Generalised energy equipartition in electrical circuits

TL;DR

This paper extends the energy equipartition theorem to electrical circuits with thermal noise, linking quantum and classical regimes and revealing a superstatistical structure in the energy distribution.

Contribution

It introduces a generalized energy equipartition theorem for resistive circuits with inductance and capacitance, incorporating quantum effects and superstatistics.

Findings

Energy distribution modulated by admittance's real part

Superstatistical structure in energy contributions

Classical limit recovered as Planck's constant approaches zero

Abstract

In this brief note, we demonstrate a generalised energy equipartition theorem for a generic electrical circuit with Johnson-Nyquist (thermal) noise. From quantum mechanical considerations, the thermal modes have an energy distribution dictated by Planck's law. For a resistive circuit with some inductance, it is shown that the real part of the admittance is proportional to a probability distribution function which modulates the contributions to the system's mean energy from various frequencies of the Fourier spectrum. Further, we analyse the case with a capacitor connected in series with an inductor and a resistor. The results resemble superstatistics, i.e. a superposition of two statistics and can be reformulated in the energy representation. The correct classical limit is obtained as $\hbar \rightarrow 0$.