High-temperature ratchets with sawtooth potentials

The concept of the effective potential is suggested as an efficient instrument to get a uniform analytical description of stochastic high-temperature on-off flashing and rocking ratchets. The analytical representation for the average particle velocity, obtained within this technique, allows description of ratchets with sharp potentials (and potentials with jumps in particular). For sawtooth potentials, the explicit analytical expressions for the average velocity of on-off flashing and rocking ratchets valid for arbitrary frequencies of potential energy fluctuations are derived; the difference in their high-frequency asymptotics is explored for the smooth and cusped profiles, and profiles with jumps. The origin of the difference as well as the appearance of the jump behavior in ratchet characteristics are interpreted in terms of self-similar universal solutions which give the continuous description of the effect. It is shown how the jump behavior in motor characteristics arises from the competition between the characteristic times of the system.

Figure 1

The coordinate dependences of the first derivative of the effective potential, Eq. (19), corresponding to the sawtooth potential (18) at $l=0.2L$, for different values of dimensionless frequency parameter $z=L/\sqrt{D\tau }$ (the main plot) and for $z\to 0$ [the uniform normalized representation, Eq. (20), depicted in the inset].

Figure 2

The frequency dependences of the quantities ${\mathrm{\Phi }}_f$ (the solid curves) and ${\mathrm{\Phi }}_r$ (the dotted curves) which determine the average velocities of flashing and rocking ratchets, Eq. (21), with a sawtooth potential relief for several values of the parameter $\xi$: $\xi =0$ (squares), $\xi =0.1$ (rhombs), and $\xi =0.3$ (triangles). The frequency dependence of the normalized average velocity for flashing ratchets is in the inset. The curves, from top to bottom, are in the order of increasing the parameter $\xi$ from $\xi =0$ (the curve with markers) to $\xi =0.001$, 0.005, 0.01, and 0.03.

Figure 3

The behavior of self-similar functions (29) (the inset) and the demonstration of the onset of jump behavior (the main plot): the $\xi$ dependences of the quantities ${\mathrm{\Phi }}_f$ and ${\mathrm{\Phi }}_r$ (the solid and dotted curves, respectively) for different values of the parameter $z$: $z=10$ (squares), $z=100$ (triangles).