Results of some simple ‘thermodynamic’ experiments on self-gravitating n-body systems are reported for a variety of boundary conditions. Systems placed in specularly reflecting enclosures did not show any unusual behavior, even though a variety of conditions was tried in an attempt to start a ‘gravothermal castastrophe’. Similarly, there was no tendency to transfer energy between ‘hot’ and ‘cool’ subclosures within a given cluster. However, systems in ‘isothermal’ enclosures gave up energy to the enclosure at a surprisingly high rate, and sustained the energy-transfer rate as long as the experiment was continued. An explanation of these different behaviors was sought and found in an examination of the premises that underlie certain attempts to construct a thermodynamics for self-gravitating systems. Conventional application of the H-theorem implies violations of the n-body equations of motion and predictions not consistent with observation. Both the ‘gravothermal catastrophe’ and the experiments in an ‘isothermal’ enclosure share this violation of the equations of motion. A new formulation that allows for all the interactions in an n-body system shows that isolated n-body systems need not form binaries or condense into other subaggregates. The virial theorem follows as an ensemble average over the micro-canonical ensemble.