Rescaling invariance and anomalous energy transport in a small vertical column of grains

It is well known that energy dissipation and finite size can deeply affect the dynamics of granular matter, often making usual hydrodynamic approaches problematic. Here we report on the experimental investigation of a small model system, made of ten beads constrained into a 1D geometry by a narrow vertical pipe and shaken at the base by a piston excited by a periodic wave. Recording the beads motion with a high frame rate camera allows to investigate in detail the microscopic dynamics and test hydrodynamic and kinetic models. Varying the energy, we explore different regimes from fully fluidized to the edge of condensation, observing good hydrodynamic behavior down to the edge of fluidization, despite the small system size. Density and temperature fields for different system energies can be collapsed by suitable space and time rescaling, and the expected constitutive equation holds very well when the particle diameter is considered. At the same time, the balance between dissipated and fed energy is not well described by commonly adopted dependence due to the up-down symmetry breaking. Our observations, supported by the measured particle velocity distributions, show a different phenomenological temperature dependence, which yields equation solutions in agreement with experimental results.