  We present a derivation from first principles of the coupled equations of
motion of an active self-diffusiophoretic Janus motor and the hydrodynamic
densities of its fluid environment that are nonlinearly displaced from
equilibrium. The derivation makes use of time-dependent projection operator
techniques defined in terms of slowly varying coarse-grained microscopic
densities of the fluid species number, total momentum, and energy. The exact
equations of motion are simplified using time-scale arguments, resulting in
Markovian equations for the Janus motor linear and angular velocities with
average forces and torques that depend on the fluid densities. For a large
colloid, the fluid equations are separated into bulk and interfacial
contributions, and the conditions under which the dynamics of the fluid
densities can be accurately represented by bulk hydrodynamic equations subject
to boundary conditions on the colloid are determined. We show how the results
for boundary conditions based on continuum theory can be obtained from the
molecular description and provide Green-Kubo expressions for all transport
coefficients, including the diffusiophoretic coupling and the slip coefficient.
