Energy pumping in electrical circuits under avalanche noise

We theoretically study energy pumping processes in an electrical circuit with avalanche diodes, where non-Gaussian athermal noise plays a crucial role. We show that a positive amount of energy (work) can be extracted by an external manipulation of the circuit in a cyclic way, even when the system is spatially symmetric. We discuss the properties of the energy pumping process for both quasistatic and finite-time cases, and analytically obtain formulas for the amounts of the work and the power. Our results demonstrate the significance of the non-Gaussianity in energetics of electrical circuits.

Figure 1

Schematic of the electrical circuit with a capacitor with a potential $U(q,\vec{a})$, resistances ($R,R^{\prime },R^{\prime \prime }$), voltages ($V$), and avalanche diodes ($D$). Because of the reverse bias voltages for the avalanche diodes, the intermittent noise appears and affects the charge in the capacitor.

Figure 2

Schematic of the rectangular protocol. We assume $a_0=O\left(1\right)$, $a_1=O\left(1\right)$, $a_1-a_0=O\left(1\right)$, and $b_0=O\left(\epsilon \right)$. We can extract a positive amount of work from the nonequilibrium fluctuation along the clockwise protocol.

Figure 3

Numerical validation of the work formula (9) for the quasistatic processes. From the Monte Carlo simulation, we obtain stochastic trajectories and calculate the ensemble average of the extracted work. We calculate the work with the total time of the operation $\tau =3.0\times {10}^4$ and take its ensemble average with 6600 samples. Here we assume the discretized time step is ${10}^{-2}$. The time scaled protocol for the simulation $[\tilde{a}\left(\tilde{s}\right),\tilde{b}\left(\tilde{s}\right)]\equiv [a\left(\tau \tilde{s}\right),b\left(\tau \tilde{s}\right)]$ is given as follows: $\tilde{a}\left(\tilde{s}\right)=a_1(0\le \tilde{s}\le 1/4),4a_1(1/2-\tilde{s})+2a_0(\tilde{s}-1/4)(1/4\le \tilde{s}\le 1/2),a_0(1/2\le \tilde{s}\le 3/4),4a_1(\tilde{s}-3/4)+4a_0(1-\tilde{s})(3/4\le \tilde{s}\le 1)$ and $\tilde{b}\left(\tilde{s}\right)=4b_0(1/4-\tilde{s})(0\le \tilde{s}\le 1/4),0(1/4\le \tilde{s}\le 1/2),4b_0(\tilde{s}-1/2)(1/2\le \tilde{s}\le 3/4),b_0(3/4\le \tilde{s}\le 1)$.

Figure 4

Scaled optimal rectangular protocol (21) and (22) on the condition of $\tilde{a}\left(0\right)=\tilde{a}(3/4)=\tilde{a}\left(1\right)=a_0$, $\tilde{a}(1/4)=\tilde{a}(1/2)=a_1$, $\tilde{b}\left(0\right)=\tilde{b}(1/4)=\tilde{b}\left(1\right)=b_0$, and $\tilde{b}(1/2)=\tilde{b}(3/4)=0$.

Figure 5

Numerical demonstration of the validity of the power formula (23). On the basis of the method of Monte Carlo, we numerically obtain trajectories with the fourth Runge Kutta method and take the ensemble average of the extracted power with the discretized time step as $\Delta t=0.005$. The ensemble number depends on the parameter $I$. For example, the ensemble number is approximately equal to $1.14\times {10}^7$ for $I=0.7$.

Figure 6

Schematics of the potential manipulation by inserting a medium with nonlinear permittivity. The medium with nonlinear permittivity ${\varepsilon }_N\left(q\right)$ is inserted as shown in this figure to control the potential of the capacitor.