  About a century ago, Jeans (1919) discovered that if binary stars reach a
state approximating energy equipartition, for example through many dynamical
encounters that exchange energy, their eccentricity distribution can be
described by : dN/de = 2e. This is referred to as the thermal eccentricity
distribution, and has been widely used for initial conditions in theoretical
investigations of binary stars. However, observations suggest that the
eccentricity distributions of most observed binaries, and particularly those
with masses < 5 Msun, are flatter than thermal and follow more closely to a
uniform distribution. Nonetheless, it is often argued that dynamical
interactions in a star cluster would quickly thermalize the binaries, which
could justify imposing a thermal eccentricity distribution at birth for all
binaries. In this paper we investigate the validity of this assumption. We
develop our own rapid semi-analytic model for binary evolution in star
clusters, and also compare with detailed N-body and Monte Carlo star cluster
models. We show that, for nearly all binaries, dynamical encounters fail to
convert an initially uniform eccentricity distribution to thermal within a star
cluster's lifetime. Thus, if a thermal eccentricity distribution is observed,
it is likely imprinted upon formation rather than through subsequent long-term
dynamical processing. Theoretical investigations that initialize all binaries
with a thermal distribution will make incorrect predictions for the evolution
of the binary population. Such models may overpredict the merger rate for
binaries with modest orbital separations by a factor of about two.
