Due to the pandemic, the summer of 2019 was regrettably and unexpectedly my last Wooster summer research program, but the team was amazing. Niklas Manz and I obtained Sherman-Fairchild funding to work with Margaret McGuire ’20, Yang (Fish) Yu ’21, and Chase Fuller ’19 to computationally study reaction-diffusion phenomena.  All three of their projects have now been published, with them as lead authors. Fish was lead on Disruption and Recovery, published in Physica A; Margaret was lead on Geographic Tongue, published in Chaos; and Chase was lead on Diffusion Diodes, published today in the International Journal of Unconventional Computing.

With important later contributions by co-author Daniel Cohen-Cobos ’23, Chase used the Tyson-Fife model of the Belousov-Zhabotinsky reaction to numerically investigate the propagation of reaction-diffusion waves through narrow, quasi-one-dimensional channels. He created “soft” obstacles where the inhibitor’s diffusion was larger than the activator’s diffusion, so the system exhibited unidirectional or one-way propagation – a diffusion diode. Furthermore, he discovered a nonlinear compensation relationship between higher activator diffusion (causing increased wave speed) and light illumination (causing decreased wave speed) that enables normal propagation. This effect should facilitate the creation of very energy efficient on-off switches for chemical computation circuits, where small changes in light levels cause the diffusion diodes to pass or block waves.

I hired Kiyomi from Hawai’i for our NSF REU summer program in spring 2020 amidst fears of the pandemic that eventually postponed the program two years. When she finally arrived in summer 2022, I had already retired from Wooster, where my last year was completely remote, classes via Teams, but several of those included Daniel. Despite the pandemic, I was fortunate to collaborate with both Daniel and Kiyomi — as well as colleagues in Chemical Physics, Physics, Astrophysics, and Computer Science — on an article just published in Frontiers in Physics. Based on Daniel’s senior thesis and Kiyomi’s summer research, the article is the very-Wooster, very-interdisciplinary Chemistry Does General Relativity: Reaction-Diffusion Waves Can Model Gravitational Lensing.

Gravitational lensing is a general relativistic (GR) phenomenon where a massive object redirects light, deflecting, magnifying, and sometimes multiplying its source. In the research, we used the chemistry of Belousov-Zhabotinsky (BΖ) reaction-diffusion (RD) waves to model this astronomical effect in a table-top experiment. We began by experimentally passing BΖ RD waves through non-planar quasi-two-dimensional molds. We next reproduced the waveforms in computer simulations of planar RD waves with variable diffusion. We then varied the diffusion parameter so the effective wave speed of planar waves matched the GR predictions for light deflection near a massive object. We thereby recovered Einstein’s famous light deflection formula, as summarized by the figure below.

When I bought my house, I knew I would soon need to replace its heat pump, which was almost 20 years old. Earlier this month, with my old pump laboring under a cold snap, I upgraded to a new version, which boasts a history of elegant inventions.

Powered by electricity, heat pumps circulate a low-boiling-point hydrofluorocarbon (HFC) refrigerant between two heat exchangers, one inside the house and one outside it, separated by a compressor and an expansion valve, cycling the refrigerant between liquid and gas phases. When the refrigerant flows one way, it evaporates in the inside heat exchanger, absorbing energy and cooling the interior (while it condenses in the outside heat exchanger, liberating energy). Alternately, when the refrigerant flows the other way, it condenses in the inside heat exchanger, liberating energy and heating the interior (while it evaporates in the outside heat exchanger, absorbing energy).

The heart of my heat pump is its scroll compressor, invented by Léon Creux over a century ago, where one interleaved scroll orbits another, channeling and squeezing the injected fuel, as in the 2D animation below. What a wonderful and unexpected use of spiral curves! With less moving parts than a traditional reciprocating compressor, a scroll compressor can be more efficient, smoother, quieter, and more reliable.

For the last two years, the Nonlinear Artificial Intelligence Lab and I have labored to incorporate diversity in machine learning. Diversity conveys advantages in nature, yet homogeneous neurons typically comprise the layers of artificial neural networks. In software, we constructed neural networks from neurons that learn their own activation functions (relating inputs to outputs), quickly diversify, and subsequently outperform their homogeneous counterparts on image classification and nonlinear regression tasks. Sub-networks instantiate the neurons, which meta-learn especially efficient sets of nonlinear responses.

Our examples included conventional neural networks classifying digits and forecasting a van der Pol oscillator and physics-informed Hamiltonian neural networks learning Hénon-Heiles stellar orbits. 

As a final real-world example, I video recorded my wall-hanging pendulum clock, ticking beside me as I write this. Engineered to be nearly Hamiltonian, and assembled with the help of a friend, the pendulum’s Graham escapement periodically interrupts the fall of its weight as gravity compensates dissipation. Using software, we tracked the ends of its compound pendulum, and extracted its angles and angular velocities at equally spaced times. We then trained a Hamiltonian neural network to forecast its phase space orbit, as summarized by the figure below. Once again, meta-learning produced especially potent neuronal activation functions that worked best when mixed.

A particle confined to an impassable box is a paradigmatic and exactly solvable one-dimensional quantum system modeled by an infinite square well potential. Working with Bill Ditto, Elliott Holliday and I recently explored some of its infinitely many generalizations to two dimensions, including particles confined to regions that exhibit integrable, ergodic, or chaotic classical billiard dynamics, using physics-informed neural networks. In particular, we generalized an unsupervised learning algorithm to find the particles’ eigenvalues and eigenfunctions, even in cases where the eigenvalues are degenerate. During training, the neural network adjusts its weights and biases, one of which is the energy eigenvalue, so that its output approximately solves the stationary Schrödinger equation with normalized and mutually orthogonal eigenfunctions.

The motion of one of the simplest dynamical systems, a torqued, damped, nonlinear pendulum, can be infinitely complicated.

Consider a simple pendulum of length l and mass m rigidly connected to an axle of radius r wrapped by a rope that hangs down one side with a mass M climbing up and down it, as in the attached animation.

If the climber’s height

x = x_0  + \frac{a}{\omega^2} \sin \omega t,

varies sinusoidally (relative to the axle), then its acceleration

\ddot x = – a \sin \omega t,

also varies sinusoidally, so the total force on the climber

M\ddot x = \sum_\text{down} f = Mg-T

implies upper rope tension

T = Mg + ma \sin \omega t,

where 0 < a < g. If the axle and rope have negligible inertia, then the total torque on the axle

m l^2 \ddot \theta = \sum_\text{CCW}\tau = – mgl \sin\theta + rT – \gamma \dot\theta,

where \gamma is the axle viscosity. The full motion equation

m l^2 \ddot \theta = – mgl \sin\theta + rMg + rma \sin \omega t – \gamma \dot\theta

reduces to

\ddot \theta = – \sin\theta + 0.7155 + 0.4 \sin 0.25 t – 0.75 \dot\theta

for parameters that describe the animation’s chaotic motion.

Quantum field theory predicts that the temperature of empty space should depend on the observer’s motion, increasing proportionally with acceleration. Here I attempt an accessible introduction to this striking effect, related to Hawking radiation and discovered independently by Fulling, Davies, and Unruh, assuming only sophomore-level physics (including hyperbolic functions) with some assistance from Mathematica.

Hyperbolic Motion

Constant acceleration in Newtonian mechanics is parabolic, while constant acceleration in Einsteinian mechanics is hyperbolic and asymptotic to light speed c = 1 in natural units. For 1+1-dimensional Minkowski spacetime, the difference in squared space and time displacements is the square of proper time displacement,

d\tau^2 = dt^2 – dx^2.

For constant proper acceleration, this has the solution

dt = d\tau \cosh a\tau,vdx = d\tau \sinh a\tau,

with velocity

v = \frac{dx}{dt} = \tanh a\tau \le 1,

where for small times v \sim a \tau \sim a t. If t = 0 and a x = 1 at \tau = 0, then integration gives

a t = \sinh a \tau,\\a x = \cosh a\tau.

Hyperbolic identities then imply

a^2 x^2 – a^2 t^2 = 1

and

a x + a t = e^{a\tau}.

Quantum Vacuum

Due to Heisenberg indeterminacy, electromagnetic fluctuations fill the vacuum. Consider a single such sinusoidal wave of angular frequency \omega_0 = k_0 in natural units. If you move at constant velocity, you observe the wave doppler-shifted to a different frequency. But if you move at constant acceleration, you observe the wave doppler-shifted to a range of frequencies corresponding to your range of velocities. For an accelerated observer at time \tau,

\phi[x,t] = \exp\left[i(k_0 x + \omega_0 t)\right] = \exp\left[i\omega_0(x + t)\right] \\= \exp\left[{i \frac{\omega_0}{a} e^{a \tau}} \right] = \phi[\tau].

Expand this waveform as a sum of harmonics

\phi[\tau] = \int_{-\infty}^{\infty}\,\frac{d\omega}{2\pi} \Phi[\omega] \exp[-i \omega \tau]

where the Fourier components

\Phi[\omega] = \int_{-\infty}^{\infty}d\tau\, \phi[\tau] \exp[+i \omega \tau].

To regularize this divergent integral, subtract a tiny imaginary part i \epsilon from the angular frequency \omega to incorporate a decaying exponential factor e^{-\epsilon \tau} in the integrand, and zero it after integrating. Find

\Phi[\omega] = \lim_{\epsilon\rightarrow 0} \int_{-\infty}^{\infty}d\tau\, \exp\left[ i \frac{\omega_0}{a} e^{a\tau} \right] \exp[+i (\omega – i \epsilon) \tau]\\~ \\= \lim_{\epsilon\rightarrow 0}\, \exp\left[ i\frac{\pi}{2} \left(\frac{\epsilon + i\omega}{a}\right) \right]\left(\frac{a}{\omega_0}\right)^{(\epsilon + i \omega)/a}\frac{1}{a} \Gamma\left[\frac{\epsilon + i \omega}{a} \right] \\= \exp\left[-\frac{\pi}{2} \frac{\omega}{a}\right]\left(\frac{a}{\omega_0}\right)^{i \omega / a}\frac{1}{a} \Gamma\left[i \frac{\omega}{a} \right],

where \Gamma[n+1] = n! analytically continues the factorial function to the complex plane. The spectrum is the absolute square of the Fourier transform,

S[\omega] = \left|\Phi[\omega]\right|^2 = \frac{\pi}{a\omega} \left(\coth\left[ \pi \frac{\omega}{a} \right]-1 \right) \\= \frac{2\pi}{\omega a} \frac{1}{e^{2\pi \omega/a} – 1} \propto \frac{1}{e^{\hbar \omega/k T}-1},

where the Planck factor suggests Bose-Einstein statistics and a thermal photon bath of temperature kT = a \hbar / 2\pi. In SI units,

T = \frac{a \hbar}{2\pi k c} \sim 40~\text{zK}~\left(\frac{a}{g_E} \right),

where g_E is Earth’s surface gravity, and a zeptokelvin is very cool.

Hawking-Unruh Temperature

Just prior to the 1970s work of Fulling, Davies, and Unruh, Stephen Hawking famously predicted that despite their reputations black holes should radiate with an effective temperature

T = \frac{\kappa \hbar}{2\pi k c},

where \kappa is the black hole’s surface gravity (observed at infinity). The Unruh and Hawking results may be linked by the equivalence principle, which equates acceleration and gravity, and by event horizons. In General Relativity, the black hole horizon is a boundary that causally disconnects the interior from the exterior. Similarly, when you accelerate, a Rindler horizon appears a distance c^2/a \sim 1~\text{ly} \left(g_E / a \right) behind you, causally disconnecting you from a region of spacetime whose photons you can outrun (so long as your acceleration continues).