Kleiber’s law (metabolic scaling)

Empirical regularityBiologyComplexity

\htmlClass{sjt-B}{B} \htmlClass{sjt-propto}{\propto} \htmlClass{sjt-M}{M}^{\htmlClass{sjt-exp}{3/4}}

Tap a symbol above, or choose one here:

B

Basal metabolic rate

Meaning

Energy an animal uses per unit time at rest (watts).

Its job in this equation

Output: what scales.

Read it, progressively

1. Gist

Bigger animals burn more energy, but less per kilogram.

What it does not establish

It is an empirical regularity, not a law of physics. Exponent estimates depend on which species are included and how the fit is done. A power law in metabolism and a power law in neural-network loss share functional form (Level 2), not an established mechanism.

Play with it

Power-law explorer

L(N) = 1.70 + 12.0\,N^{-0.080}

α, exponent0.080A, scale12.0L∞, irreducible floor1.70Probe model size N10^9.0

L at probe3.987

L at 10× params3.602

10× buys (reducible part)×0.832

10× buys (absolute)−0.385

Illustrative constants, not a fit to real models. On log–log axes the reducible part, $AN^{-\alpha}$ , is a straight line with slope −α. The full curve bends toward the floor $L_\infty$ , the loss that no amount of scale removes in this model. On linear axes nearly all of the change happens at the far left, which is why scaling papers plot on log axes.

Loading interactive…

§ Module 3: Biology as an Information System Next equation: An illustrative neural scaling law →