With this substitution, our final result for the heating due to inelastic loss is If we assume that the ensemble stays in cross-dimensional equilibrium, T = T$_{z}$ = T$_{x}$ , then the overall heating rate is simply or equivalently, dT$_{z}$ dt = n 4 ( − K$_{0}$T$_{z}$ + 2 K$_{1}$T$_{x}$ ) T$_{z}$ (21) dT$_{x}$ dt = n 4 ( K$_{0}$T$_{z}$ ) T$_{x}$. (22) dT dt = 1 3 dT$_{z}$ dt + 2 3 dT$_{x}$ dt = n 12 ( K$_{0}$ + 2 K$_{1}$ ) T $^{2}$, (23) dT dt / ( T $^{2}$n ) = β/T 12 . (24) 4