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A major puzzle concerning the wide stellar binaries (semimajor axes $a\gtrsim10^3$\,AU) in the Solar neighborhood is the origin of their observedsuperthermal eccentricity distribution function (DF), which iswell-approximated by $P(e)\propto e^\alpha$ with $\alpha\approx 1.3$. This DFevolves under the combined influence of (i) tidal torques from the Galacticdisk and (ii) scattering by passing stars, molecular clouds, and substructure.Recently, it was demonstrated that Galactic tides alone cannot produce asuperthermal eccentricity DF from an initially isotropic, non-superthermal one,under the restrictive assumptions that the eccentricity DF was initially ofpower law form and then was rapidly phase-mixed toward a steady state by thetidal perturbation. In this paper we first prove analytically that thisconclusion is valid at all times, regardless of these assumptions. We thenadopt a thin Galactic disk model and numerically integrate the equations ofmotion for several ensembles of tidally perturbed wide binaries to study thetime evolution in detail. We find that even non-power law DFs can be describedby an effective power law index $\alpha_\mathrm{eff}$ which accuratelycharacterizes both their initial and final states, and that a DF with initial(effective or exact) power law index $\alpha_\mathrm{i}$ is transformed byGalactic tides into another power law with index $\alpha_\mathrm{f}\approx(1+\alpha_\mathrm{i})/2$ on a timescale $ \sim4\,\mathrm{Gyr}\,(a/10^4\mathrm{AU})^{-3/2}$. In a companion paper, weinvestigate separately the effect of stellar scattering. As the GAIA datacontinues to improve, these results will place strong constraints on widebinary formation channels.

Eccentricity dynamics of wide binaries -- I. The effect of Galactic tides