  The Yang-Mills gradient flow is considered on the four dimensional torus T^4
for SU(N) gauge theory coupled to N_f flavors of massless fermions in arbitrary
representations. The small volume dynamics is dominated by the constant gauge
fields. The expectation value of the field strength tensor squared is
calculated for positive flow time t by treating the non-zero gauge modes
perturbatively and the zero modes exactly. The finite volume correction to the
infinite volume result is found to contain both algebraic and exponential
terms. The leading order result is then used to define a one parameter family
of running coupling schemes in which the coupling runs with the linear size of
the box. The new scheme is tested numerically in SU(3) gauge theory coupled to
N_f = 4 flavors of massless fundamental fermions. The calculations are
performed at several lattice spacings with a controlled continuum
extrapolation. The continuum result agrees with the perturbative 2-loop
prediction for small renormalized coupling as expected.
