676 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
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
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
On the formation/dissolution of equilibrium droplets
We consider liquid-vapor systems in finite volume at parameter
values corresponding to phase coexistence and study droplet formation due to a
fixed excess of particles above the ambient gas density. We identify
a dimensionless parameter and a
\textrm{universal} value \Deltac=\Deltac(d), and show that a droplet of the
dense phase occurs whenever \Delta>\Deltac, while, for \Delta<\Deltac, the
excess is entirely absorbed into the gaseous background. When the droplet first
forms, it comprises a non-trivial, \textrm{universal} fraction of excess
particles. Similar reasoning applies to generic two-phase systems at phase
coexistence including solid/gas--where the ``droplet'' is crystalline--and
polymorphic systems. A sketch of a rigorous proof for the 2D Ising lattice gas
is presented; generalizations are discussed heuristically.Comment: An announcement of a forthcoming rigorous work on the 2D Ising model;
to appear in Europhys. Let
Colligative properties of solutions: I. Fixed concentrations
Using the formalism of rigorous statistical mechanics, we study the phenomena
of phase separation and freezing-point depression upon freezing of solutions.
Specifically, we devise an Ising-based model of a solvent-solute system and
show that, in the ensemble with a fixed amount of solute, a macroscopic phase
separation occurs in an interval of values of the chemical potential of the
solvent. The boundaries of the phase separation domain in the phase diagram are
characterized and shown to asymptotically agree with the formulas used in
heuristic analyses of freezing point depression. The limit of infinitesimal
concentrations is described in a subsequent paper.Comment: 28 pages, 1 fig; see also math-ph/0407035 (both to appear in JSP
Superconductivity and charge carrier localization in ultrathin bilayers
/ (LSCO15/LCO) bilayers
with a precisely controlled thickness of N unit cells (UCs) of the former and M
UCs of the latter ([LSCO15\_N/LCO\_M]) were grown on (001)-oriented {\slao}
(SLAO) substrates with pulsed laser deposition (PLD). X-ray diffraction and
reciprocal space map (RSM) studies confirmed the epitaxial growth of the
bilayers and showed that a [LSCO15\_2/LCO\_2] bilayer is fully strained,
whereas a [LSCO15\_2/LCO\_7] bilayer is already partially relaxed. The
\textit{in situ} monitoring of the growth with reflection high energy electron
diffraction (RHEED) revealed that the gas environment during deposition has a
surprisingly strong effect on the growth mode and thus on the amount of
disorder in the first UC of LSCO15 (or the first two monolayers of LSCO15
containing one plane each). For samples grown in pure
gas (growth type-B), the first LSCO15 UC next to the SLAO
substrate is strongly disordered. This disorder is strongly reduced if the
growth is performed in a mixture of and gas
(growth type-A). Electric transport measurements confirmed that the first UC of
LSCO15 next to the SLAO substrate is highly resistive and shows no sign of
superconductivity for growth type-B, whereas it is superconducting for growth
type-A. Furthermore, we found, rather surprisingly, that the conductivity of
the LSCO15 UC next to the LCO capping layer strongly depends on the thickness
of the latter. A LCO capping layer with 7~UCs leads to a strong localization of
the charge carriers in the adjacent LSCO15 UC and suppresses superconductivity.
The magneto-transport data suggest a similarity with the case of weakly hole
doped LSCO single crystals that are in a so-called {"{cluster-spin-glass
state}"
Territorial Developments Based on Graffiti: a Statistical Mechanics Approach
We study the well-known sociological phenomenon of gang aggregation and
territory formation through an interacting agent system defined on a lattice.
We introduce a two-gang Hamiltonian model where agents have red or blue
affiliation but are otherwise indistinguishable. In this model, all
interactions are indirect and occur only via graffiti markings, on-site as well
as on nearest neighbor locations. We also allow for gang proliferation and
graffiti suppression. Within the context of this model, we show that gang
clustering and territory formation may arise under specific parameter choices
and that a phase transition may occur between well-mixed, possibly dilute
configurations and well separated, clustered ones. Using methods from
statistical mechanics, we study the phase transition between these two
qualitatively different scenarios. In the mean-field rendition of this model,
we identify parameter regimes where the transition is first or second order. In
all cases, we have found that the transitions are a consequence solely of the
gang to graffiti couplings, implying that direct gang to gang interactions are
not strictly necessary for gang territory formation; in particular, graffiti
may be the sole driving force behind gang clustering. We further discuss
possible sociological -- as well as ecological -- ramifications of our results
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