Heat capacities only exist in one dimension

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: there is no... Read more 08 Mar 2026 - 3 minute read

Can a pendulum have temperature?

Phase space image from https://tikz.net/dynamics_phaseportrait/. 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 ave... Read more 20 Sep 2025 - 36 minute read

The math behind adjoints: PDE-constrained optimization for design

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 multipliers in $\ma... Read more 06 Jan 2025 - 34 minute read

Infinite children? A nice problem in probability theory

Consider the following 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 ask... Read more 26 Dec 2024 - 9 minute read

The interaction between heat and elasticity

When subject to an increase in temperature, a solid object will usually expand, and this expansion will act to increase internal stresses. In this way, variations in temperature affect solids’ mechanical properties. On the other hand, inside a solid body, heat transfer occurs mostly by diffusion from hot to colder zones. This depends not only o... Read more 16 Aug 2024 - 11 minute read