Surprises in Brownian Motion: Brownian Donkeys and Parrondo's Paradox

Abstract

While the behavior of equilibrium systems is rather well documented and understood on the basis of general principles, the study of nonequilibrium systems appears to be much more intricate. The two topics presented in this thesis are dealing with nonequilibrium Brownian motion, leading to surprising and counterintuitive behavior. The first part discusses Brownian donkeys. Such systems are characterised by a special response behavior, referred to as absolute negative mobility: when applying an external force, the particles always move in a direction opposite to that of the force. In other words, the particles climb up a potential gradient. The energy needed to do this is extracted by the Brownian particle from the nonequilibrium fluctuations. We present several explicit constructions involving either a single or a collection of Brownian particle(s). The second part deals with the Parrondo paradox. In short, this paradox states that the random or periodic alternation between fair games need no longer be fair. After reviewing the original paradox we introduce the concept of strategy, in which the player(s) can now decide at each turn on the choice of the next game. We present a systematic study of several strategies, including the so-called greedy and optimal one. For the original games, the search for the optimal strategy turns out to be difficult, if not impossible. We introduce a new version of the paradox, the so-called primary Parrondo paradox, whose dynamics involves only superstable fixed points, and for which a full analytical solution for any strategy, together with the identification of the optimal strategy, becomes possible.