49 research outputs found
The effects of forcing and dissipation on phase transitions in thin granular layers
Recent experimental and computational studies of vibrated thin layers of
identical spheres have shown transitions to ordered phases similar to those
seen in equilibrium systems. Motivated by these results, we carry out
simulations of hard inelastic spheres forced by homogenous white noise. We find
a transition to an ordered state of the same symmetry as that seen in the
experiments, but the clear phase separation observed in the vibrated system is
absent. Simulations of purely elastic spheres also show no evidence for phase
separation. We show that the energy injection in the vibrated system is
dramatically different in the different phases, and suggest that this creates
an effective surface tension not present in the equilibrium or randomly forced
systems. We do find, however, that inelasticity suppresses the onset of the
ordered phase with random forcing, as is observed in the vibrating system, and
that the amount of the suppression is proportional to the degree of
inelasticity. The suppression depends on the details of the energy injection
mechanism, but is completely eliminated when inelastic collisions are replaced
by uniform system-wide energy dissipation.Comment: 10 pages, 5 figure
Geometry of Valley Growth
Although amphitheater-shaped valley heads can be cut by groundwater flows
emerging from springs, recent geological evidence suggests that other processes
may also produce similar features, thus confounding the interpretations of such
valley heads on Earth and Mars. To better understand the origin of this
topographic form we combine field observations, laboratory experiments,
analysis of a high-resolution topographic map, and mathematical theory to
quantitatively characterize a class of physical phenomena that produce
amphitheater-shaped heads. The resulting geometric growth equation accurately
predicts the shape of decimeter-wide channels in laboratory experiments,
100-meter wide valleys in Florida and Idaho, and kilometer wide valleys on
Mars. We find that whenever the processes shaping a landscape favor the growth
of sharply protruding features, channels develop amphitheater-shaped heads with
an aspect ratio of pi
Unsteady Crack Motion and Branching in a Phase-Field Model of Brittle Fracture
Crack propagation is studied numerically using a continuum phase-field
approach to mode III brittle fracture. The results shed light on the physics
that controls the speed of accelerating cracks and the characteristic branching
instability at a fraction of the wave speed.Comment: 11 pages, 4 figure
Phase field model of premelting of grain boundaries
We present a phase field model of solidification which includes the effects
of the crystalline orientation in the solid phase. This model describes grain
boundaries as well as solid-liquid boundaries within a unified framework. With
an appropriate choice of coupling of the phase field variable to the gradient
of the crystalline orientation variable in the free energy, we find that high
angle boundaries undergo a premelting transition. As the melting temperature is
approached from below, low angle grain boundaries remain narrow. The width of
the liquid layer at high angle grain boundaries diverges logarithmically. In
addition, for some choices of model coupling, there may be a discontinuous jump
in the width of the fluid layer as function of temperature.Comment: 6 pages, 9 figures, RevTeX
Estimation of prokaryotic supergenome size and composition from gene frequency distributions
BACKGROUND: Because prokaryotic genomes experience a rapid flux of genes, selection may act at a higher level than an individual genome. We explore a quantitative model of the distributed genome whereby groups of genomes evolve by acquiring genes from a fixed reservoir which we denote as supergenome. Previous attempts to understand the nature of the supergenome treated genomes as random, independent collections of genes and assumed that the supergenome consists of a small number of homogeneous sub-reservoirs. Here we explore the consequences of relaxing both assumptions. RESULTS: We surveyed several methods for estimating the size and composition of the supergenome. The methods assumed that genomes were either random, independent samples of the supergenome or that they evolved from a common ancestor along a known tree via stochastic sampling from the reservoir. The reservoir was assumed to be either a collection of homogeneous sub-reservoirs or alternatively composed of genes with Gamma distributed gain probabilities. Empirical gene frequencies were used to either compute the likelihood of the data directly or first to reconstruct the history of gene gains and then compute the likelihood of the reconstructed numbers of gains. CONCLUSIONS: Supergenome size estimates using the empirical gene frequencies directly are not robust with respect to the choice of the model. By contrast, using the gene frequencies and the phylogenetic tree to reconstruct multiple gene gains produces reliable estimates of the supergenome size and indicates that a homogeneous supergenome is more consistent with the data than a supergenome with Gamma distributed gain probabilities
Properties of Ridges in Elastic Membranes
When a thin elastic sheet is confined to a region much smaller than its size
the morphology of the resulting crumpled membrane is a network of straight
ridges or folds that meet at sharp vertices. A virial theorem predicts the
ratio of the total bending and stretching energies of a ridge. Small strains
and curvatures persist far away from the ridge. We discuss several kinds of
perturbations that distinguish a ridge in a crumpled sheet from an isolated
ridge studied earlier (A. E. Lobkovsky, Phys. Rev. E. 53 3750 (1996)). Linear
response as well as buckling properties are investigated. We find that quite
generally, the energy of a ridge can change by no more than a finite fraction
before it buckles.Comment: 13 pages, RevTeX, acknowledgement adde
Critical examination of cohesive-zone models in the theory of dynamic fracture
We have examined a class of cohesive-zone models of dynamic mode-I fracture,
looking both at steady-state crack propagation and its stability against
out-of-plane perturbations. Our work is an extension of that of Ching, Langer,
and Nakanishi (CLN) (Phys. Rev. E, vol. 53, no. 3, p. 2864 (1996)), who studied
a non-dissipative version of this model and reported strong instability at all
non-zero crack speeds. We have reformulated the CLN theory and have discovered,
surprisingly, that their model is mathematically ill-posed. In an attempt to
correct this difficulty and to construct models that might exhibit realistic
behavior, we have extended the CLN analysis to include dissipative mechanisms
within the cohesive zone. We have succeeded to some extent in finding
mathematically well posed systems; and we even have found a class of models for
which a transition from stability to instability may occur at a nonzero crack
speed via a Hopf bifurcation at a finite wavelength of the applied
perturbation. However, our general conclusion is that these cohesive-zone
models are inherently unsatisfactory for use in dynamical studies. They are
extremely difficult mathematically, and they seem to be highly sensitive to
details that ought to be physically unimportant.Comment: 19 pages, REVTeX 3.1, epsf.sty, also available at
http://itp.ucsb.edu/~lobkovs
Low temperature dynamics of kinks on Ising interfaces
The anisotropic motion of an interface driven by its intrinsic curvature or
by an external field is investigated in the context of the kinetic Ising model
in both two and three dimensions. We derive in two dimensions (2d) a continuum
evolution equation for the density of kinks by a time-dependent and nonlocal
mapping to the asymmetric exclusion process. Whereas kinks execute random walks
biased by the external field and pile up vertically on the physical 2d lattice,
then execute hard-core biased random walks on a transformed 1d lattice. Their
density obeys a nonlinear diffusion equation which can be transformed into the
standard expression for the interface velocity v = M[(gamma + gamma'')kappa +
H]$, where M, gamma + gamma'', and kappa are the interface mobility, stiffness,
and curvature, respectively. In 3d, we obtain the velocity of a curved
interface near the orientation from an analysis of the self-similar
evolution of 2d shrinking terraces. We show that this velocity is consistent
with the one predicted from the 3d tensorial generalization of the law for
anisotropic curvature-driven motion. In this generalization, both the interface
stiffness tensor and the curvature tensor are singular at the
orientation. However, their product, which determines the interface velocity,
is smooth. In addition, we illustrate how this kink-based kinetic description
provides a useful framework for studying more complex situations by modeling
the effect of immobile dilute impurities.Comment: 11 pages, 10 figure
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Viral Diversity Threshold for Adaptive Immunity in Prokaryotes
Bacteria and archaea face continual onslaughts of rapidly diversifying viruses and plasmids. Many prokaryotes maintain adaptive immune systems known as clustered regularly interspaced short palindromic repeats (CRISPR) and CRISPR-associated genes (Cas). CRISPR-Cas systems are genomic sensors that serially acquire viral and plasmid DNA fragments (spacers) that are utilized to target and cleave matching viral and plasmid DNA in subsequent genomic invasions, offering critical immunological memory. Only 50% of sequenced bacteria possess CRISPR-Cas immunity, in contrast to over 90% of sequenced archaea. To probe why half of bacteria lack CRISPR-Cas immunity, we combined comparative genomics and mathematical modeling. Analysis of hundreds of diverse prokaryotic genomes shows that CRISPR-Cas systems are substantially more prevalent in thermophiles than in mesophiles. With sequenced bacteria disproportionately mesophilic and sequenced archaea mostly thermophilic, the presence of CRISPR-Cas appears to depend more on environmental temperature than on bacterial-archaeal taxonomy. Mutation rates are typically severalfold higher in mesophilic prokaryotes than in thermophilic prokaryotes. To quantitatively test whether accelerated viral mutation leads microbes to lose CRISPR-Cas systems, we developed a stochastic model of virus-CRISPR coevolution. The model competes CRISPR-Cas-positive (CRISPR-Cas+) prokaryotes against CRISPR-Cas-negative (CRISPR-Cas−) prokaryotes, continually weighing the antiviral benefits conferred by CRISPR-Cas immunity against its fitness costs. Tracking this cost-benefit analysis across parameter space reveals viral mutation rate thresholds beyond which CRISPR-Cas cannot provide sufficient immunity and is purged from host populations. These results offer a simple, testable viral diversity hypothesis to explain why mesophilic bacteria disproportionately lack CRISPR-Cas immunity. More generally, fundamental limits on the adaptability of biological sensors (Lamarckian evolution) are predicted
Erosion of a granular bed driven by laminar fluid flow
Motivated by examples of erosive incision of channels in sand, we investigate
the motion of individual grains in a granular bed driven by a laminar fluid to
give us new insights into the relationship between hydrodynamic stress and
surface granular flow. A closed cell of rectangular cross-section is partially
filled with glass beads and a constant fluid flux flows through the cell.
The refractive indices of the fluid and the glass beads are matched and the
cell is illuminated with a laser sheet, allowing us to image individual beads.
The bed erodes to a rest height which depends on . The Shields
threshold criterion assumes that the non-dimensional ratio of the
viscous stress on the bed to the hydrostatic pressure difference across a grain
is sufficient to predict the granular flux. Furthermore, the Shields criterion
states that the granular flux is non-zero only for . We find
that the Shields criterion describes the observed relationship when the bed height is offset by approximately half a grain diameter.
Introducing this offset in the estimation of yields a collapse of the
measured Einstein number to a power-law function of
with exponent . The dynamics of the bed height relaxation are
well described by the power law relationship between the granular flux and the
bed stress.Comment: 12 pages, 5 figure