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.

I asked Siri to wake me at 7:15 this morning so I could watch SpaceX’s second Integrated Flight Test of Super Heavy Starship, the biggest and most powerful rocket ever built. Unfortunately, my house suffered a rare power outage an hour or two earlier, so I found myself lying in bed watching the coverage on my iPhone.

The sunrise launch was truly spectacular, a major improvement over the first test earlier this year, when multiple engines failed at launch. This morning, for the first time in any test, all 33 Raptormethaloxengines ignited together and ran throughout the boost phase! With a thrill and a shiver, I recalled the excited “Voice of Apollo” Jack King announcing a good start to Apollo 11 with inspiring words that have reverberated throughout my life, “All engine(s) running” (without the “s”).

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 impassablebox 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.

When I was a kid, Saturn was the ringed planet. But today, we know that all of the outer planets have rings. The James Webb Space Telescope has now imaged each of them in infrared revealing their distinctive structures, including Jupiter‘s very faint ring (located by the arrow and dashed curve). The planet images below are roughly to scale, 11:9:4:4, where Earth is 1.

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

Just a couple of months after announcing the remarkable discovery of a single shape that forces a non-periodic tiling of the plane, Smith, Myers, Kaplan, and Goodman-Strauss have announced an improved aperiodicmonotile or ein stein. (Ein stein is “one stone” in German.)

The hat and turtle shapes tile the plane only non-periodically, but with their mirror reflections, which would be practically troublesome for one-sided tiles. Smith and colleagues realized that if reflections are forbidden an intermediate equilateral shape tiles the plane only non-periodically. Furthermore, perturbing the shape’s sides can block periodic tiling using it and its reflection, thereby generating specter shapes that tile the plane only non-periodically, whether reflections are allowed or forbidden.

Because specters don’t need their reflections, and vampires are said to not reflect in mirrors, the authors playfully suggest calling them vampire ein steins. You could tile your bathroom with a single specter shape, even though bathroom tile is glazed on only one side!

Below are three different colorings of a tiling by a stegosaurus shape, a particularly simple equilateral specter with only \pm 60^\circ and \pm 90^\circ turns (and a single 0^\circ turn), which I created in Mathematica and Illustrator. The darker tile pairs in the first coloring are called mystics

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 constantproper 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,

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

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

where the Planck factor suggests Bose-Einstein statistics and a thermal photon bath of temperaturekT = 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’ssurface 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).

One hundred years ago today the Physical Review published research on light scattering electrons that would earn its author, Wooster graduate Arthur Compton, a PhysicsNobel Prize.

By relativistically conserving spacetimemomentum, as in the diagram below, and treating light as particles now called photons, Compton discovered that deflecting an electron through an angle \theta stretches the light’s wavelength by

where \lambda_c = h / mc \sim 1/40 \textup{~\AA} is the Compton wavelength of the electron. Compton’s experiment helped convince the physics community of wave-particle duality. Today, sophomores Compton scatter in our Modern Physics lab.

After half a century confined to low-Earth orbit, and as soon as late next year, humans will once again leave Earth and voyage to Moon. The reality of this exciting adventure crystallized earlier this month when NASA announced the diverse and inspiring Artemis II crew: Clockwise from left in the photo below are Christina Koch, Victor Glover, Jeremy Hansen, and Reid Wiseman.

Christina holds the record for longest space flight by a woman, during which she participated in the first all-woman spacewalk. She is now scheduled to become the first woman to fly around Moon. Along with two electrical engineer degrees, Christina obtained a bachelor’s degree in physics from NC State, and I often think of her as I walk to my present office in NC State’s physics building (although I understand that physics moved into the renovated Riddick shortly after she graduated).

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Thanks, Mark! I enjoy reading your posts as well.

Nice post, John! Thanks for writing these. I always enjoy them.

Thanks, Mark! I enjoy reading your posts as well.