Recent experimental progress on cold atoms has made it important to understand the physics of isolated systems. The energy of an isolated system may be changed by attaching it to a heat bath, then detaching it after equilibration. The final energy obtained will vary between experiments, due to the thermal fluctuations while the bath is attached. We consider a different approach, where the system's energy is changed by a series of small non-equilibrium perturbations. A simple expression for the final width of the energy distribution is obtained, depending only on the system's heat capacity, and the (possibly varying) rate of energy injection to the system. The expression is used to characterize the energy distribution of several systems.