Statistical Patterns in the Equations of Physics and the Emergence of a Meta-Law of Nature

TL;DR

The paper addresses whether physics equations exhibit universal statistical patterns in operator usage. It analyzes three physics corpora and uses simulation-based inference to compare Zipf, Zipf-Mandelbrot, and exponential frequency-rank laws. The main finding is that an exponential law $f(r) \sim e^{-\beta r}$ with $\beta \approx 0.3$ best describes operator frequencies across all corpora, decisively out-performing Zipf, with Planck's formula illustrating the pattern at work. The results suggest a possible meta-law of Nature governing symbolic physics and offer priors that could improve symbolic regression and the discovery of new physical laws.

Abstract

Physics, as a fundamental science, aims to understand the laws of Nature and describe them in mathematical equations. While the physical reality manifests itself in a wide range of phenomena with varying levels of complexity, the equations that describe them display certain statistical regularities and patterns, which we begin to explore here. By drawing inspiration from linguistics, where Zipf's law states that the frequency of any word in a large corpus of text is roughly inversely proportional to its rank in the frequency table, we investigate whether similar patterns for the distribution of operators emerge in the equations of physics. We analyse three corpora of formulae and find, using sophisticated implicit-likelihood methods, that the frequency of operators as a function of their rank in the frequency table is best described by an exponential law with a stable exponent, in contrast with Zipf's inverse power-law. Understanding the underlying reasons behind this statistical pattern may shed light on Nature's modus operandi or reveal recurrent patterns in physicists' attempts to formalise the laws of Nature. It may also provide crucial input for symbolic regression, potentially augmenting language models to generate symbolic models for physical phenomena. By pioneering the study of statistical regularities in the equations of physics, our results open the door for a meta-law of Nature, a (probabilistic) law that all physical laws obey.

Figures (5)

• Figure 1: Binary expression tree for Newton's law of gravitation.
• Figure 2: Distribution of expression complexity in the three corpora, which approximately corresponds to the number of operarors appearing in the equation.
• Figure 3: Comparison between the Feynman Lectures corpus and randomly labelled binary trees. The black points correspond to the true data, whereas red points indicate the operator frequency distribution for randomly generated corpora of formulae of the same size and complexity distribution as the Feynman Lectures corpus. In purple, we give the expected piece-wise uniform distribution in the large complexity limit.
• Figure 4: Posterior distributions of the fits to the different corpora, where we compare a Zipf (\ref{['Zipf_law']}), Zipf--Mandelbrot (\ref{['Zipf_Mandelbrot_law']}), and exponential (\ref{['Exponential_law']}) fit. It is clear that the latter provides the best fit. The solid lines indicate the posterior mean, and the coloured bands show the 68% confidence interval.
• Figure 5: Frequency of operators in the Planck formula against the frequency--rank relations from \ref{['eq:exponential_3', 'eq:exponential_4']}.