arXiv subjects

H. G. E. Hentschel

Publications and source records attributed to H. G. E. Hentschel.

25 records · Page 2Linked to original sources

arXiv2007

Converting genetic network oscillations into somite spatial pattern

In most vertebrate species, the body axis is generated by the formation of repeated transient structures called somites. This spatial periodicity in somitogenesis has been related to the temporally sustained oscillations in certain mRNAs and their associated gene products in the cells forming the presomatic mesoderm. The mechanism underlying these oscillations have been identified as due to the delays involved in the synthesis of mRNA and translation into protein molecules [J. Lewis, Current Biol. {\bf 13}, 1398 (2003)]. In addition, in the zebrafish embryo intercellular Notch signalling couples these oscillators and a longitudinal positional information signal in the form of an Fgf8 gradient exists that could be used to transform these coupled temporal oscillations into the observed spatial periodicity of somites. Here we consider a simple model based on this known biology and study its consequences for somitogenesis. Comparison is made with the known properties of somite formation in the zebrafish embryo . We also study the effects of localized Fgf8 perturbations on somite patterning.

q-bio.QM↗

arXiv2006

Direct Identification of the Glass Transition: Growing Length Scale and the Onset of Plasticity

Understanding the mechanical properties of glasses remains elusive since the glass transition itself is not fully understood, even in well studied examples of glass formers in two dimensions. In this context we demonstrate here: (i) a direct evidence for a diverging length scale at the glass transition (ii) an identification of the glass transition with the disappearance of fluid-like regions and (iii) the appearance in the glass state of fluid-like regions when mechanical strain is applied. These fluid-like regions are associated with the onset of plasticity in the amorphous solid. The relaxation times which diverge upon the approach to the glass transition are related quantitatively.

cond-mat.stat-mech↗

arXiv2006

Statistical Mechanics of the Glass Transition

The statistical mechanics of simple glass forming systems in 2 dimensions is worked out. The glass disorder is encoded via a Voronoi tessellation, and the statistical mechanics is performed directly in this encoding. The theory provides, without free parameters, an explanation of the glass transition phenomenology, including the identification of two different temperatures, $T_g$ and $T_k$, the first associated with jamming and the second associated with the appearance of a quasi-crystal at very low temperatures.

cond-mat.stat-mech↗

arXiv2003

Diffusion Limited Aggregation with Power-Law Pinning

Using stochastic conformal mapping techniques we study the patterns emerging from Laplacian growth with a power-law decaying threshold for growth $R_N^{-γ}$ (where $R_N$ is the radius of the $N-$ particle cluster). For $γ> 1$ the growth pattern is in the same universality class as diffusion limited aggregation (DLA) growth, while for $γ< 1$ the resulting patterns have a lower fractal dimension $D(γ)$ than a DLA cluster due to the enhancement of growth at the hot tips of the developing pattern. Our results indicate that a pinning transition occurs at $γ= 1/2$, significantly smaller than might be expected from the lower bound $α_{min} \simeq 0.67$ of multifractal spectrum of DLA. This limiting case shows that the most singular tips in the pruned cluster now correspond to those expected for a purely one-dimensional line. Using multifractal analysis, analytic expressions are established for $D(γ)$ both close to the breakdown of DLA universality class, i.e., $γ\lesssim 1$, and close to the pinning transition, i.e., $γ\gtrsim 1/2$.

cond-mat.stat-mech↗

arXiv2003

Anisotropic Diffusion Limited Aggregation

Using stochastic conformal mappings we study the effects of anisotropic perturbations on diffusion limited aggregation (DLA) in two dimensions. The harmonic measure of the growth probability for DLA can be conformally mapped onto a constant measure on a unit circle. Here we map $m$ preferred directions for growth of angular width $σ$ to a distribution on the unit circle which is a periodic function with $m$ peaks in $[-π, π)$ such that the width $σ$ of each peak scales as $σ\sim 1/\sqrt{k}$, where $k$ defines the ``strength'' of anisotropy along any of the $m$ chosen directions. The two parameters $(m,k)$ map out a parameter space of perturbations that allows a continuous transition from DLA (for $m=0$ or $k=0$) to $m$ needle-like fingers as $k \to \infty$. We show that at fixed $m$ the effective fractal dimension of the clusters $D(m,k)$ obtained from mass-radius scaling decreases with increasing $k$ from $D_{DLA} \simeq 1.71$ to a value bounded from below by $D_{min} = 3/2$. Scaling arguments suggest a specific form for the dependence of the fractal dimension $D(m,k)$ on $k$ for large $k$, form which compares favorably with numerical results.

cond-mat.stat-mech↗

arXiv2000

Conformal Dynamics of Fractal Growth Patterns Without Randomness

Many models of fractal growth patterns (like Diffusion Limited Aggregation and Dielectric Breakdown Models) combine complex geometry with randomness; this double difficulty is a stumbling block to their elucidation. In this paper we introduce a wide class of fractal growth models with highly complex geometry but without any randomness in their growth rules. The models are defined in terms of deterministic itineraries of iterated conformal maps, automatically generating the global conformal function which maps the exterior of the unit circle to the exterior of a n-particle growing aggregate. The complexity of the evolving interfaces is fully contained in the deterministic dynamics of the conformal map. We focus attention to a class of growth models in which the itinerary is quasiperiodic. Such itineraries can be approached via a series of rational approximants. The analytic power gained is used to introduce a scaling theory of the fractal growth patterns. We explain the mechanism for the fractality of the clusters and identify the exponent that determines the fractal dimension.

cond-mat.stat-mech↗

arXiv1998

Diffusion Limited Aggregation and Iterated Conformal Maps

The creation of fractal clusters by diffusion limited aggregation (DLA) is studied by using iterated stochastic conformal maps following the method proposed recently by Hastings and Levitov. The object of interest is the function $Φ^{(n)}$ which conformally maps the exterior of the unit circle to the exterior of an $n$-particle DLA. The map $Φ^{(n)}$ is obtained from $n$ stochastic iterations of a function $ϕ$ that maps the unit circle to the unit circle with a bump. The scaling properties usually studied in the literature on DLA appear in a new light using this language. The dimension of the cluster is determined by the linear coefficient in the Laurent expansion of $Φ^{(n)}$, which asymptotically becomes a deterministic function of $n$. We find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the Laurent coefficients.

chao-dyn↗