Recycled noise rectification: An automated Maxwell’s daemon

The one-dimensional motion of a massless Brownian particle on a symmetric periodic substrate can be rectified by reinjecting its driving noise through a realistic recycling procedure. If the recycled noise is multiplicatively coupled to the substrate, the ensuing feedback system works like a passive Maxwell’s daemon, capable of inducing a net current that depends on both the delay and the autocorrelation times of the noise signals. Extensive numerical simulations show that the underlying rectification mechanism is a resonant nonlinear effect: The observed currents can be optimized for an appropriate choice of the recycling parameters with immediate application to the design of nanodevices for particle transport.

Figure 1

(Color online) Effective action of an external field of force on a Brownian particle in a narrow channel: (a) The external field splits into a direct, $E_1\left(t\right)$, and a transmitted component, $E_2\left(t\right)$, that couple to the particle with relative delay ${\tau }_c$; (b) the field couples to the particle both directly, $E_1\left(t\right)$, and modifying its environment through the local field $E_2\left(t\right)$. In both cases $E_2\left(t\right)$ modulates the channel geometry (multiplicative coupling).

Figure 2

(Color online) Autocorrelation function $C\left(t\right)$ of the process (2.1) for $Q_1=Q_2=2$ and $a=2$; the recycled noise ${\xi }_2$ is added to (filled circles) or subtracted (empty circles) from the primary noise ${\xi }_1$ after a delay time ${\tau }_d=1$. The solid curves represent the corresponding analytical predictions (2.4). Inset: $C\left(0\right)=⟨x^2⟩$ vs ${\tau }_d$; notation and the remaining simulation parameters are as in the main panel.

Figure 3

(Color online) (a), (b) Activation process in the bistable potential $V\left(x\right)=-a{x}^2∕2+b{x}^4∕4$ driven by ${\xi }_1\left(t\right)\pm {\xi }_2\left(t\right)$: $T_K^{+}$ (filled symbols) and $T_K^{-}$ (empty symbols) vs ${\tau }_d$ for ${\tau }_c=0$ and (a) $Q_1=Q_2=0.5$ (squares), 0.2 (circles); (b) $Q_1=Q_2=0.025$. (c) Probability density $P\left(x\right)$ for ${\xi }_1\left(t\right)\pm {\xi }_2\left(t\right)$ with ${\tau }_c=0$, $Q_1=Q_2=0.025$, and ${\tau }_d=0$ (circles), 0.3 (squares), and 100 (triangles). Solid lines represent the relevant best fits for $P\left(x\right)=\mathcal{N}{e}^{-V\left(x\right)∕D_{+}}$; a large discrepancy is apparent for ${\tau }_d=0.3$.

Figure 4

(Color online) Characteristic curve $v\left({\tau }_d\right)\equiv \mid ⟨\dot{x}\left({\tau }_d\right)⟩\mid$ for the process (3.2) with $a=1$, $D\equiv 0$, ${\tau }_c=0$, and different noise intensities $(Q_1,Q_2)$: (a) high noise levels, resonant tails; (b) low noise level, exponential tails versus fitting law (3.3) (dashed lines). In both panels $⟨\dot{x}\left({\tau }_d\right)⟩$ is negative.

Figure 5

(Color online) Characteristics curve $v\left({\tau }_d\right)$ for the process (3.2) with $a=1$, $D\equiv 0$, $Q_1=Q_2=1$, and different autocorrealtion times ${\tau }_c$.

Figure 6

(Color online) (a) Characteristics curve $v\left({\tau }_d\right)$ for the process (3.2) with $a=1$, $Q_1=Q_2=1$, ${\tau }_c=0$, and different $D$. (b) $v$ vs $Q_2$ with $Q_1=1$ (empty symbols) and vs $Q_1$ with $Q_2=1$ (filled symbols) for different values of $({\tau }_d,{\tau }_c)$: squares (0.2, 0.2), circles (0.0,0.2), and triangles (0.2, 2.0). The remaining simulation parameters are $D\equiv 0$ and $a=1$.