Mechanics and thermodynamics of the "Bernoulli" oscillators (uni-dimensional closed motions) Part I : Historical review and recent assessments

In this paper, we join two different theoretical approaches to the problem of finding a classical-like interpretation of quantum effects: the "fluctuating energy" model and the "hidden variables" model. We show that they merge together into a more powerful, comprehensive one. A basic assumption in the last is that a ''vacuum interaction'' is responsible for energy fluctuations, and splits the classical motion into two different components. These are as it was first identified, on general grounds, by Kapitza in a famous theorem: a "super-oscillation" of the particle position and velocity around a center; and the motion of the center itself, respectively. These motions define what we call the hidden degree(s) of freedom HDF, figuring in the energy theorem through a peculiar potential that we show correlated with a mass effect. The implicated functions for each energy level are called "the mass eigenfunctions". Classical oscillators submitted to these vacuum perturbations exhibit a quantum-like behavior. We name them the Bernoulli oscillators, because their properties came out in a frame where the mass-flow theorem takes a dominant role. A brief historical review and recent assessments are given in the present Part I of the work. Two concrete examples will be solved numerically in the following Part II; the results will also allow us to give insight into the classical limit peculiar to the model, so that this last will be found there expounded afterwards.