Non-linear Nyquist theorem: A conjecture

TL;DR

This paper proposes a conjecture for a nonlinear Nyquist theorem, extending the classical relation between fluctuations and resistance to nonlinear systems, supported by specific system tests and with potential practical applications.

Contribution

It introduces an explicit conjectured formula linking equilibrium fluctuations to nonlinear admittance, extending Nyquist's theorem beyond linear systems.

Findings

Conjectured explicit formula for nonlinear Nyquist relation.

Supported by tests in specific nonlinear systems.

Potential applications in nonlinear electronic device analysis.

Abstract

Thermodynamics of equilibrium states is well established. However, in nonequilibrium few general results are known. One prime and important example is that of Nyquist theorem. It relates equilibrium tiny voltage fluctuations across a conductor with its resistance. In linear systems it was proved in its generality in a beautiful piece of work by Callen and Welton (in 1950s\cite{cw}). However Callen-Welton's formalism has not been extended to nonlinear systems up to now, although alternative methods exist (like Kubo's approach) that leads to formal and implicit expressions {\it at nonlinear order} with no practical consequence. Here--using a brute-force method--we conjecture "a non-linear Nyquist theorem". This is an explicit formula much like Nyquist's original one. Our conjecture is based upon tests of the conjectured explicit formula in specific systems. We conjecture that higher moments of {\it equilibrium} fluctuations bear a relation to {\it nonlinear} admittance very similar to Nyquist's relation. Thus one can easily compute nonlinear admittance from the character of {\it equilibrium} fluctuations. Our relation will have great practical applicability, for example for electronic devices that operate under nonlinear response.