Brownian excursion the Airy function and formula_22 is the confluent hypergeometric function. Janson and Louchard (2007) show that and They also give higher-order expansions in both cases. Janson (2007) gives moments of formula_19 and many other area functionals. In particular, Brownian excursions also arise in connection with queuing problems, railway traffic, and the heights of random rooted binary trees. Brownian excursion In probability theory a Brownian excursion process is a stochastic process that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In
Brownian ratchet for example lifting a weight ("m") against gravity. The energy necessary to do this work apparently would come from the heat bath, without any heat gradient. Were such a machine to work successfully, its operation would violate the second law of thermodynamics, one form of which states: "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." Although at first sight the Brownian ratchet seems to extract useful work from Brownian motion, Feynman demonstrated that if the entire device is at the same temperature, the