Charging capacitors using diodes at different temperatures. I Theor

TL;DR

This paper models energy harvesting using nonlinear diodes and capacitors at different temperatures, analyzing the system's long-term charge distribution evolution through perturbation methods and numerical simulations.

Contribution

It introduces a Chapman-Enskog approach to describe the slow evolution of charge difference probability density in a thermal diode system.

Findings

The system rapidly reaches a quasi stationary state with constant total charge.

The charge difference evolves slowly, forming a pulse that sharpens over time.

Perturbation results agree well with numerical simulations.

Abstract

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.