Charging capacitors using diodes at different temperatures. I. Theory

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., freestanding 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 quasistationary 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 quasistationary state to the final stationary state (thermal equilibrium for equal temperatures). The following paper [J. M. Mangum et al., Phys. Rev. E 113, 014124 (2026)10.1103/9ftf-fs52] 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 quasistationary 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.