Distributing Action instead of Energy

According to Oudet 2006, the condition of thermal equilibrium cannot leave out thermal perturbations. So we are led to reformulate the standard canonical ensemble apparatus for 1D oscillators distributing action quanta instead of energy. In this frame, describing action fluctuations by ±h/2 is also straightforward and optimal match is found with our fluctuation thermodynamics. The Lagrange multiplier and perturbated specific heat come out as functions of temperature, quantum energy, quantum entropy. We define the thermodynamic potentials in the action domain and demonstrate a general distribution form (Bose-Einstein like) fit for even strongly anharmonic oscillators (at least, with specific heat cv > 0). We show equivalence with quantum behavior as the result of fluctuations in presence of the modal constraint. We extend previous definitions and computing of out-of-equilibrium entropy to the general anharmonic case. As stated in previous work already, the thermal equilibrium condition is the equality between the entropy difference across the fluctuation interval and the corresponding thermodynamical step.