10 100 M/m 10 −3 10 −2 FIG. 9: (Color online) Dimensionless compressibility − mL − $^{1}$( ∂L/∂M )$_{v}$$_{W}$ (filled symbols) and scaled position fluctuations of the piston σ 2 $_{Z}$mg 0 /Lb ( α, α$_{P}$ ) T$_{P}$ ( empty symbols) as a function of the mass ratio, M/m . The different symbols correspond to different values of the coefficient of restitution for the collision between particles, α , as indicated in the insert. The straight line has a slope − 3 / 4, and it is a guide for the eye. Nevertheless, the analysis carried out in ref. [19] indicates that the range of hydrodynamic gradients considered here exceeds the limit of validity of the Navier-Stokes approximation. Some of the main results can be summarized as follows: i) for large mass of the movable piston compared with the mass of the gas particles, the volume fluctuations are Gaussian with very good accuracy, ii) the square root of the second moment of their distribution scales with the square of the velocity of the vibrating wall at the bottom, i.e. in the same way as the amplitudes of the hydrodynamic fields in the gas, iii) by requiring the same relation between volume fluctuations and compressibility as in equilibrium systems to be verified, an “effective temperature” can be defined, iv) the effective temperature turns out to be proportional to the second moment of the velocity fluctuations of the piston, with a proportionality parameter that depends on the inelasticity of both the particle-particle 18