Electrical autonomous Brownian gyrator

We study experimentally and theoretically the steady-state dynamics of a simple stochastic electronic system featuring two resistor-capacitor circuits coupled by a third capacitor. The resistors are subject to thermal noises at real temperatures. The voltage fluctuation across each resistor can be compared to a one-dimensional Brownian motion. However, the collective dynamical behavior, when the resistors are subject to distinct thermal baths, is identical to that of a Brownian gyrator, as first proposed by Filliger and Reimann [Phys. Rev. Lett. 99, 230602 (2007)]. The average gyrating dynamics is originated from the absence of detailed balance due to unequal thermal baths. We look into the details of this stochastic gyrating dynamics, its dependences on the temperature difference and coupling strength, and the mechanism of heat transfer through this simple electronic circuit. Our work affirms the general principle and the possibility of a Brownian ratchet working near room temperature scale.

Figure 1

Electrical autonomous Brownian gyrator. (a) Schematics of the experimental system, featuring a capacitively coupled RC circuit agitated by two heat baths. (b) A snapshot of concurrent $V_1\left(t\right)$ and $V_2\left(t\right)$ over 30 ms with $C_c=1.0\mathrm{nF}$ and $T_1=120\mathrm{K}$. (c) A virtual particle evolving in the 2D phase space formed by $V_1$ and $V_2$. A small segment of its trajectory [corresponding to the data in (b)] is shown by the orange line. The dashed lines indicate the $q_1$ and $q_2$ axes; see text. The virtual particle is influenced by two heat baths and experiences two random noises ${\xi }_1$ and ${\xi }_2$ from directions parallel to $q_1$ and $q_2$ axes, respectively, as noises are depicted by two sets of wavy arrows. The tilt ellipses designate potential contours with a minimum in the origin.

Figure 2

Behavior of Brownian gyrator. The figures present the main results of this work. The value of $C_c=1.0\mathrm{nF}$ is used here. White contour: the equipotential contours of the coupled RC circuit. Color map: the steady-state distribution $P_{\mathrm{ss}}\left(\vec{V}\right)$. Vector field: the probability flux, ${\vec{J}}_{\mathrm{ss}}\left(\vec{V}\right)$. The inset color map shows the curl of the probability flux, $\mathbf{\nabla }\times {\vec{J}}_{\mathrm{ss}}$. The experimental results are listed in (a) and (b). (a) Equilibrium case ($T_1=T_2=296$ K). The contour lines in $P_{\mathrm{ss}}$ and $U$ mutually agree, and ${\vec{J}}_{\mathrm{ss}}$ hardly exhibits any flowing trend as the detailed balance is valid. (b) NESS case ($T_1=120$ K). Contour lines of $P_{\mathrm{ss}}$ are tilted with respect to those in $U$, and ${\vec{J}}_{\mathrm{ss}}$ reveals a circulating trend about the origin. (c) Theoretical counterparts of (b).

Figure 3

Energy flow in a semiadiabatic cycle. (a) A closed cycle in the phase space formed by four chosen semiadiabatic paths (I, II, III, IV) about the origin. The red (blue) arrows indicate the processes adiabatic to $T_1$ ($T_2$), while the system is subject to energy exchanges with the $T_2$ ($T_1$) heat bath only. (b) Major current flows (black solid arrows) and energy flows (gray dashed arrows) in the circuit for the four processes. (c) An illustrated example of paving a closed cycle by semiadiabatic processes.

Figure 4

Rotation speed of gyrator. (a) The measured time evolution in the angle of the virtual Brownian particle position $\varphi \left(t\right)$ for various $T_1$ ($C_c=1.0\mathrm{nF}$ being used). The inset provides a zoom-in detail for fluctuating motions. (b) $\langle \dot{\varphi }\rangle$ vs $\mathrm{\Delta }T$ evaluated from the average slope of $\varphi \left(t\right)$ (solid square) and the average rotating speed derived from ${\vec{v}}_{\mathrm{flow}}$ (open circle) ($C_c=1.0\mathrm{nF}$ being used). Theoretical result is provided as the dashed curve. (c) $\langle \dot{\varphi }\rangle$ vs $C_c$ for $T_1=120\mathrm{K}$. Symbols show the results from measurement while the dashed curve gives the theoretical prediction. (d) The averaged leading angle $\langle \alpha \rangle$, defined by the advanced phase of oscillation in $V_1$ over that in $V_2$, for a virtual particle circulating along the flux field $[T_1=120\mathrm{K}$; the same symbols as in (c)].