20 70 120 170 M/m 0 0.5 1 T$_{eff}$/T$_{P}$ FIG. 7: (Color online) Ratio of the effective temperature, T$_{eff}$ , defined in Eq. (12) to the temper- ature parameter of the piston, T$_{P}$ , as a function of the mass ratio for the same systems as in Fig. 5. linearly with 1 − α $^{2}$, at least in the considered interval, 0 . 85 ≤ α ≤ 0 . 98. It is worth to note that some care is needed when extrapolating to the elastic limit α → 1 these results. In this limit, a stationary state is only possible if, in addition, the vibrating wall is arrested, i.e. also the limit v$_{W}$ → 0 is taken. But all the previous discussion has been carried out at constant velocity of the vibrating wall. This explains why extrapolation of the linear fitting in Fig. 8 does not lead to b = 1, and this does not mean any kind of contradiction. In Fig. 9, the dimensionless compressibility, − mL − 1 ( ∂L/∂M )$_{v}$$_{W}$ and the scaled second moment of the position fluctuations σ 2 $_{Z}$mg 0 /Lb ( α, α$_{P}$ ) T$_{P}$ have been plotted as functions of M/m . A logarithmic representation is employed. Data for several values of the coefficient α have been included. For all the data α$_{P}$ = 0 . 99. If T$_{eff}$ had been used instead of b ( α, α$_{P}$ ) T$_{P}$ , the two plotted quantities would be the same by definition, i.e. the filled and empty symbols would agree in all the cases. It is seen that the dependence on M/m of the dimensionless compressibility is accurately described by a power law of the form ( M/m ) − 3 / $^{4}$, indicated in 16