Modelling human endurance: power laws vs critical power

Abstract

The power–duration relationship describes the time to exhaustion for exercise at different intensities. It is believed to be a “fundamental bioenergetic property of living systems” that this relationship is hyperbolic. Indeed, the hyperbolic (a.k.a. critical-power) model which formalises this belief is the dominant tool for describing and predicting high-intensity exercise performance, e.g. in cycling, running, rowing or swimming. However, the hyperbolic model is now the focus of a heated debate in the literature because it unrealistically represents efforts that are short (< 2 min) or long (> 15 min). We contribute to this debate by demonstrating that the power–duration relationship is more adequately represented by an alternative, power-law model. In particular, we show that the often-observed good fit of the hyperbolic model between 2 and 15 min should not be taken as proof that the power–duration relationship is hyperbolic. Rather, in this range, a hyperbolic function just happens to approximate a power law fairly well. We also prove mathematical results which suggest that the power-law model is a safer tool for pace selection than the hyperbolic model and that the former more naturally models fatigue than the latter.

Exchangeable Quantities and Power Laws: Τhe Case of Pores in Solids

“…[16]; musical notes, colors, and shapes in music and painting [16,17]; food along the trophic chain of pelagic species (phytoplankton, zooplankton, and fishes) [18,19]; athletic equipment, tools, or objects (balls, disks, javelins, etc.) used in competitive sports in respect to the resulting power law medal ranking [20], as well as to the achieved human endurance [21]; and, last but not least, energy and heat between the atmosphere and ocean, which drives the power law distribution of sea-ice floes in Arctic seas [22,23].…”

“…The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/foundations4020012/s1, Table S1: The differential pore properties D i (diameter, nm) and N i (pore density, pores/g) of the three (3) porous spinels CoAl 2 O 4 T = 500, 700, and 900 • C discussed in this work are from refs [21][22][23][24][25]32]. The partial pressure 0 < (P/Po) = P i < 1 where the D i and N i values were determined during nitrogen porosimetry is also shown; Table S2: Pore properties D i (diameter, nm) and N i (pore density, pores/g) of eight (8) porous volcanic rocks or mimics plus one (1) sample of porous soil (last double column) discussed in the present work.…”