Fourier transforms can see the future

Motivation Consider the ODE for a driven harmonic oscillator with no dissipation: \[\boxed{\ddot x + \omega^2 x=f(t),\quad x(0)=x_0,\quad \dot x(0)=v_0,}\label{eq:ode}\tag{1.1}\] where we pick $f$ to be completely generic. One approach we may attempt in order to solve this equation is to use Fourier transforms. By writing $x(t)$ as \[x(t)=\... Read more 14 Mar 2026 - 13 minute read

Heat capacities only exist in one dimension

Recap As commonly seen in textbooks, the heat capacity (at constant volume, say) is defined as \[C_V=\left(\frac{\partial Q}{\partial T}\right)_V.\] This may look like an innocuous definition, but strictly speaking, it does not make sense. The heat $Q$ is not a state function; only the 1-form $\delta Q$ is well-defined, and it is not exact: t... Read more 08 Mar 2026 - 3 minute read

Can a pendulum have temperature?

Shut up and calculate? A very standard question in introductory statistical mechanics courses is: Problem: Consider a 3D harmonic oscillator with mass $m$ and sprint constant $k$ in contact with a heat bath at temperature $T$. (a) Calculate the partition function $Z(\beta)$. (b) Show that the average energy for the oscillator is $3k_BT$. Th... Read more 20 Sep 2025 - 37 minute read

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 it. We will navigate from good old Lagrange multi... Read more 06 Jan 2025 - 35 minute read

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 will have? I was once asked this question dur... Read more 26 Dec 2024 - 9 minute read