  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
