The flashing Brownian ratchet and Parrondo’s paradox

Abstract

A Brownian ratchet is a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and biologists for nearly 25 years. The flashing Brownian ratchet is the process that motivated Parrondo’s paradox, in which two fair games of chance, when alternated, produce a winning game. Parrondo’s games are relatively simple, being discrete in time and space. The flashing Brownian ratchet is rather more complicated. We show how one can study the latter process numerically using a random walk approximation.

The Tilted Flashing Brownian Ratchet

“…A nearly identical figure appeared in Harmer and Abbott [3].) flashing Brownian ratchet [4], using a random walk approximation. We will see, in particular, that panels (d)-(f ) of the conceptual Figure 1, although again largely accurate, can be improved as well.…”

“…The result is directed motion, as shown in panels (a)-(c) of Figure 1 (from Harmer et al [2]). This conceptual figure, specifically panels (a)-(c), although largely accurate, can be improved, as demonstrated by Ethier and Lee [4] using a random walk approximation.…”

“…As shown in [4], game B is fair (asymptotically), whereas the random mixture A slightly stronger form of the Parrondo effect occurs when two losing games combine to win. That can be achieved by replacing the cumulative-profit process of game A by the simple random walk on Z with…”

“…Other sources with equations similar to (4) Given time parameters τ 1 , τ 2 > 0, the tilted flashing Brownian ratchet is a time-inhomogeneous one-dimensional diffusion process that evolves as a Brownian motion with drift −κ (i.e., B t − κt) on [0, τ 1 ] (potential "off"), then as the tilted Brownian ratchet (3) on [τ 1 , τ 1 + τ 2 ] (potential "on"), and so on, alternating between these two regimes. Such a process Y t is governed by the SDE…”

Paradoxical Survival: Examining the Parrondo Effect across Biology

“…We first give an overview of the origins and fundamental principles of the paradox, in a largely nontechnical perspective suitable for supporting the subsequent discussions. The paradox was first conceptualized as an abstraction of the phenomenon of flashing Brownian ratchets, [2][3][4][5][6][7][8][9] wherein diffusive particles exhibit unexpected drift when exposed to alternating periodic potentials. It has since been applied across a multitude of neighboring disciplines in the physical sciences and engineering-related fields, [10,11] such as diffusive and granular flow dynamics, [12][13][14] information thermodynamics, [15][16][17][18] chaos theory, [19][20][21][22][23][24][25] switching problems, [26][27][28] and quantum phenomena; [29][30][31][32][33][34][35][36][37][38] but a plethora of exciting applications has also been found in biology, which this paper focuses upon.…”