berezinskii-Kosterlitz-Thouless type of the surface transition expected for
this incommensurate CDW. This scenario is checked by extracting the temperature
dependence of the order % %parameter correlation functions. The regime of 2D
fluctuations is analyzed according to a Berezinskii-Kosterlitz-Thouless type of
surface transition, expected for this incommensurate 2D CDW, by extracting the
temperature dependence of the order parameter correlation functions.Comment: 5 pages, 2 figure
Particle Diffusion and Acceleration by Shock Wave in Magnetized Filamentary Turbulence
We expand the off-resonant scattering theory for particle diffusion in
magnetized current filaments that can be typically compared to astrophysical
jets, including active galactic nucleus jets. In a high plasma beta region
where the directional bulk flow is a free-energy source for establishing
turbulent magnetic fields via current filamentation instabilities, a novel
version of quasi-linear theory to describe the diffusion of test particles is
proposed. The theory relies on the proviso that the injected energetic
particles are not trapped in the small-scale structure of magnetic fields
wrapping around and permeating a filament but deflected by the filaments, to
open a new regime of the energy hierarchy mediated by a transition compared to
the particle injection. The diffusion coefficient derived from a quasi-linear
type equation is applied to estimating the timescale for the stochastic
acceleration of particles by the shock wave propagating through the jet. The
generic scalings of the achievable highest energy of an accelerated ion and
electron, as well as of the characteristic time for conceivable energy
restrictions, are systematically presented. We also discuss a feasible method
of verifying the theoretical predictions. The strong, anisotropic turbulence
reflecting cosmic filaments might be the key to the problem of the acceleration
mechanism of the highest energy cosmic rays exceeding 100 EeV (10^{20} eV),
detected in recent air shower experiments.Comment: 39 pages, 2 figures, accepted for publication in Ap
Glassy dynamics of kinetically constrained models
We review the use of kinetically constrained models (KCMs) for the study of
dynamics in glassy systems. The characteristic feature of KCMs is that they
have trivial, often non-interacting, equilibrium behaviour but interesting slow
dynamics due to restrictions on the allowed transitions between configurations.
The basic question which KCMs ask is therefore how much glassy physics can be
understood without an underlying ``equilibrium glass transition''. After a
brief review of glassy phenomenology, we describe the main model classes, which
include spin-facilitated (Ising) models, constrained lattice gases, models
inspired by cellular structures such as soap froths, models obtained via
mappings from interacting systems without constraints, and finally related
models such as urn, oscillator, tiling and needle models. We then describe the
broad range of techniques that have been applied to KCMs, including exact
solutions, adiabatic approximations, projection and mode-coupling techniques,
diagrammatic approaches and mappings to quantum systems or effective models.
Finally, we give a survey of the known results for the dynamics of KCMs both in
and out of equilibrium, including topics such as relaxation time divergences
and dynamical transitions, nonlinear relaxation, aging and effective
temperatures, cooperativity and dynamical heterogeneities, and finally
non-equilibrium stationary states generated by external driving. We conclude
with a discussion of open questions and possibilities for future work.Comment: 137 pages. Additions to section on dynamical heterogeneities (5.5,
new pages 110 and 112), otherwise minor corrections, additions and reference
updates. Version to be published in Advances in Physic
Growing timescales and lengthscales characterizing vibrations of amorphous solids
Low-temperature properties of crystalline solids can be understood using
harmonic perturbations around a perfect lattice, as in Debye's theory.
Low-temperature properties of amorphous solids, however, strongly depart from
such descriptions, displaying enhanced transport, activated slow dynamics
across energy barriers, excess vibrational modes with respect to Debye's theory
(i.e., a Boson Peak), and complex irreversible responses to small mechanical
deformations. These experimental observations indirectly suggest that the
dynamics of amorphous solids becomes anomalous at low temperatures. Here, we
present direct numerical evidence that vibrations change nature at a
well-defined location deep inside the glass phase of a simple glass former. We
provide a real-space description of this transition and of the rapidly growing
time and length scales that accompany it. Our results provide the seed for a
universal understanding of low-temperature glass anomalies within the
theoretical framework of the recently discovered Gardner phase transition.Comment: 12 pages, 20 figures. Accepted for publication in PNA
Phase Transitions for Gödel Incompleteness
Gödel's first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers
have been looking for natural examples of such assertions and breakthroughs have been obtained in the seventies by Jeff Paris (in part jointly with Leo Harrington and Laurie Kirby) and Harvey Friedman who produced first mathematically interesting
independence results in Ramsey theory (Paris) and well-order and well-quasi-order theory (Friedman).
In this article we investigate Friedman style principles of combinatorial well-foundedness for the ordinals below epsilon_0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.
For these independence principles we classify (as a part of a general research program) their phase transitions, i.e. we classify exactly the bounding conditions which lead from
provability to unprovability in the induced combinatorial
well-foundedness principles.
As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel's coding of finite sequences from his classical 1931 paper on his incompleteness results.
This choice makes the investigation highly non trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions.
For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem by Parameswaran which apparently is far remote from Gödel's theorem
Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and SOS
The "SOS" in the title does not refer to the international distress signal,
but to "solid-on-solid" (SOS) surface growth. The catastrophic cascades are
those observed by Buldyrev {\it et al.} in interdependent networks, which we
re-interpret as multiplex networks with agents that can only survive if they
mutually support each other, and whose survival struggle we map onto an SOS
type growth model. This mapping not only reveals non-trivial structures in the
phase space of the model, but also leads to a new and extremely efficient
simulation algorithm. We use this algorithm to study interdependent agents on
duplex Erd\"os-R\'enyi (ER) networks and on lattices with dimensions 2, 3, 4,
and 5. We obtain new and surprising results in all these cases, and we correct
statements in the literature for ER networks and for 2-d lattices. In
particular, we find that is the upper critical dimension, that the
percolation transition is continuous for but -- at least for -- not in the universality class of ordinary percolation. For ER networks we
verify that the cluster statistics is exactly described by mean field theory,
but find evidence that the cascade process is not. For we find a first
order transition as for ER networks, but we find also that small clusters have
a nontrivial mass distribution that scales at the transition point. Finally,
for with intermediate range dependency links we propose a scenario
different from that proposed in W. Li {\it et al.}, PRL {\bf 108}, 228702
(2012).Comment: 19 pages, 32 figure
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