Researchers at the California Institute of Technology have uncovered a new mathematical perspective on a core principle in physics, the Boltzmann distribution, by approaching it from a question originating in economic theory. The Boltzmann distribution is a statistical rule used in physics to predict the likelihood of a system occupying certain states, such as how molecules in a gas distribute themselves at a given temperature. Rather than tracking every particle’s chaotic path, it describes the probability of outcomes and underlies models in fields as diverse as thermodynamics, chemistry, and even economics.
Caltech economist and mathematician Omer Tamuz and his collaborator Fedor Sandomirskiy posed a deceptively simple question: What other mathematical laws could describe systems made of independent, unrelated parts without producing nonsensical or spurious connections? In economics that corresponds to models where unrelated choices—think choosing breakfast cereal versus dish soap—should not influence each other. In physics, it corresponds to systems whose parts truly act independently.
To explore this, the researchers used a playful analogy involving pairs of dice. Standard dice follow well-known probability patterns. “Crazy” dice, with nonstandard numberings, can produce the same combined outcomes as normal dice when their sums are considered, even though individual behavior differs. This provided a way to test whether alternative distribution laws could mimic the statistical independence seen in both economic preferences and physical systems. Through a comprehensive analysis using infinite sets of these alternative constructions, the team showed that no other distribution satisfies the requirement of modeling unrelated systems consistently.
The result, published in Mathematische Annalen, mathematically proves that the Boltzmann distribution is the only law capable of accurately describing systems composed of unrelated components without creating false dependencies. This finding not only strengthens confidence in a foundational physical law but also illustrates how abstract mathematical reasoning can bridge economics and physics, revealing deep common structure across disciplines.