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.

Keywords: Brownian motion; Brownian ratchet; Parrondo's paradox; flashing Brownian ratchet; random walk.

Figure 1.

‘Brownian ratchet mechanism. The sawtooth and flat potentials are labeled with U_on and U_off, respectively, while the distribution of Brownian particles is shown via the normal curves. This sequence of flashing between on and off potentials shows there is a net movement of particles to the right.’ (Reprinted from Harmer et al. [, p. 706] with the permission of AIP Publishing.)

Figure 2.

‘The flashing ratchet at work. The figure represents three snapshots of the potential and the density of particles. Initially (upper figure), the potential is on and all the particles are located around one of the minima of the potential. Then the potential is switched off and the particles diffuse freely, as shown in the centred figure, which is a snapshot of the system immediately before the potential is switched on again. Once the potential is connected again, the particles in the darker region move to the right-hand minimum whereas those within the small gray region move to the left. Due to the asymmetry of the potential, the ensemble of particles move, on average, to the right.’ (Reprinted from Parrondo & Dinís [, p. 148] with the permission of Taylor & Francis Ltd.)

Figure 3.

The periodic drift μ with $\alpha =\frac{1}{3}$ and L=3 is plotted on the interval [−6,6]. Each interval [j,j+1) (in black) is replaced by its midpoint $j+\frac{1}{2}$, which we relabel as j (in red) to discretize space. To discretize time as well, we replace the Brownian motion by a simple symmetric random walk on Z, and we replace the Brownian ratchet by an asymmetric random walk on Z whose periodic state-dependent transition probabilities are determined by a discretized version of μ.

Figure 4.

We mathematically model the third panel of figure 1. A Brownian motion, starting at 0, runs for time τ₁=2.4. Then, starting from where the Brownian motion ended, a Brownian ratchet with $\alpha =\frac{1}{4}$, L=4, γ=3λ/8 and λ=1,2,3,4,5 (from top to bottom) runs for time τ₂=2.4. The black curve is an approximation to the density of the flashing Brownian ratchet at time τ₁+τ₂, via the random walk approximation with n=100. The blue curve is the sawtooth potential.

Figure 5.

We mathematically model the full figure 1. Starting from the stationary distribution $\overline{\pi }$ with support [−3,1), a Brownian motion runs for time τ₁=2.4. Then, starting from where the Brownian motion ended, a Brownian ratchet with $\alpha =\frac{1}{4}$, L=4, γ=3λ/8 and λ=5, runs for time τ₂=2.4. The black curves in all three panels are based on the random walk approximation with n=100. The blue curves represent the sawtooth potential. The vertical axes in the first and third panels are comparable, whereas the vertical axis in the second panel has been stretched for clarity.