We develop a stochastic simplex framework for width partitioning in braided river cross sections, modeling normalized channel widths as random compositional vectors. Two complementary models are introduced: the symmetric Dirichlet distribution, which models width allocation as exchangeable stochastic fluctuations around a uniform baseline with no preferred channel, and the isotropic logistic-normal distribution, which accommodates systematic width concentration through a single scale parameter. Both models carry independent physical content: the Dirichlet characterizes morphodynamic states in which all channels are statistically equivalent, while the logistic-normal identifies and quantifies organized concentration regimes. We derive exact closed-form expressions for the expected concentration index and width concentration fraction under both models, establish a parameter-free structural breakdown criterion, and analyze the temporal evolution of model parameters across 11 annual surveys of the Brahmaputra-Jamuna River (Bangladesh) spanning 1976–2007. The analysis reveals a decadal-scale morphodynamic transition: for 1976–1985, width is distributed among channels in a statistically uniform fashion consistent with the Dirichlet model. From 1995 onwards, a progressive increase in the logistic-normal scale parameter σ * , optimized independently for each survey year and fitted by a logistic regression ( R 2 = 0.997 ), indicates systematic growth of channel width concentration. A parameter-free structural breakdown criterion certifies the failure of the Dirichlet model between 1995 and 1999, where the logistic-normal model recovers goodness-of-fit values above 0.95. Both σ * and the Dirichlet parameter c * retain memory of the concentration phase above pretransition levels throughout the observational window.
We solve exactly the transient two-dimensional advection diffusion problem generated by one and two semi-infinite internal Dirichlet lines, representing slender dendrites advancing into a supersaturated liquid while fixing the solute concentration on their surfaces. For a single dendrite, the mixed boundary value problem is reduced to a Wiener-Hopf equation, factorized explicitly, and inverted to the time domain by the Cagniard-de Hoop method, yielding the single-layer density in closed form in terms of the complex error function. For two parallel dendrites separated by a lateral distance h , the interaction enters through the transcendental kernel ( 1 + e − κ h ) − 1 , whose lower Wiener-Hopf factor is represented by a logarithmic Cauchy integral, and the resulting finite distance feeding field is reduced to a regularized oscillatory integral. The exact two-dendrite solution reveals a structural obstruction to purely local growth laws: the leading inverse square-root tip singularity is universal, with identical amplitudes for both dendrites independently of spacing, even though the neighbor modifies the feeding field at every finite distance behind the tip. Competition therefore cannot be encoded in any local tip amplitude alone. This obstruction forces a nonlocal coupling variable, which we identify as a finite tip zone functional that retains the subleading near tip structure responsible for competitive depletion. The interaction further resolves into a discrete hierarchy of exponentially screened modes with dominant screening length h / π , which sets the natural scale of competitive suppression. These exact transport results provide the analytical backbone for a reduced theory of dendrite arrest and winner selection, and are validated against independent finite-difference computations.
A solidifying dendrite advancing toward an oblique impermeable wall senses the wall through reflection of its own diffusive depletion field. The strength of this reflection depends on both wall obliquity and the unsteady kinematics of the closing gap, and cannot be captured by any quasistatic frozen offset construction. We solve the moving boundary diffusion problem exactly in the laboratory frame (wall fixed, dendrite advancing at constant speed) by recasting it as a Wiener-Hopf-Hankel system on a pair of spectral branches. The resulting upper triangular operator is inverted sequentially: first as a Fredholm integral equation on the lower branch, then as a scalar Wiener-Hopf problem on the upper branch, reaching a closed form solution via Cagniard-de Hoop. A Padé-Abrahams rational reduction of the spectral radical reduces the entire scheme to a low cost algebraic system at each Laplace node. The exact unsteady solution inverts the classical starvation picture. Measured against an isolated dendrite at the same elapsed time, the wall correction is weakly suppressive while the tip is more than about two diffusion lengths from the wall, where the reflected depletion returns with little retardation and acts as ordinary blockage, but the suppression never exceeds one percent. As the gap closes the correction changes sign and the feeding rises to 31 percent above the isolated reference at grazing incidence, because the reflected field carries a memory of wider past gaps and delivers more solute than the instantaneous geometry would allow. The enhancement is governed by the oblique Péclet number Pe θ = v d 0 sin θ / D and is strongest at grazing incidence, where it exceeds the preceding suppression by a factor of 30.
The aggregation or clustering of proteins plays an important role in the formation of postsynaptic domains (PSDs) at excitatory and inhibitory synapses in neurons. PSDs are rich in scaffolding proteins that can transiently trap transmembrane neurotransmitter receptors, allowing them to regulate the strength of synaptic connections during learning and memory. Recently, a two-dimensional diffusion-mediated aggregation model of PSD formation was developed in which the spatial locations of the clusters are determined by a set of fixed anchoring sites. The system is kept out of equilibrium by the recycling of particles between the cell membrane and interior. This results in a nontrivial stationary state consisting of multiple stable protein clusters. In this paper, we use matched asymptotic methods to reduce the underlying reaction-diffusion model with moving interior boundaries to a corresponding finite-dimensional nonlinear system. The latter couples the cluster radii to the spatially averaged protein concentration in the bulk domain. We assume that the diffusivity D is O ( 1 / ν ) , where ν = − 1 / ln ε and ε is a small parameter that characterizes the size of the clusters relative to the size of the bulk domain. The reduced dynamical system allows us to explore both the existence and stability of the multicluster stationary state while maintaining the effects of diffusion-mediated interactions between clusters.
