• Fourier transforms can see the future

  Motivation Consider the ODE for a driven harmonic oscillator with no dissipation: where we pick to be completely generic. One approach we may attempt in order to solve this equation is to use Fourier transforms. By writing as we see that time derivatives will just bring out factors of from the… Read more

  March 14th, 2026 · 14 min read

• Heat capacities only exist in one dimension

  Recap As commonly seen in textbooks, the heat capacity (at constant volume, say) is defined as This may look like an innocuous definition, but strictly speaking, it does not make sense. The heat is not a state function; only the 1-form is well-defined, and it is not exact: there is no function such… Read more

  March 8th, 2026 · 4 min read
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  Can a pendulum have temperature?

  Phase space image from https://tikz.net/dynamicsphaseportrait/. Shut up and calculate? A very standard question in introductory statistical mechanics courses is: Problem: Consider a 3D harmonic oscillator with mass and sprint constant in contact with a heat bath at temperature . (a) Calculate the… Read more

  September 20th, 2025 · 39 min read
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  The math behind adjoints: PDE-constrained optimization for design

  Introduction This article introduces PDE-constrained optimization, a discipline with deep applications in engineering design. After all, any physical law is represented by a PDE, and engineering cares a lot about finding optimal designs -- lower costs, higher safety, better performance, you name… Read more

  January 6th, 2025 · 36 min read
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  Infinite children? A nice problem in probability theory

  The problem A couple decides to have a male child no matter what. If their first child is a son, they stop; if it is a daughter, they continue having more children. There is no upper limit on how many daughters they can have before having a son. What is the average number of sons and daughters they… Read more

  December 26th, 2024 · 10 min read