Why Do Some Phenomena Follow Three Laws? ✛ ✎

Original: "Physical Law Ecology: mapping multi-mechanism ecologies as the zeroth step of data-driven scientific discovery"
arXiv:2609.08536v1 · 2026-09-08 · arXiv non-exclusive · independent analysis · 1 min
Rubber, boiling water, and galaxies obey several equations at once.

Stretch a rubber band. At first it stretches easily, then resistance grows, and under strong stretching the band becomes stiff and may even turn white. These three stages are governed by three different mechanisms: Hooke's law, entropic elasticity, and strain-induced crystallization. Each is described by its own equation, and they don't switch abruptly but smoothly pass influence from one to the next, like a relay baton.

Scientists have long searched for a single best equation for each phenomenon. But the new system, called Physical Law Ecology, takes a different approach: it treats all possible equations as species in an ecosystem, competing for dominance under different conditions. The algorithm itself determines how many such "species" are needed and which one dominates where. In four different systems, rubber, boiling, galaxy rotation, and droplet evaporation, it found exactly three mechanisms.

In galaxies, the system detected three gravitational laws: the usual Newtonian one, a transitional one, and a "constant field" of acceleration. The characteristic acceleration at which the regime changes turned out to be different for gas-poor and gas-rich galaxies. This means there is no single curve for all galaxies.

This approach helps make more accurate predictions in engineering and science. Instead of arguing about which law is "correct," you can see how several laws work together and understand where each one matters most.

A candle, which we see as a single flame, is actually governed by four different equations: heat conduction in the wax, diffusion combustion in the blue zone, radiation in the bright part, and convection in the tongue above the flame.

w_i(s,c) = \frac{\exp(-n \cdot MSE_i / 2\sigma^2)}{\sum_j \exp(-n \cdot MSE_j / 2\sigma^2)}
The smaller the equation's error on the data, the higher its weight; normalization ensures the weights sum to one.
K^* = \arg\min_K \left[ \sum_{s,c} n_{s,c} \ln(MSE^*_{s,c}) + K \cdot p_{max} \cdot \ln(N) \right]
Balances descriptive accuracy against model complexity; the minimum determines the number of equations.
Concepts
principle of least action Kepler's first law quantum law ecological pyramid rule Cope's rule bayesian evidence neural network scientific discovery physics physics informed neural network control theory biology Gause's competitive exclusion principle geodesic equation statistical mechanics evolution quantum mechanics initial conditions Bayesian inference computational fluid dynamics time evolution