Stochastic Theory of Environmental Effects in Nonlinear Electrical Circuits

Abstract

We present a stochastic approach to calculate the full statistics of classical voltage fluctuations across an arbitrary, nonlinear, dissipative device embedded in a circuit in the presence of a bias. We show how the feedback resulting from the circuit, made of an ohmic resistor and a capacitor, affects the cumulants of the voltage, and in particular resolves Brillouin's paradox to satisfy thermodynamics. We apply our results to the case of a tunnel junction and a diode.

Figures (4)

• Figure 1: Schematics of the circuit.
• Figure 2: Rescaled I-V characteristics of a tunnel junction. The full black line corresponds to the ideal I-V characteristic $\langle I\rangle=G\langle U\rangle$, the dashed black line to the fully blockaded junction and the colored lines correspond different values of $RG$.
• Figure 3: Rescaled variance of voltage fluctuations in a tunnel junction, $\langle\!\langle U^2\rangle\!\rangle/\Delta^2$ as a function of $u_0/\Delta$ (dashed line) and $\langle U\rangle /\Delta$ (solid line) for $RG=5$ and $k_{\mathrm{B}} T=1.1e\Delta/3$. The value at equilibrium (zero bias voltage) corresponds to $\langle\!\langle U^2\rangle\!\rangle=k_{\mathrm{B}} T/C$ for both curves.
• Figure 4: Rescaled environmental corrections for the DC voltage in a diode for various values of $RG$ at room temperature. For large positive values of the bias $V$, the corrections to the average voltage approach $\Delta/2$.