Thermodynamic description of worldwide distribution of energy and carbon emission

What the math gives to ML

The paper introduces a Rayleigh–Jeans allocation law in which mode mass is inversely proportional to an energy gap, rather than exponentially proportional to a logit. The transferable asset is a constrained, heavy-tailed routing distribution whose concentration is controlled by a chemical potential approaching the lowest-energy state, producing an explicit condensation mechanism. A practical neural adaptation is to replace softmax attention or dense MoE routing with a positive inverse-affine kernel and solve its temperature and chemical potential from normalization plus a prescribed mean-energy constraint. This is not a guaranteed improvement, but it is a concrete alternative to exponential attention that can be tested for sharper routing, lower entropy, and improved compute allocation.