Brownian motors: Current fluctuations and rectification efficiency

With this work, we investigate an often neglected aspect of Brownian motor transport, namely, the role of fluctuations of the noise-induced current and its consequences for the efficiency of rectifying noise. In doing so, we consider a Brownian inertial motor that is driven by an unbiased monochromatic, time-periodic force and thermal noise. Typically, we find that the asymptotic, time-, and noise-averaged transport velocities are small, possessing rather broad velocity fluctuations. This implies a corresponding poor performance for the rectification power. However, for tailored profiles of the ratchet potential and appropriate drive parameters, we can identify a drastic enhancement of the rectification efficiency. This regime is marked by persistent, unidirectional motion of the Brownian motor with few back-turns only. The corresponding asymmetric velocity distribution is then rather narrow, with a support that predominantly favors only one sign for the velocity.

Figure 1

(Color online) Fluctuation behavior of an inertial Brownian motor vs the driving strength $a$. (a) Averaged dimensionless velocity $⟨v⟩$ of the inertial Brownian motor in Eq. (3) with rescaled units; (b) variance of the corresponding velocity fluctuations ${\sigma }_v$; (c) fluctuations of the rescaled kinetic energy ${\sigma }_E^2∕{⟨E⟩}^2$; (d) rectification efficiency in Eq. (8). All quantities have been computed for the rescaled potential $V\left(x\right)=-V_0[\sin \left(2\pi x\right)+0.25\sin (4\pi x)]$, where $V_0\simeq 0.454$ normalizes the barrier height to unity, see the inset in (a). The force corresponding to this potential ranges from −2.14 to 4.28. The angular frequencies at the well bottom and at the barrier top, respectively, equal each other, reading 5.28. Bottom panel (e) shows velocity distributions for selected driving amplitudes, i.e., $a=0.68$,1.5,2.14,3.25. All these distributions were normalized by setting their maximum to 1. The remaining rescaled parameters read friction $\mathrm{\gamma }=0.5$, angular driving frequency $\omega =3.6$, and weak thermal noise of strength $D_0=0.01$.

Figure 2

(Color online) Tailoring the shape of the potential. (a) Averaged dimensionless velocity $⟨v⟩$ of the inertial Brownian motor under nonadiabatic driving conditions; (b) corresponding velocity fluctuations ${\sigma }_v$; (c) fluctuations of the rescaled kinetic energy ${\sigma }_E^2∕{⟨E⟩}^2$; (d) efficiency. All quantities are plotted vs the external driving amplitude $a$ for the asymmetric ratchet potential $V\left(x\right)=V_0\left[\sin \left(2\pi x\right)+0.245\sin \left(4\pi x\right)+0.04\sin \left(6\pi x\right)\right]$, where $V_0\simeq 0.461$ normalizes the barrier height to unity [see inset in (a)]. The forces stemming from such a potential range between −4.67 and 1.83. The two angular frequencies at the well bottom and at the barrier top are the same, reading 5.34. The corresponding velocity distributions $P\left(v\right)$ are displayed in panel (e) for the indicated driving amplitudes, i.e., $a=0$,4,6,7. All $P\left(v\right)$ curves have been normalized by setting their maximum to 1. The remaining parameters are $\mathrm{\gamma }=0.9$, $\omega =4.9$, and $D_0=0.01$.

Figure 3

(Color online) Fluctuation behavior of an inertial Brownian motor vs the noise strength $D_0$. (a) Averaged dimensionless velocity $⟨v⟩$; (b) the corresponding velocity variance ${\sigma }_v$; (c) fluctuations of the rescaled kinetic energy ${\sigma }_E^2∕{⟨E⟩}^2$; (d) the corresponding efficiency. Numerical results obtained for the same ratchet potential as in Fig. 1. In the bottom panel (e), the velocity distribution $P\left(v\right)$ is shown for $D_0=0.01$,0.1,2,10. All $P\left(v\right)$ curves are normalized as in Fig. 1. The remaining rescaled parameters are $\mathrm{\gamma }=0.5$, $\omega =3.6$, and $a=0.8$.

Figure 4

(Color online) Tailoring the shape of the potential. (a) Averaged dimensionless velocity $⟨v⟩$ of the inertial Brownian motor under nonadiabatic driving conditions; (b) corresponding velocity fluctuations ${\sigma }_v$; (c) fluctuations of the rescaled kinetic energy ${\sigma }_E^2∕{⟨E⟩}^2$; (d) efficiency. All quantities are plotted vs the driving amplitude $a$ for the asymmetric ratchet potential of Fig. 2. The corresponding velocity distributions $P\left(v\right)$ are shown in panel (e) for selected driving amplitudes, i.e., $a=0$,3.25,3.5,4. 5. All $P\left(v\right)$ curves are normalized as in Fig. 2. The remaining parameters are $\mathrm{\gamma }=0.9$, $\omega =4.9$, and $D_0=0.001$.