Generation of virtual potentials by controlled feedback in electric circuit systems

Electric circuits influenced by thermal noise are analogous to confined Brownian particles and can be an alternative and convenient scheme for studying stochastic thermodynamics. Here we experimentally demonstrate an effective technique of generating tunable potentials for Brownian dynamics in an electric circuit, realized by external controlled feedback. We present two illustrative examples of one-dimensional virtual potentials: static harmonic potential and time-varying double-well potential. The thermal noises of both cases undergo equivalent Brownian dynamics as if they were in the authentic potentials as long as the feedback is fast enough to respond to the designed potentials. The results show that the electric circuit provides a simple, effective, and programmable scheme to study the feedback-controlled virtual potential.

Figure 1

Feedback-controlled virtual potential. (a) Schematic of an analogous particle confined in a designed virtual potential $U_d$. (b) Schematic diagram of the experiment setup to realize feedback-controlled virtual potential. The system mainly consists of an RC circuit circled by a dashed line, and a FPGA board for sampling the position $q$, calculating the required feedback, and producing the feedback voltage $V_f$. $\xi$ represents the thermal noise from the resistor. A voltage preamplifier is used to magnify the voltage signal $V\left(t\right)$ with signal level of the order of microvolts before sampling by FPGA. A low-pass filter is introduced after the preamplifier to prevent the aliasing effect caused by the high-frequency white noise from the preamplifier. (c) The timing diagram of the feedback system. The FPGA samples $q^{\left(n\right)}$ at $t^{\left(n\right)}$ and begins to apply the feedback voltage $V_f^{\left(n\right)}$ after a delay time $t_d$ due to the required signal processing time. $q^{\left(n\right)}$ is sampled every time interval $t_u$, and each $V_f^{\left(n\right)}$ lasts for the same time interval $t_u$.

Figure 2

Virtual harmonic potential. (a) Trajectories of analogous particle with various designed stiffness $k_d$. (b) Probability distribution $P\left(q\right)$ and (c) power spectral density $S_q\left(f\right)$ from the trajectories in (a). The black lines in (c) are given from Eq. (A9). (d) Plot of the variance $\langle q^2\rangle$ as a function of $k_d$. The symbols are the data. The blue solid line indicates $\langle q^2\rangle =k_{B}T/k_d$, the prediction of equipartition theorem for real harmonic potential confinement. The red dashed line is the fitting curve Eq. (A4) from the result of the delay differential equation.

Figure 3

Virtual double-well potential. (a) Trajectories of the analogous particle in double-well potential with fixed $q_m=2.12\mathrm{fC}$ and various $E_b$. (b) Probability distribution $P\left(q\right)$ of the trajectories in (a). (c) Derived potential $U_d\left(q\right)=-k_{B}TlnP\left(q\right)$ from the distribution in (b). The symbols represent the derived potentials, and black lines are the fit to the double-well potential. The designed values and fitting parameters of $q_m$ and $E_b$ are listed in Table 1. (d) Dwelling time ${\tau }_D$ with various $E_b$ in a log-log plot. The black line denotes the Kramers relation ${\tau }_D\propto exp(E_b/k_{B}T)$.

Figure 4

Demonstration of tilting double well and stochastic resonance. (a) PDF $P\left(q\right)$ of time-independent tilting double well for $E_b=3.12k_{B}T$ and $A_t=-0.06$ (blue inverted triangle) and $+0.06$ (red triangle), respectively. (b) Effective potential $U_d\left(q\right)$ from the distributions in (a). The symbols denote the effective potential, and the black lines plot the designed static tilting double-well potential. (c) Trajectory response of the analogous particle in a periodic tilt double-well potential for $A_t=0.2$, $f_t=10\mathrm{Hz}$, and various $E_b$. (d) The map of $1/2f_t\langle {\tau }_D\rangle$ with fixed $A_t=0.2$ for various $E_b$ and $f_t$. The white color-mapping denotes $f_t=1/2\langle {\tau }_D\rangle$, which is the region that stochastic resonance occurs.