Discrete--time ratchets, the Fokker--Planck equation and Parrondo's paradox

Parrondo's games manifest the apparent paradox where losing strategies can be combined to win and have generated significant multidisciplinary interest in the literature. Here we review two recent approaches, based on the Fokker-Planck equation, that rigorously establish the connection between Parrondo's games and a physical model known as the flashing Brownian ratchet. This gives rise to a new set of Parrondo's games, of which the original games are a special case. For the first time, we perform a complete analysis of the new games via a discrete-time Markov chain (DTMC) analysis, producing winning rate equations and an exploration of the parameter space where the paradoxical behaviour occurs.Comment: 17 pages, 5 figure

Red, Blue and Green Electrons

: If the Fokker{Planck equation for the phase space density of electrons in a storage ring is given, the corresponding equation for the polarization density has a related, simple and elegant form which can be deduced by a simple illuminating argument. 1 Introduction If the phase space density W orb (~u; s) of electrons in a storage ring evolves according to a Fokker{ Planck equation: @W orb @s = L FP;orb W orb ; (1) where s is the distance around the ring, ~u is the vector of the six canonical phase space coordinates and L FP;orb is the Fokker{Planck operator for the orbital motion, how can we write a corresponding equation for the transport of spin? The solution is to work with the polarization density ~ P(~u; s) which is dened as 2=h ~ S where ~ S is the phase space density per particle of the spin angular momentum. It can then be shown [1] that ~ P(~u; s) satises the equation: @ ~ P @s = L FP;orb ~ P + ~ (~u; s) ~ P ; (2) where ~ ~u; s) is the spin p..