  We investigate a many-body localization transition based on a Boltzmann
transport theory. Introducing weak localization corrections into a Boltzmann
equation, Hershfield and Ambegaokar re-derived the Wolfle-Vollhardt
self-consistent equation for the diffusion coefficient [Phys. Rev. B {\bf 34},
2147 (1986)]. We generalize this Boltzmann equation framework, introducing
electron-electron interactions into the Hershfield-Ambegaokar Boltzmann
transport theory based on the study of Zala-Narozhny-Aleiner [Phys. Rev. B {\bf
64}, 214204 (2001)]. Here, not only Altshuler-Aronov corrections but also
dephasing effects are taken into account. As a result, we obtain a
self-consistent equation for the diffusion coefficient in terms of the disorder
strength and temperature, which extends the Wolfle-Vollhardt self-consistent
equation in the presence of electron correlations. Solving our self-consistent
equation numerically, we find a many-body localization insulator-metal
transition, where a metallic phase appears from dephasing effects dominantly
instead of renormalization effects at high temperatures. Although this
mechanism is consistent with that of recent seminal papers [Ann. Phys. (N. Y).
{\bf 321}, 1126 (2006); Phys. Rev. Lett. {\bf 95}, 206603 (2005)], we find that
our three-dimensional metal-insulator transition belongs to the first order
transition, which differs from the Anderson metal-insulator transition
described by the Wolfle-Vollhardt self-consistent theory. We speculate that a
bimodal distribution function for the diffusion coefficient is responsible for
this first order phase transition.
