Statistical physics of hair fibre bundles and the shape of a ponytail
There are 100,000 hair fibres on a typical head of hair, hence calculating perceived properties like 'volume' and compressibility are problems in statistical physics. To address this, a density functional theory for the distribution of hair in a fibre assembly has been developed, treating individual elements as elastic filaments with random intrinsic curvatures.
This formalism has been applied to the iconic problem of a ponytail, for which the combined effects of bending elasticity, gravity, and orientational disorder are recast as a differential equation for the envelope -- the ponytail shape equation. Aside from compressibility, this problem is characterised by a single dimensionless measure of the ponytail length, christened the Rapunzel number. From laboratory measurements of model ponytails, the balance of forces in various regions of the ponytail can be identified and related to the measured random curvatures of individual hairs.
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