## Stochastic & Statistical Physics of Circuits

Noise in electrical circuits is a statistical physical process where a large number of electrons undergo fluctuations. Noise in linear elements is understood in the framework of equilibrium statistical physics (e.g., resistor’s Nyquist noise). In contrast, noise in nonlinear dynamic circuits such as self-sustained oscillators is harder to grasp from the fundamental viewpoint. Part of my research lies in understanding noise processes in nonlinear dynamic circuits in fundamental physical terms by applying the stochastic calculus methods of non-equilibrium statistical physics, in order to attain design insights and fundamental limits.

Self-sustained oscillators, widely used as frequency references, sustain oscillation on limit cycles. Ambient noise perturbs oscillation, causing phase error on the limit cycle, which grows without bound. The phase thus undergoes Brownian motion, diffusing along the limit cycle. This phase diffusion, known as phase noise, broadens oscillator’s spectrum, and is among the most important aspects of oscillator’s dynamics and performance. Phase noise has been extensively studied since 1960s. Nonetheless, one general problem (phase noise in oscillators where energy and noise are spatially distributed) and one fundamental problem (inherent nonlinearity in phase diffusion process) had yet to be understood:

### Phase Noise in Distributed Oscillator

In distributed oscillators such as standing wave oscillators and modelocked pulse oscillators, the oscillating signal is an EM wave in a medium like a transmission line. Their phase noise had not been well understood, as an infinite number of voltage/current variables representing the EM wave are distributed along the medium, and are perturbed by distributed noise sources. I developed a time-domain analysis that can deal with this general situation. It expresses phase diffusion rate in terms of the EM wave’s spatiotemporal shape and noise intensity, and offers a physically meaningful result. In standing wave oscillators with a monotone sinusoidal wave, a thermodynamics argument based on energy equipartition theorem produces a sensible result. In modelocked pulse oscillators where many inter-locked harmonics coexist, the effect of the EM pulse’s shape overwhelms energy equipartition, rendering a shorter pulse with less phase noise. Confirmed by measurements, the work offers a calculation method, physical understanding, and design insights for phase noise in distributed oscillators.

### Phase Diffusion & Lamb-Shift-Like Spectrum Shift

The second problem concerns the fundamental phase noise model. Phase diffusion is governed by an inherently nonlinear stochastic equation. An approximate linear model has been used, and the nonlinearity’s effect has been unknown. By developing a perturbation method suited to the problem, I obtained the first analytical solution to the nonlinear stochastic equation, clarifying when and how the linear approximation fails, and revealing physical consequences of the nonlinearity: reduced phase noise and blue shift of oscillation frequency. Not only are these results helpful for circuits and lasers, but they can be useful in evaluating the timing accuracy of biological oscillators such as neural circuits. Moreover, the frequency blue shift is very surprising, as noise perturbs phase in plus and minus directions with no bias. My further study revealed that this frequency shift in classical oscillator due to interplay between noise fluctuations and nonlinearity is analogous to the Lamb shift in quantum electrodynamics, where an atomic energy level is shifted via interplay between vacuum fluctuations and nonlinear dipole coupling. This is fundamentally interesting, showcasing the beautiful formal connection between stochastic classical systems and quantum systems.