Charging capacitors using diodes at different temperatures. I Theor

Nonlinear elements in a rectifying circuit can be used to harvest energy from thermal fluctuations either steadily or transitorily. We study an energy harvesting system comprising a small variable capacitor (e.g., free standing graphene) wired to two diodes and two storage capacitors that may be kept at different temperatures (or at a single one) and use two current loops. The system reaches very rapidly a quasi stationary state with constant overall charge while the difference of the charges at the storage capacitors evolves much more slowly to its stationary value. In this paper, we extract an exponentially small factor out of the solution of the Fokker-Planck equation and use a Chapman-Enskog procedure to describe the long evolution of the marginal probability density for the charge difference, from the quasi stationary state to the final stationary state (thermal equilibrium for equal temperatures). The second paper of this series shows that the results of the perturbation procedure compare well with direct numerical simulations. For a specific form of the diodes’ nonlinear mobilities, we can approximate the quasi stationary state by Gaussian functions and further study the evolution of the marginal probability density. The latter adopts the shape of a slowly expanding pulse (comprising left and right moving wave fronts whose fore edges become sharper as time elapses) in the space of charge differences that leaves the final stationary state behind it.

💡 Research Summary

The paper presents a comprehensive theoretical analysis of an energy‑harvesting circuit that exploits thermal fluctuations through nonlinear rectification. The system consists of a freely suspended graphene sheet acting as a variable capacitor, two diodes with nonlinear current‑voltage characteristics, and two storage capacitors that may be held at different temperatures T₁ and T₂. Fluctuations of the graphene change the electrode separation, thereby modulating the capacitance C₀ = εA/d(t) and generating a displacement current. This current is split by the diodes, each described by a sigmoid conductance µ(u)=1/(1+e^{‑u/w}), and charges the storage capacitors C₁ and C₂.

The joint probability density ρ(q₁,q₂,t) of the charges on the storage capacitors obeys a two‑dimensional Fokker‑Planck equation (FPE). The drift term contains the product of the diode conductance and the electrostatic force, while the diffusion term is proportional to the local temperature, reflecting the thermal noise. The FPE captures the essential non‑equilibrium physics arising from the temperature difference and the diode nonlinearity.

To expose the multiple time scales, the authors introduce nondimensional variables: ξ = (q₁‑q₂)/(C₀V₀) and η = (q₁+q₂)/(C₀V₀(1+ε)), where ε = C₀/(2C₁) ≪ 1 and w ≪ 1. In this scaling, ξ represents a fast mode with variance of order ε, while η is a slow mode of order unity. In the limit ε → 0 the equilibrium density reduces to a Gaussian sharply peaked at ξ = 0, i.e., ρ_eq ∝ exp