The Thermodynamics of a Single Bead in a Vibrating Container
  Statistics of the 1d dynamics of a single particle in a box of length L is studied under different conditions of periodic excitation: vibrated container, or 1 wall moving periodically or two walls vibrating in opposite phases. Theoretical predictions for the mean typical speed <v> are derived using some Random Phase Approximation (RPA) for small b/L ratio, as functions of the amplitude b and frequency f of vibration and of the collision restitution coefficient r=vout/vin. They are compared to numerical simulations. It is concluded that RPA is valid for b/L>0.005, that long time memory, i.e. non ergodicity, and/or resonance develop at large restitution coefficient r (r>0.95) and/or at large b/L, that <v> scales always as f, and scales as b except for large b/L values, i.e. b/L>0.02, and that the relative velocity <V>=[<v>/(bf)] is about 1 when r<0.4-0.5 . Relative standard deviation DV/V is found to be approximately constant, i.e. DV/V is about 0.3 except when resonance occurs; in this case, memory effect and non ergodicity effects are observed and become too important . Pacs # : 5.40 ; 45.70 ; 64 ; 83.70.Fn 
P. Evesque