The Brandeis Dice Problem and Statistical Mechanics

TL;DR

This paper explores alternative solutions to Jaynes' Brandeis Dice Problem, comparing MaxEnt and Bayesian approaches, and discusses their implications for understanding statistical mechanics analogies.

Contribution

It introduces two new solutions to the dice problem, highlighting differences and similarities with Jaynes' approach and implications for statistical mechanics analogies.

Findings

MaxEnt and Bayesian solutions differ for the original problem.

Solutions converge in the thermodynamic limit as dice sides increase.

Statistical mechanics involves more than MaxEnt, with physical properties beyond mathematical analogies.

Abstract

Jaynes invented the Brandeis Dice Problem as a simple illustration of the MaxEnt (Maximum Entropy) procedure that he had demonstrated to work so well in Statistical Mechanics. I construct here two alternative solutions to his toy problem. One, like Jaynes' solution, uses MaxEnt and yields an analogue of the canonical ensemble, but at a different level of description. The other uses Bayesian updating and yields an analogue of the micro-canonical ensemble. Both, unlike Jaynes' solution, yield error bars, whose operational merits I discuss. These two alternative solutions are not equivalent for the original Brandeis Dice Problem, but become so in what must, therefore, count as the analogue of the thermodynamic limit, $M$-sided dice with $M\rightarrow\infty$. Whereas the mathematical analogies between the dice problem and Stat Mech are quite close, there are physical properties that the former lacks but that are crucial to the workings of the latter. Stat Mech is more than just MaxEnt.