  A major puzzle concerning the wide stellar binaries (semimajor axes $a\gtrsim
10^3$\,AU) in the Solar neighborhood is the origin of their observed
superthermal eccentricity distribution function (DF), which is
well-approximated by $P(e)\propto e^\alpha$ with $\alpha\approx 1.3$. This DF
evolves under the combined influence of (i) tidal torques from the Galactic
disk and (ii) scattering by passing stars, molecular clouds, and substructure.
Recently, it was demonstrated that Galactic tides alone cannot produce a
superthermal eccentricity DF from an initially isotropic, non-superthermal one,
under the restrictive assumptions that the eccentricity DF was initially of
power law form and then was rapidly phase-mixed toward a steady state by the
tidal perturbation. In this paper we first prove analytically that this
conclusion is valid at all times, regardless of these assumptions. We then
adopt a thin Galactic disk model and numerically integrate the equations of
motion for several ensembles of tidally perturbed wide binaries to study the
time evolution in detail. We find that even non-power law DFs can be described
by an effective power law index $\alpha_\mathrm{eff}$ which accurately
characterizes both their initial and final states, and that a DF with initial
(effective or exact) power law index $\alpha_\mathrm{i}$ is transformed by
Galactic tides into another power law with index $\alpha_\mathrm{f}\approx
(1+\alpha_\mathrm{i})/2$ on a timescale $ \sim
4\,\mathrm{Gyr}\,(a/10^4\mathrm{AU})^{-3/2}$. In a companion paper, we
investigate separately the effect of stellar scattering. As the GAIA data
continues to improve, these results will place strong constraints on wide
binary formation channels.
