The convex hull for a random acceleration process in two dimensions

TL;DR

This paper provides exact formulas for the average perimeter and area of the convex hull of a two-dimensional random acceleration process, linking geometric properties to extreme value statistics of the process.

Contribution

It introduces an exact mapping that relates convex hull properties to extreme value statistics for a non-Markovian stochastic process.

Findings

Exact expressions for mean perimeter and area of convex hulls

Mapping relates convex hull properties to extreme value statistics

Results applicable to modeling semi-flexible polymers

Abstract

We compute exactly the mean perimeter <L(T)> and the mean area <A(T)> of the convex hull of a random acceleration process of duration T in two dimensions. We use an exact mapping that relates, via Cauchy's formulae, the computation of the perimeter and the area of the convex hull of an arbitrary two dimensional stochastic process [x(t); y(t)] to the computation of the extreme value statistics of the associated one dimensional component process x(t). The latter can be computed exactly for the one dimensional random acceleration process even though the process in non-Markovian. Physically, our results are relevant in describing theaverage shape of a semi-flexible ideal polymer chain in two dimensions.