  We describe a number conserving approach to the dynamics of Bose-Einstein
condensed dliute atomic gases. This builds upon the works of Gardiner [C. W.
Gardiner, Phys. Rev. A 56, 1414 (1997)], and Castin and Dum [Y. Castin and R.
Dum, Phys. Rev. A 57, 3008 (1998)]. We consider what is effectively an
expansion in powers of the ratio of non-condensate to condensate particle
numbers, rather than inverse powers of the total number of particles. This
requires the number of condensate particles to be a majority, but not
necessarily almost equal to the total number of particles in the system. We
argue that a second-order treatment of the relevant dynamical equations of
motion is the minimum order necessary to provide consistent coupled condensate
and non-condensate number dynamics for a finite total number of particles, and
show that such a second-order treatment is provided by a suitably generalized
Gross-Pitaevskii equation, coupled to the Castin-Dum number-conserving
formulation of the Bogoliubov-de Gennes equations. The necessary equations of
motion can be generated from an approximate third-order Hamiltonian, which
effectively reduces to second order in the steady state. Such a treatment as
described here is suitable for dynamics occurring at finite temperature, where
there is a significant non-condensate fraction from the outset, or dynamics
leading to dynamical instabilities, where depletion of the condensate can also
lead to a significant non-condensate fraction, even if the non-condensate
fraction is initially negligible.
