360 research outputs found
Transcritical flow of a stratified fluid: The forced extended Korteweg-de Vries model
Transcritical, or resonant, flow of a stratified fluid over an obstacle is studied using a forced extended Korteweg-de Vries model. This model is particularly relevant for a two-layer fluid when the layer depths are near critical, but can also be useful in other similar circumstances. Both quadratic and cubic nonlinearities are present and they are balanced by third-order dispersion. We consider both possible signs for the cubic nonlinear term but emphasize the less-studied case when the cubic nonlinear term and the dispersion term have the same-signed coefficients. In this case, our numerical computations show that two kinds of solitary waves are found in certain parameter regimes. One kind is similar to those of the well-known forced Korteweg-de Vries model and occurs when the cubic nonlinear term is rather small, while the other kind is irregularly generated waves of variable amplitude, which may continually interact. To explain this phenomenon, we develop a hydraulic theory in which the dispersion term in the model is omitted. This theory can predict the occurence of upstream and downstream undular bores, and these predictions are found to agree quite well with the numerical computations. © 2002 American Institute of Physics.published_or_final_versio
Phase Separation of Crystal Surfaces: A Lattice Gas Approach
We consider both equilibrium and kinetic aspects of the phase separation
(``thermal faceting") of thermodynamically unstable crystal surfaces into a
hill--valley structure. The model we study is an Ising lattice gas for a simple
cubic crystal with nearest--neighbor attractive interactions and weak
next--nearest--neighbor repulsive interactions. It is likely applicable to
alkali halides with the sodium chloride structure. Emphasis is placed on the
fact that the equilibrium crystal shape can be interpreted as a phase diagram
and that the details of its structure tell us into which surface orientations
an unstable surface will decompose. We find that, depending on the temperature
and growth conditions, a number of interesting behaviors are expected. For a
crystal in equilibrium with its vapor, these include a low temperature regime
with logarithmically--slow separation into three symmetrically--equivalent
facets, and a higher temperature regime where separation proceeds as a power
law in time into an entire one--parameter family of surface orientations. For a
crystal slightly out of equilibrium with its vapor (slow crystal growth or
etching), power--law growth should be the rule at late enough times. However,
in the low temperature regime, the rate of separation rapidly decreases as the
chemical potential difference between crystal and vapor phases goes to zero.Comment: 16 pages (RevTex 3.0); 12 postscript figures available on request
([email protected]). Submitted to Physical Review E. SFU-JDSDJB-94-0
Extended Universality of the Surface Curvature in Equilibrium Crystal Shapes
We investigate the universal property of curvatures in surface models which
display a flat phase and a rough phase whose criticality is described by the
Gaussian model. Earlier we derived a relation between the Hessian of the free
energy and the Gaussian coupling constant in the six-vertex model. Here we show
its validity in a general setting using renormalization group arguments. The
general validity of the relation is confirmed numerically in the RSOS model by
comparing the Hessian of the free energy and the Gaussian coupling constant in
a transfer matrix finite-size-scaling study. The Hessian relation gives clear
understanding of the universal curvature jump at roughening transitions and
facet edges and also provides an efficient way of locating the phase
boundaries.Comment: 19 pages, RevTex, 3 Postscript Figures, To appear in Phys. Rev.
Cooling-rate effects in a model of (ideal?) glass
Using Monte Carlo simulations we study cooling-rate effects in a
three-dimensional Ising model with four-spin interaction. During coarsening,
this model develops growing energy barriers which at low temperature lead to
very slow dynamics. We show that the characteristic zero-temperature length
increases very slowly with the inverse cooling rate, similarly to the behaviour
of ordinary glasses. For computationally accessible cooling rates the model
undergoes an ideal glassy transition, i.e., the glassy transition for very
small cooling rate coincides a thermodynamic singularity. We also study cooling
of this model with a certain fraction of spins fixed. Due to such heterogeneous
crystalization seeds the final state strongly depends on the cooling rate.Only
for sufficiently fast cooling rate does the system end up in a glassy state
while slow cooling inevitably leads to a crystal phase.Comment: 11 pages, 6 figure
Quenched bond dilution in two-dimensional Potts models
We report a numerical study of the bond-diluted 2-dimensional Potts model
using transfer matrix calculations. For different numbers of states per spin,
we show that the critical exponents at the random fixed point are the same as
in self-dual random-bond cases. In addition, we determine the multifractal
spectrum associated with the scaling dimensions of the moments of the spin-spin
correlation function in the cylinder geometry. We show that the behaviour is
fully compatible with the one observed in the random bond case, confirming the
general picture according to which a unique fixed point describes the critical
properties of different classes of disorder: dilution, self-dual binary
random-bond, self-dual continuous random bond.Comment: LaTeX file with IOP macros, 29 pages, 14 eps figure
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
Metastable lifetimes in a kinetic Ising model: Dependence on field and system size
The lifetimes of metastable states in kinetic Ising ferromagnets are studied
by droplet theory and Monte Carlo simulation, in order to determine their
dependences on applied field and system size. For a wide range of fields, the
dominant field dependence is universal for local dynamics and has the form of
an exponential in the inverse field, modified by universal and nonuniversal
power-law prefactors. Quantitative droplet-theory predictions are numerically
verified, and small deviations are shown to depend nonuniversally on the
details of the dynamics. We identify four distinct field intervals in which the
field dependence and statistical properties of the lifetimes are different. The
field marking the crossover between the weak-field regime, in which the decay
is dominated by a single droplet, and the intermediate-field regime, in which
it is dominated by a finite droplet density, vanishes logarithmically with
system size. As a consequence the slow decay characteristic of the former
regime may be observable in systems that are macroscopic as far as their
equilibrium properties are concerned.Comment: 18 pages single spaced. RevTex Version 3. FSU-SCRI-94-1
Unsteady undular bores in fully nonlinear shallow-water theory
We consider unsteady undular bores for a pair of coupled equations of
Boussinesq-type which contain the familiar fully nonlinear dissipationless
shallow-water dynamics and the leading-order fully nonlinear dispersive terms.
This system contains one horizontal space dimension and time and can be
systematically derived from the full Euler equations for irrotational flows
with a free surface using a standard long-wave asymptotic expansion.
In this context the system was first derived by Su and Gardner. It coincides
with the one-dimensional flat-bottom reduction of the Green-Naghdi system and,
additionally, has recently found a number of fluid dynamics applications other
than the present context of shallow-water gravity waves. We then use the
Whitham modulation theory for a one-phase periodic travelling wave to obtain an
asymptotic analytical description of an undular bore in the Su-Gardner system
for a full range of "depth" ratios across the bore. The positions of the
leading and trailing edges of the undular bore and the amplitude of the leading
solitary wave of the bore are found as functions of this "depth ratio". The
formation of a partial undular bore with a rapidly-varying finite-amplitude
trailing wave front is predicted for ``depth ratios'' across the bore exceeding
1.43. The analytical results from the modulation theory are shown to be in
excellent agreement with full numerical solutions for the development of an
undular bore in the Su-Gardner system.Comment: Revised version accepted for publication in Phys. Fluids, 51 pages, 9
figure
Test of the Kolmogorov-Johnson-Mehl-Avrami picture of metastable decay in a model with microscopic dynamics
The Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory for the time evolution of
the order parameter in systems undergoing first-order phase transformations has
been extended by Sekimoto to the level of two-point correlation functions.
Here, this extended KJMA theory is applied to a kinetic Ising lattice-gas
model, in which the elementary kinetic processes act on microscopic length and
time scales. The theoretical framework is used to analyze data from extensive
Monte Carlo simulations. The theory is inherently a mesoscopic continuum
picture, and in principle it requires a large separation between the
microscopic scales and the mesoscopic scales characteristic of the evolving
two-phase structure. Nevertheless, we find excellent quantitative agreement
with the simulations in a large parameter regime, extending remarkably far
towards strong fields (large supersaturations) and correspondingly small
nucleation barriers. The original KJMA theory permits direct measurement of the
order parameter in the metastable phase, and using the extension to correlation
functions one can also perform separate measurements of the nucleation rate and
the average velocity of the convoluted interface between the metastable and
stable phase regions. The values obtained for all three quantities are verified
by other theoretical and computational methods. As these quantities are often
difficult to measure directly during a process of phase transformation, data
analysis using the extended KJMA theory may provide a useful experimental
alternative.Comment: RevTex, 21 pages including 14 ps figures. Submitted to Phys. Rev. B.
One misprint corrected in Eq.(C1
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