11,538 research outputs found
Turing conditions for pattern forming systems on evolving manifolds
The study of pattern-forming instabilities in reaction-diffusion systems on
growing or otherwise time-dependent domains arises in a variety of settings,
including applications in developmental biology, spatial ecology, and
experimental chemistry. Analyzing such instabilities is complicated, as there
is a strong dependence of any spatially homogeneous base states on time, and
the resulting structure of the linearized perturbations used to determine the
onset of instability is inherently non-autonomous. We obtain general conditions
for the onset and structure of diffusion driven instabilities in
reaction-diffusion systems on domains which evolve in time, in terms of the
time-evolution of the Laplace-Beltrami spectrum for the domain and functions
which specify the domain evolution. Our results give sufficient conditions for
diffusive instabilities phrased in terms of differential inequalities which are
both versatile and straightforward to implement, despite the generality of the
studied problem. These conditions generalize a large number of results known in
the literature, such as the algebraic inequalities commonly used as a
sufficient criterion for the Turing instability on static domains, and
approximate asymptotic results valid for specific types of growth, or specific
domains. We demonstrate our general Turing conditions on a variety of domains
with different evolution laws, and in particular show how insight can be gained
even when the domain changes rapidly in time, or when the homogeneous state is
oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to
higher-order spatial systems are also included as a way of demonstrating the
generality of the approach
Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane
We present an overview of pattern formation analysis for an analogue of the
Swift-Hohenberg equation posed on the real hyperbolic space of dimension two,
which we identify with the Poincar\'e disc D. Different types of patterns are
considered: spatially periodic stationary solutions, radial solutions and
traveling waves, however there are significant differences in the results with
the Euclidean case. We apply equivariant bifurcation theory to the study of
spatially periodic solutions on a given lattice of D also called H-planforms in
reference with the "planforms" introduced for pattern formation in Euclidean
space. We consider in details the case of the regular octagonal lattice and
give a complete descriptions of all H-planforms bifurcating in this case. For
radial solutions (in geodesic polar coordinates), we present a result of
existence for stationary localized radial solutions, which we have adapted from
techniques on the Euclidean plane. Finally, we show that unlike the Euclidean
case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf
bifurcation to traveling waves which are invariant along horocycles of D and
periodic in the "transverse" direction. We highlight our theoretical results
with a selection of numerical simulations.Comment: Dedicated to Klaus Kirchg\"assne
The thermodynamics for a hadronic gas of fireballs with internal color structures and chiral fields
The thermodynamical partition function for a gas of color-singlet bags
consisting of fundamental and adjoint particles in both and
group representations is reviewed in detail. The constituent particle species
are assumed to satisfy various thermodynamical statistics. The gas of bags is
probed to study the phase transition for a nuclear matter in the extreme
conditions. These bags are interpreted as the Hagedorn states and they are the
highly excited hadronic states which are produced below the phase transition
point to the quark-gluon plasma. The hadronic density of states has the
Gross-Witten critical point and exhibits a third order phase transition from a
hadronic phase dominated by the discrete low-lying hadronic mass spectrum
particles to another hadronic phase dominated by the continuous Hagedorn
states. The Hagedorn threshold production is found just above the highest known
experimental discrete low-lying hadronic mass spectrum. The subsequent Hagedorn
phase undergoes a first order deconfinement phase transition to an explosive
quark-gluon plasma. The role of the chiral phase transition in the phases of
the discrete low-lying mass spectrum and the continuous Hagedorn mass spectrum
is also considered. It is found crucial in the phase transition diagram.
Alternate scenarios are briefly discussed for the Hagedorn gas undergoes a
higher order phase transition through multi-processes of internal color-flavor
structure modification.Comment: 110 pages and 13 figures. Added references to the introductio
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
Loop exponent in DNA bubble dynamics
Dynamics of DNA bubbles are of interest for both statistical physics and
biology. We present exact solutions to the Fokker-Planck equation governing
bubble dynamics in the presence of a long-range entropic interaction. The
complete meeting time and meeting position probability distributions are
derived from the solutions. Probability distribution functions reflect the
value of the loop exponent of the entropic interaction. Our results extend
previous results which concentrated mainly on the tails of the probability
distribution functions and open a way to determining the strength of the
entropic interaction experimentally which has been a matter of recent
discussions. Using numerical integration, we also discuss the influence of the
finite size of a DNA chain on the bubble dynamics. Analogous results are
obtained also for the case of subdiffusive dynamics of a DNA bubble in a
heteropolymer, revealing highly universal asymptotics of meeting time and
position probability functions.Comment: 24 pages, 11 figures, text identical to the published version; v3 -
updated Ref. [47] and corrected Eqs. (3.6) and (3.10
A Multiscale Diffuse-Interface Model for Two-Phase Flow in Porous Media
In this paper we consider a multiscale phase-field model for
capillarity-driven flows in porous media. The presented model constitutes a
reduction of the conventional Navier-Stokes-Cahn-Hilliard phase-field model,
valid in situations where interest is restricted to dynamical and equilibrium
behavior in an aggregated sense, rather than a precise description of
microscale flow phenomena. The model is based on averaging of the equation of
motion, thereby yielding a significant reduction in the complexity of the
underlying Navier-Stokes-Cahn-Hilliard equations, while retaining its
macroscopic dynamical and equilibrium properties. Numerical results are
presented for the representative 2-dimensional capillary-rise problem
pertaining to two closely spaced vertical plates with both identical and
disparate wetting properties. Comparison with analytical solutions for these
test cases corroborates the accuracy of the presented multiscale model. In
addition, we present results for a capillary-rise problem with a non-trivial
geometry corresponding to a porous medium
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