  Active colloidal particles that are propelled by a self-diffusiophoretic
mechanism are often described by Langevin equations that are either postulated
on physical grounds or derived using the methods of fluctuating hydrodynamics.
While these descriptions are appropriate for colloids of micrometric and larger
size, they will break down for very small active particles. A fully microscopic
derivation of Langevin equations for self-diffusiophoretic particles powered by
chemical reactions catalyzed asymmetrically by the colloid is given in this
paper. The derivation provides microscopic expressions for the translational
and rotational friction tensors, as well as reaction rate coefficients
appearing in the Langevin equations. The diffusiophoretic force and torque are
expressed in terms of nonequilibrium averages of fluid fields that satisfy
generalized transport equations. The results provide a description of active
motion on small scales where descriptions in terms of coarse grained continuum
fluid equations combined with boundary conditions that account for the presence
of the colloid may not be appropriate.
