WITH APPLICATION OF MICROSCOPIC MODEL OF A TRAPPED BOSE-CONDENSATE GAS AT FINITE TEMPERATURES. WE DERIVE AN EQUATION OF MOTION FOR THE CONDENSEDE WAVE FUNCTION AND A QUANTUM KINETIC EQUATION FOR EXCITED ATOMS. IN THE QUANTUM KINETIC EQUATION COLLISIONS BETWEEN THE CONDENSATE AND NON-CONDENSATE ARE NOW INCLUDED. THE CONTINUITY EQUATION CONTAINS A SOURCE TERM WHICH IS RELATED TO THE INTGRATION OF COLLISION BETWEEN CONDENSATE AND NON-CONDENSATE.
WE ASSUME THAT THE NON-CONDENSATE COLLSION RATE IS SUFFICIENTLY RAPID TO ENSURE THAT THE NON-CONDENSATE DISTRIBUTION FUNCTION CAN BE APPROXIMATED BY A LOCAL EQUILIBRIUM BOSE DISTRIBUTION. THEREFORE THE NEW EQUATIONS OF MOTION ARE ACHIEVED SO THAT THE NON-CONDENSATE ATOMS ARE IN LOCAL THERMAL EQUILIBRIUM WITH EACH OTHER BUT ARE NOT YET IN COMPLETE EQUILIBRIUM WITH THE OTHER BOSONIC PARAMETERS. THE SOURCE TERMS APPEARING IN THESE EQUATIONS PLAY A KEY ROLE IN DESCRIBING THE EQUILIBRATION OF THE LOCAL CHEMICAL POTENTIALS ASSOCIATED WITH THE CONDENSATE AND NON-CONDENSATE ATOMS.