The heat theorem (i.e. the second law of thermodynamics or the existence of entropy) is a manifestation of a general property of hamilto- Boltzmann's Heat TheoremIn equilibrium statistical mechanics states are identificed with time invariant probability distributions µ on phase space. Thermodynamic functions, identified with time averages of mechanical observables, are expressed as integrals F of suitable mechanical obeservables F . The averages depend on control parameters α, like volume, energy, kinetic energy. Under changes b dα of the control parameters the thermodynamic quantities change so that the variation of the average energy and the variation of the volume are dU and dV and are related to the time averages of the kinetic energy