  We derive the first law of binary point-particle mechanics for generic bound
(i.e. eccentric) orbits at the fourth post-Newtonian (4PN) order, accounting
for the non-locality in time of the dynamics due to the occurence of a
gravitational-wave tail effect at that order. Using this first law, we show how
the periastron advance of the binary system can be related to the averaged
redshift of one of the two bodies for a slightly non-circular orbit, in the
limit where the eccentricity vanishes. Combining this expression with existing
analytical self-force results for the averaged redshift, we recover the known
4PN expression for the circular-orbit periastron advance, to linear order in
the mass ratio.
