983 research outputs found
Phase coexistence of gradient Gibbs states
We consider the (scalar) gradient fields --with denoting
the nearest-neighbor edges in --that are distributed according to the
Gibbs measure proportional to \texte^{-\beta H(\eta)}\nu(\textd\eta). Here
is the Hamiltonian, is a symmetric potential,
is the inverse temperature, and is the Lebesgue measure on the linear
space defined by imposing the loop condition
for each plaquette
in . For convex , Funaki and Spohn have shown that
ergodic infinite-volume Gibbs measures are characterized by their tilt. We
describe a mechanism by which the gradient Gibbs measures with non-convex
undergo a structural, order-disorder phase transition at some intermediate
value of inverse temperature . At the transition point, there are at
least two distinct gradient measures with zero tilt, i.e., .Comment: 3 figs, PTRF style files include
Orbital order in classical models of transition-metal compounds
We study the classical 120-degree and related orbital models. These are the
classical limits of quantum models which describe the interactions among
orbitals of transition-metal compounds. We demonstrate that at low temperatures
these models exhibit a long-range order which arises via an "order by disorder"
mechanism. This strongly indicates that there is orbital ordering in the
quantum version of these models, notwithstanding recent rigorous results on the
absence of spin order in these systems.Comment: 7 pages, 1 eps fi
Trapping in the random conductance model
We consider random walks on among nearest-neighbor random conductances
which are i.i.d., positive, bounded uniformly from above but whose support
extends all the way to zero. Our focus is on the detailed properties of the
paths of the random walk conditioned to return back to the starting point at
time . We show that in the situations when the heat kernel exhibits
subdiffusive decay --- which is known to occur in dimensions --- the
walk gets trapped for a time of order in a small spatial region. This shows
that the strategy used earlier to infer subdiffusive lower bounds on the heat
kernel in specific examples is in fact dominant. In addition, we settle a
conjecture concerning the worst possible subdiffusive decay in four dimensions.Comment: 21 pages, version to appear in J. Statist. Phy
Colligative properties of solutions: II. Vanishing concentrations
We continue our study of colligative properties of solutions initiated in
math-ph/0407034. We focus on the situations where, in a system of linear size
, the concentration and the chemical potential scale like and
, respectively. We find that there exists a critical value \xit such
that no phase separation occurs for \xi\le\xit while, for \xi>\xit, the two
phases of the solvent coexist for an interval of values of . Moreover, phase
separation begins abruptly in the sense that a macroscopic fraction of the
system suddenly freezes (or melts) forming a crystal (or droplet) of the
complementary phase when reaches a critical value. For certain values of
system parameters, under ``frozen'' boundary conditions, phase separation also
ends abruptly in the sense that the equilibrium droplet grows continuously with
increasing and then suddenly jumps in size to subsume the entire system.
Our findings indicate that the onset of freezing-point depression is in fact a
surface phenomenon.Comment: 27 pages, 1 fig; see also math-ph/0407034 (both to appear in JSP
Dilution Effects in Two-dimensional Quantum Orbital System
We study dilution effects in a Mott insulating state with quantum orbital
degree of freedom, termed the two-dimensional orbital compass model. This is a
quantum and two-dimensional version of the orbital model where the interactions
along different bond directions cause frustration between different orbital
configurations. A long-range correlation of a kind of orbital at each row or
column, termed the directional order, is studied by means of the quantum
Monte-Carlo method. It is shown that decrease of the ordering temperature due
to dilution is much stronger than that in spin models. Quantum effect enhances
the effective dimensionality in the system and makes the directional order
robust against dilution. We discuss an essential mechanism of the dilute
orbital systems.Comment: 5pages, 4 figure
Mean-field driven first-order phase transitions in systems with long-range interactions
We consider a class of spin systems on with vector valued spins
(\bS_x) that interact via the pair-potentials J_{x,y} \bS_x\cdot\bS_y. The
interactions are generally spread-out in the sense that the 's exhibit
either exponential or power-law fall-off. Under the technical condition of
reflection positivity and for sufficiently spread out interactions, we prove
that the model exhibits a first-order phase transition whenever the associated
mean-field theory signals such a transition. As a consequence, e.g., in
dimensions , we can finally provide examples of the 3-state Potts model
with spread-out, exponentially decaying interactions, which undergoes a
first-order phase transition as the temperature varies. Similar transitions are
established in dimensions for power-law decaying interactions and in
high dimensions for next-nearest neighbor couplings. In addition, we also
investigate the limit of infinitely spread-out interactions. Specifically, we
show that once the mean-field theory is in a unique ``state,'' then in any
sequence of translation-invariant Gibbs states various observables converge to
their mean-field values and the states themselves converge to a product
measure.Comment: 57 pages; uses a (modified) jstatphys class fil
Optimal designs for rational function regression
We consider optimal non-sequential designs for a large class of (linear and
nonlinear) regression models involving polynomials and rational functions with
heteroscedastic noise also given by a polynomial or rational weight function.
The proposed method treats D-, E-, A-, and -optimal designs in a
unified manner, and generates a polynomial whose zeros are the support points
of the optimal approximate design, generalizing a number of previously known
results of the same flavor. The method is based on a mathematical optimization
model that can incorporate various criteria of optimality and can be solved
efficiently by well established numerical optimization methods. In contrast to
previous optimization-based methods proposed for similar design problems, it
also has theoretical guarantee of its algorithmic efficiency; in fact, the
running times of all numerical examples considered in the paper are negligible.
The stability of the method is demonstrated in an example involving high degree
polynomials. After discussing linear models, applications for finding locally
optimal designs for nonlinear regression models involving rational functions
are presented, then extensions to robust regression designs, and trigonometric
regression are shown. As a corollary, an upper bound on the size of the support
set of the minimally-supported optimal designs is also found. The method is of
considerable practical importance, with the potential for instance to impact
design software development. Further study of the optimality conditions of the
main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory
and additional example
Indirect Self-Modulation Instability Measurement Concept for the AWAKE Proton Beam
AWAKE, the Advanced Proton-Driven Plasma Wakefield Acceleration Experiment,
is a proof-of-principle R&D experiment at CERN using a 400 GeV/c proton beam
from the CERN SPS (longitudinal beam size sigma_z = 12 cm) which will be sent
into a 10 m long plasma section with a nominal density of approx. 7x10^14
atoms/cm3 (plasma wavelength lambda_p = 1.2mm). In this paper we show that by
measuring the time integrated transverse profile of the proton bunch at two
locations downstream of the AWAKE plasma, information about the occurrence of
the self-modulation instability (SMI) can be inferred. In particular we show
that measuring defocused protons with an angle of 1 mrad corresponds to having
electric fields in the order of GV/m and fully developed self-modulation of the
proton bunch. Additionally, by measuring the defocused beam edge of the
self-modulated bunch, information about the growth rate of the instability can
be extracted. If hosing instability occurs, it could be detected by measuring a
non-uniform defocused beam shape with changing radius. Using a 1 mm thick
Chromox scintillation screen for imaging of the self-modulated proton bunch, an
edge resolution of 0.6 mm and hence a SMI saturation point resolution of 1.2 m
can be achieved.Comment: 4 pages, 4 figures, EAAC conference proceeding
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