Coconuts and Islanders: A Statistics-First Guide to the Boltzmann Distribution

The Boltzmann distribution is one of the key equations of thermal physics and is widely used in machine learning as well. Here I derive a Boltzmann distribution in a simple pedagogical example using only tools from a first-year probability course. The example is called "coconuts and islanders" and was taught to me by my father, Shoucheng Zhang (1963 - 2018), to whom these notes are dedicated. By focusing on this simple story, which can be easily simulated on a computer, I aim to provide a more accessible and intuitive presentation of the Boltzmann distribution. Yet I hope this exposition also inspires deep thinking about statistical physics. For instance, I show that the coconuts and islanders story illuminates a connection between the "fundamental assumption of statistical mechanics"---all microstates are equally probable---and the statistical property of detailed balance.