arXiv subjects

Peter March

Publications and source records attributed to Peter March.

5 recordsLinked to original sources

arXiv2026

A Simple Model of Sea Spray

We propose a stochastic model of sea spray as a shot noise process attached to a homogeneous Poisson process whose intensity measure is the sea spray generation function, that is to say the average number of droplets of a given radius ejected into the atmosphere per unit area per unit time. We assume the impulse response function defining the shot noise is determined by the trajectory of the center of a spherical droplet whose radius may change with time. We also assume the droplet trajectory is vertical and droplets don't interact with one another. Under these assumptions we show the total mass of airborne sea spray droplets is a stationary stochastic process whose mean and covariance can be written explicitly as functions of the physical parameters. We illustrate the model through examples of increasing physical relevance the most general of which includes gravity, Stokes drag forces, and Langmuir's D-squared law of evaporation. A consequence of this particular example is an estimate of the height of the marine boundary layer at sufficiently large wind speeds.

math.PR↗

arXiv2026

Second Order Differential Operators on Graphs

The commutator of a pair of vector fields on a graph is not a vector field in general, but rather a second order differential operator. We investigate this departure from the classical case of vectors fields on a manifold by examining the geometry of balls of radius two, concentrating on the set of paths of length two connecting a given vertex with the center of the ball. There is a natural surjection from the space of sections of the second tangent bundle to the space of second order differential operators whose kernel reflects the geometry of these balls. Using this map we draw several conclusions about second order differential operators including canonical forms, formulas for their adjoints, and a necessary and sufficient condition for a commutator to be a vector field.

math.CO↗

arXiv2024

Helmholtz-Hodge Decomposition on Graphs

We propose a definition of the curl of a vector field X on a finite simple graph as the projection of X onto the orthogonal complement of circulation-free vector fields, where a vector field is circulation-free provided its line integral around every simple circuit vanishes. We justify the definition by observing that X and curl X have the same circulation and curl of the gradient and divergence of the curl vanish. This shows the gradient, curl, and divergence operators form an exact sequence, in analogy with the classical case of vector fields on Euclidean domains and yields the Helmholtz-Hodge decomposition of a vector field on a graph as the sum of a gradient, a curl, and a harmonic field. Along the way, we also prove analogues of the divergence theorem, Green's identities, and Helmholtz's theorem. A consequence of our definition is that the curl is a non-local operator, in sharp contrast to the classical case and existing notions of curl on a graph.

math.DG↗

arXiv2024

Bochner's Identity on Graphs

We prove an identity on a graph analogous to Bochner's identity on a Riemannian manifold. An auxiliary graph called the complete tangent graph intervenes in the term corresponding to Ricci curvature.

math.DG↗

arXiv2017

Convergence of the Freely Rotating Chain to the Kratky-Porod Model of Semi-flexible Polymers

The freely rotating chain is one of the classic discrete models of a polymer in dilute solution. It consists of a broken line of N straight segments of fixed length such that the angle between adjacent segments is constant and the N-1 torsional angles are independent, identically distributed, uniform random variables. We provide a rigorous proof of a folklore result in the chemical physics literature stating that under an appropriate scaling, as N tends to infinity, the freely rotating chain converges to a random curve defined by the property that its derivative with respect to arclength is a brownian motion on the unit sphere. This is the Kratky-Porod model of semi-flexible polymers. We also investigate limits of the model when a stiffness parameter, called the persistence length, tends to zero or infinity. The main idea is to introduce orthogonal frames adapted to the polymer and to express conformational changes in the polymer in terms of stochastic equations for the rotation of these frames.

math.PR↗