  We present a comprehensive mathematical model and experimental measurements
for the evolution of a trapped particle ensemble driven by collisions with a
room-temperature background vapor. The model accommodates any trap geometry,
confining potential, initial trapped distribution, and other experimental
details; it only depends on the the probability distribution function $P_t(E)$
for the collision-induced energy transfer to the trapped ensemble. We describe
how to find $P_t(E)$ using quantum scattering calculations and how it can be
approximated using quantum diffractive universality. We then compare our model
to experimental measurements of a $^{87}$Rb ensemble energy evolution exposed
to a room temperature background gas of Ar by means of a single parameter fit
for the total collision rate $\Gamma$. We extracted a collision rate of $\Gamma
= 0.646(1)\ \text{s}^{-1}$. This is compared to a value of $0.664(4)\
\text{s}^{-1}$ found by the commonly used method of zero-trap depth
extrapolation, a $2.8\%$ correction that is a result of our model fully taking
ensemble loss and heating into account. Finally, we report a five-fold increase
in the precision of our collision rate extraction from the experimental data.
