Brownian web is what we now call the Brownian web. Formally speaking, it is a collection of one-dimensional coalescing Brownian motions starting from every space-time point in formula_1. The fact that the Brownian web consists of an "uncountable" number of Brownian motions is what makes the construction highly non-trivial. Arratia gave a construction but was unable to prove convergence of coalescing random walks to a limiting object and characterize such a limiting object. Tóth and Werner in their study of the true self-repelling motion obtained many detailed properties of this limiting object and its dual but did not prove convergence of coalescing
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