Abstract
World records in athletics provide a measure of physical1
as well as physiological human performance2, 3. Here we analyse
running records and show that the mean speed as a function of race time can
be described by two scaling laws that have a breakpoint at about 150–170
seconds (corresponding to the 
1,000 m race). We interpret this as being
the transition time between anaerobic and aerobic energy expenditure by athletes.
Records measured under standard external conditions represent the most
reliable and up-to-date index of human performance, and attempts have been
made to model running records since the beginning of the century1, 4, 5.
It has been suggested1 that the record times, 
, and distances,
d, of men's running can be represented by a single power law, 
dn, in races from 100 to 10,000 metres,
where n is about 1.1 and depends on the epoch. This power law implies
that there is an invariance of statistical features of the system over all
distances and there are no evident characteristic scales as in critical phenomena.
We find, however, that both running and swimming can be characterized by
two distinct critical phenomena when the record mean speed, u, which
is equal to d/, is considered. Unlike d, which is set,
u relates to an athlete's energy expenditure. The exponent of
dn is almost 1, so the mean speed
is u=d/-
, where the exponent 
=1-1/n.
The fact that n is slightly larger than 1 amplifies every small deviation
in the plot of d against in a plot of u against .
Using record data for swimming and running for men and women, we find that
there are two regimes described by two scaling laws of the kind
u-, where has two different
values. These separate abruptly at a characteristic time, *,
which lies in the range 2.2–2.8 min (Fig. 1) that
separates short races (200 m
d1,000 m and 50 md200
m for running and swimming, respectively) and long races (1,500 md
42,195 m and 400 md1,500 m for running and swimming,
respectively). Two different critical phenomena (and no more) simultaneously
coexist in speed sports.
Figure 1: Plots of world-record mean speeds against the record time (at November 1999).
a,b, Running, and c,d, swimming records: for men (a,c
), we consider 11 races (200 m, 400 m, 800 m, 1,000 m, 1,500 m, the mile,
3,000 m, 5,000 m, 10,000 m, 1 hour, and marathon); the same races are considered
for women (b,d), apart from the 1 hour race. Lines represent the best
fits. The scaling exponents and characteristic times *
of the breakpoints are shown; characteristic times have been determined
by using a 
2 minimization on a broken power law. Triangles
in a,b represent the 100 m race, which is excluded from the analysis
because the mean speed is strongly affected by the standing start of athletes.
The transition between the two scaling exponents corresponds to
the switch from the anaerobic metabolism that is needed for short sprints
to the aerobic metabolism used to supply energy for long-distance races. Table 1 shows values for the two exponents an
and ae (for anaerobic and aerobic, respectively) for
men and women in athletics at different epochs. This transition is consistently
evident since athletics records began.
The slopes of the two scaling laws can be used as a test to compare how evolved and how close to the limit performances are for athletes of different countries of origin, age or sex, for example. The smaller the scaling exponents, the better the performance in longer races, in the sense that the dissipated power will decrease more slowly. In running, men and women show comparable efficiency (their exponents are the same within error): the commonly held belief that women are better than men in long-distance races is not confirmed by our analysis. In swimming, small differences probably result from a difference in buoyancy, which helps women's swimming performance. The slopes are significantly steeper for running than for swimming, perhaps because swimming demands a small aerobic contribution for short races.
