The convex hull for a random acceleration process in two dimensions

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 is non-Markovian. Physically, our results are relevant in describing the average shape of a semi-flexible ideal polymer chain in two dimensions.