Analytical Solution to Transport in Brownian Ratchets via the Gambler’s Ruin Model

We present an analogy between the classic gambler’s ruin problem and the thermally activated dynamics in periodic Brownian ratchets. By considering each periodic unit of the ratchet as a site chain, we calculated the transition probabilities and mean first passage time for transitions between energy minima of adjacent units. We consider the specific case of Brownian ratchets driven by Markov dichotomous noise. The explicit solution for the current is derived for any arbitrary temperature, and is verified numerically by Langevin simulations. The conditions for current reversal in the ratchet are obtained and discussed.

Figure 1

Schematic diagram of an $L$-periodic ratchet potential.

Figure 2

Temperature-driven reversal of ratchets current. Close agreement between analytical Monte Carlo prediction and Langevin dynamical simulation. The simulation parameters are $R=0.005$, $L=1.0$, $F=0.6$, $\theta =0.42$, $\gamma =1$, and ${\tau }_c=0.15$, 0.25, 0.5 from top to bottom. Error bars are smaller than the symbol size. Inset: Extracted zero-current curve with respect to $\gamma /{\tau }_c$.

Figure 3

The zero-current surface with respect to parameters $\beta$, $\gamma /{\tau }_c$, and $F$.