Gaussian white noise as a resource for work extraction

We show that uncorrelated Gaussian noise can drive a system out of equilibrium and can serve as a resource from which work can be extracted. We consider an overdamped particle in a periodic potential with an internal degree of freedom and a state-dependent friction, coupled to an equilibrium bath. Applying additional Gaussian white noise drives the system into a nonequilibrium steady state and causes a finite current if the potential is spatially asymmetric. The model thus operates as a Brownian ratchet, whose current we calculate explicitly in three complementary limits. Since the particle current is driven solely by additive Gaussian white noise, this shows that the latter can potentially perform work against an external load. By comparing the extracted power to the energy injection due to the noise, we discuss the efficiency of such a ratchet.

Figure 1

The first-order contribution to the dimensionless current $J_z^{\left(1\right)}$ for slow switching, Eq. (14) as a function of the noise intensity $D$ (a), potential depth $U_0$ (b), friction coefficient ${\gamma }_2$ (c) and asymmetry parameter $x_0$ (d). The respective remaining parameters are as follows: $D=\gamma T,{\gamma }_2=2\gamma ,{\gamma }_1=\gamma ,U_0=2T,x_0=0.2L$. Note that for $D\to 0$, the current approaches 0 as $D^2$.

Figure 2

The first-order contribution to the dimensionless current $J_z^{\left(1\right)}$ for fast switching, Eq. (25) as a function of the noise intensity $D$ (a), potential depth $U_0$ (b), friction coefficient ${\gamma }_2$ (c), and asymmetry parameter $x_0$ (d). The respective remaining parameters are as follows: $D=\gamma T,{\gamma }_2=2\gamma ,{\gamma }_1=\gamma ,U_0=2T,x_0=0.2L$.

Figure 3

The crossover value ${\rho }_c$ Eq. (27) between slow and fast switching as a function of the noise intensity $D$ (a), potential depth $U_0$ (b), friction coefficient ${\gamma }_2$ (c), and asymmetry parameter $x_0$ (d). The respective remaining parameters are as follows: $D=\gamma T,{\gamma }_2=2\gamma ,{\gamma }_1=\gamma ,U_0=2T,x_0=0.2L$.

Figure 4

The first-order correction to the current for a potential $U\left(x\right)=U_0\{sin(2\pi x/L)+\delta [sin(4\pi x/L)/2+sin(6\pi x/L)/3]\}$, for slow (a) and fast (b) switching, shown as a function of the asymmetry parameter $\delta$. The remaining parameters are $D=\gamma T,{\gamma }_2=2\gamma ,{\gamma }_1=\gamma ,U_0=2T$. Apart from being overall much larger, the fast switching current is also negative for $\delta \lesssim -1$, while the current for slow switching is positive. This implies a current reversal at some intermediate switching rate.

Figure 5

The dimensionless current computed from the small $D$ (black), small $\rho$ [blue, Eq. (A1)] and large $\rho$ [orange, Eq. (26)] expansions as a function of $\rho$ for $D=0.05\gamma T$. The remaining parameters are as follows: ${\gamma }_2=2\gamma ,{\gamma }_1=\gamma ,U_0=2T,x_0=0.2L$. The result from the small $D$ expansion agrees very well with the slow and fast switching results in the respective limits. In between, as expected, it exhibits a maximum, whose position is approximately described by the crossover value ${\rho }_c\simeq 31$ (dashed line).

Figure 6

Efficiency Eq. (32) as a function of the applied load force $F_0$ for a sawtooth potential, obtained from Langevin simulations. As discussed in the text, the efficiency reaches a maximum at half the stall force. For large applied force, the particle is dragged by the external force and the efficiency becomes negative. Here the parameters are $U_0=3, T=0.05, D=1.5, {\gamma }_1=1, {\gamma }_2=2.5, L=1$, and $x_0=0.1$. The points are the average over five simulation runs of 1000 trajectories each, and the error bars are the standard deviation between the individual runs.