  Models for human running performances of various complexities and underlying
principles have been proposed, often combining data from world record
performances and bio-energetic facts of human physiology. Here we present a
novel, minimal and universal model for human running performance that employs a
relative metabolic power scale. The main component is a self-consistency
relation for the time dependent maximal power output. The analytic approach
presented here is the first to derive the observed logarithmic scaling between
world (and other) record running speeds and times from basic principles of
metabolic power supply. Various female and male record performances (world,
national) and also personal best performances of individual runners for
distances from 800m to the marathon are excellently described by this model,
with mean errors of (often much) less than 1%. The model defines endurance in a
way that demonstrates symmetry between long and short racing events that are
separated by a characteristic time scale comparable to the time over which a
runner can sustain maximal oxygen uptake. As an application of our model, we
derive personalized characteristic race speeds for different durations and
distances.
