Viscous Brane Cosmology with a Brane-Bulk Energy Interchange Term
  We assume a flat brane located at y=0, surrounded by an AdS space, and consider the 5D Einstein equations when the energy flux component of the energy-momentum tensor is related to the Hubble parameter through a constant Q. We calculate the metric tensor, as well as the Hubble parameter on the brane, when Q is small. As a special case, if the brane is tensionless, the influence from Q on the Hubble parameter is absent. We also consider the emission of gravitons from the brane, by means of the Boltzmann equation. Comparing the energy conservation equation derived herefrom with the energy conservation equation for a viscous fluid on the brane, we find that the entropy change for the fluid in the emission process has to be negative. This peculiar effect is related to the fluid on the brane being a non-closed thermodynamic system. The negative entropy property for non-closed systems is encountered in other areas in physics also, in particular, in connection with the Casimir effect at finite temperature. 
Iver Brevik, Jon-Mattis Borven, and Sebastian Ng