Stochastic theory of nonlinear electrical circuits in thermal equilibrium

TL;DR

This paper develops a comprehensive stochastic theory for nonlinear RLC circuits in thermal equilibrium, extending Hamiltonian mechanics to include dissipation and noise, with potential applications in quantum circuit analysis.

Contribution

It generalizes Hamiltonian mechanics to nonlinear, dissipative circuits and derives exact stochastic fluctuation expressions, bridging classical and quantum circuit theories.

Findings

Exact Johnson noise expressions for nonlinear resistors.

Formalism describes thermal fluctuations in arbitrary topology circuits.

Potential extension to quantum circuits for decoherence studies.

Abstract

We revisit the theory of dissipative mechanics in RLC circuits, allowing for circuit elements to have nonlinear constitutive relations, and for the circuit to have arbitrary topology. We systematically generalize the dissipationless Hamiltonian mechanics of an LC circuit to account for resistors and incorporate the physical postulate that the resulting RLC circuit thermalizes with its environment at a constant positive temperature. Our theory explains stochastic fluctuations, or Johnson noise, which are mandated by the fluctuation-dissipation theorem. Assuming Gaussian Markovian noise, we obtain exact expressions for multiplicative Johnson noise through nonlinear resistors in circuits with convenient (parasitic) capacitors and/or inductors. With linear resistors, our formalism is describable using a Kubo-Martin-Schwinger-invariant Lagrangian formalism for dissipative thermal systems. Generalizing our technique to quantum circuits could lead to an alternative way to study decoherence in nonlinear superconducting circuits without the Caldeira-Leggett formalism.