12 research outputs found
Natural selection as coarsening
Analogies between evolutionary dynamics and statistical mechanics, such as
Fisher's second-law-like "fundamental theorem of natural selection" and
Wright's "fitness landscapes", have had a deep and fruitful influence on the
development of evolutionary theory. Here I discuss a new conceptual link
between evolution and statistical physics. I argue that natural selection can
be viewed as a coarsening phenomenon, similar to the growth of domain size in
quenched magnets or to Ostwald ripening in alloys and emulsions. In particular,
I show that the most remarkable features of coarsening---scaling and
self-similarity---have strict equivalents in evolutionary dynamics. This
analogy has three main virtues: it brings a set of well-developed mathematical
tools to bear on evolutionary dynamics; it suggests new problems in theoretical
evolution; and it provides coarsening physics with a new exactly soluble model.Comment: Submitted to J. Stat. Phys. for special issue on evolutionary
dynamic
The Lifshitz-Slyozov-Wagner equation for reaction-controlled kinetics
We rigorously derive a weak form of the Lifshitz-Slyozov-Wagner equation as
the homogenization limit of a Stefan-type problem describing
reaction-controlled coarsening of a large number of small spherical particles.
Moreover, we deduce that the effective mean-field description holds true in the
particular limit of vanishing surface-area density of particles.Comment: 15 pages, LaTeX; minor revision, change of titl
Long-time asymptotics for polymerization models
This study is devoted to the long-term behavior of nucleation, growth and
fragmentation equations, modeling the spontaneous formation and kinetics of
large polymers in a spatially homogeneous and closed environment. Such models
are, for instance, commonly used in the biophysical community in order to model
in vitro experiments of fibrillation. We investigate the interplay between four
processes: nucleation, polymeriza-tion, depolymerization and fragmentation. We
first revisit the well-known Lifshitz-Slyozov model, which takes into account
only polymerization and depolymerization, and we show that, when nucleation is
included, the system goes to a trivial equilibrium: all polymers fragmentize,
going back to very small polymers. Taking into account only polymerization and
fragmentation, modeled by the classical growth-fragmentation equation, also
leads the system to the same trivial equilibrium, whether or not nucleation is
considered. However, also taking into account a depolymer-ization reaction term
may surprisingly stabilize the system, since a steady size-distribution of
polymers may then emerge, as soon as polymeriza-tion dominates depolymerization
for large sizes whereas depolymerization dominates polymerization for smaller
ones-a case which fits the classical assumptions for the Lifshitz-Slyozov
equations, but complemented with fragmentation so that " Ostwald ripening "
does not happen.Comment: https://link.springer.com/article/10.1007/s00220-018-3218-
Non-self-similar behavior in the LSW theory of Ostwald ripening
The classical Lifshitz-Slyozov-Wagner theory of domain coarsening predicts
asymptotically self-similar behavior for the size distribution of a dilute
system of particles that evolve by diffusional mass transfer with a common mean
field. Here we consider the long-time behavior of measure-valued solutions for
systems in which particle size is uniformly bounded, i.e., for initial measures
of compact support.
We prove that the long-time behavior of the size distribution depends
sensitively on the initial distribution of the largest particles in the system.
Convergence to the classically predicted smooth similarity solution is
impossible if the initial distribution function is comparable to any finite
power of distance to the end of the support. We give a necessary criterion for
convergence to other self-similar solutions, and conditional stability theorems
for some such solutions. For a dense set of initial data, convergence to any
self-similar solution is impossible.Comment: 31 pages, LaTeX2e; Revised version, to appear in J. Stat. Phy
Macroscopic limit of the Becker-D\"oring equation via gradient flows
This work considers gradient structures for the Becker-D\"oring equation and
its macroscopic limits. The result of Niethammer [17] is extended to prove the
convergence not only for solutions of the Becker-D\"oring equation towards the
Lifshitz-Slyozov-Wagner equation of coarsening, but also the convergence of the
associated gradient structures. We establish the gradient structure of the
nonlocal coarsening equation rigorously and show continuous dependence on the
initial data within this framework. Further, on the considered time scale the
small cluster distribution of the Becker--D\"oring equation follows a
quasistationary distribution dictated by the monomer concentration
On thermodynamically consistent Stefan problems with variable surface energy
A thermodynamically consistent two-phase Stefan problem with
temperature-dependent surface tension and with or without kinetic undercooling
is studied. It is shown that these problems generate local semiflows in
well-defined state manifolds. If a solution does not exhibit singularities, it
is proved that it exists globally in time and converges towards an equilibrium
of the problem. In addition, stability and instability of equilibria is
studied. In particular, it is shown that multiple spheres of the same radius
are unstable if surface heat capacity is small; however, if kinetic
undercooling is absent, they are stable if surface heat capacity is
sufficiently large.Comment: To appear in Arch. Ration. Mech. Anal. The final publication is
available at Springer via http://dx.doi.org/10.1007/s00205-015-0938-y. arXiv
admin note: substantial text overlap with arXiv:1101.376
Μερικές διαφορικές εξισώσεις και προβλήματα αλλαγής φάσεων
Η παρούσα διατριβή αποτελείται από δύο εργασίες στην περιοχή των μερικών
διαφορικών εξισώσεων, οι οποίες ασχολούνται με δύο φυσικά φαινόμενα αλλαγής
φάσεων. Η πρώτη ασχολείται με το φαινόμενο της ωρίμασης Ostwald και αναπαράγει
μία εκδοχή της θεωρίας των Lifshitz, Slyozov και Wagner μέσω ομογενοποίησης
ενός προβλήματος Stefan, ενώ η δεύτερη ασχολείται με την εξαγωγή των συνθηκών
Plateau για τις γωνίες στις τριπλές συμβολές διεπιφανειών στον τριδιάστατο χώρο
από τη διανυσματική εξίσωση Allen–Cahn μέσω ενός τανυστή τάσης που συνδέεται με
την εξίσωση.The present dissertation comprises two papers in the area of partial
differential equations, which study two physical phenomena of phase
transitions. The first is about the phenomenon of Ostwald ripening and derives
a version of the Lifshitz–Slyozov–Wagner theory through the homogenization of a
Stefan problem, while the second is about the derivation of the Plateau angle
conditions at triple junctions of interfaces in three-dimensional space from
the vector-valued Allen–Cahn equation via an associated stress tensor
Μερικές διαφορικές εξισώσεις και προβλήματα αλλαγής φάσεων
Η διατριβή αυτή αποτελείται από δύο εργασίες στην περιοχή των μερικών διαφορικών εξισώσεων, οι οποίες ασχολούνται με δύο φυσικά φαινόμενα αλλαγής φάσεων. Η πρώτη ασχολείται με το φαινόμενο της ωρίμασης Ostwald και αναπαράγει μία εκδοχή της θεωρίας των Lifshitz, Slyozov και Wagner μέσω ομογενοποίησης ενός προβλήματος Stefan. Η δεύτερη εργασία ασχολείται με την εξαγωγή των συνθηκών Plateau για τις γωνίες στις τριπλές συμβολές διεπιφανειών στον τριδιάστατο χώρο από τη διανυσματική εξίσωση Allen–Cahn μέσω ενός τανυστή τάσης που συνδέεται με την εξίσωση