**The Thermodynamics of a Single Bead in a
Vibrating Container**

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**Abstract:
**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 =n =2p/w of vibration and of the collision restitution coefficient
r=v_{out}/v_{in}. 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 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>/(bw)] » 1 when r<0.4-0.5 . Relative standard
deviation DV/V is found to be approximately constant, i.e. DV/V » 0.3 except when resonance; in this case, memory
effect and non ergodicity effects are observed and become too important .* *

**Pacs # : **5.40 ; 45.70 ; 64 ;
83.70.Fn

*poudres
& grains* **12** (2) 17-42
(mars 2001)