  The ground-state phonon pressure is an analogue to the famous Casimir
pressure of vacuum produced by zero-point photons. The acoustic Casimir forces
are, however, many orders of magnitude weaker than the electromagnetic Casimir
forces, as the typical speed of sound is 100 000 times smaller than the speed
of light. Because of its weakness, zero-point acoustic Casimir pressure was
never observed, although the pressure of artificially introduced sound noise on
a narrow aperture has been reported. However, the magnitude of Casimir pressure
increases as $1/L^3$ with the decrease of the sample size $L$, and reaches
picoNewtons in the sub-micron scales. We demonstrate and measure the acoustic
Casimir pressure induced by zero-point phonons in solid helium adsorbed on a
carbon nanotube. We have also observed Casimir-like "pushing out" thermal
phonons with the decreasing temperature or the length. We also show that all
thermodynamic quantities are size-dependent, and therefore in the mesoscopic
range $L\lesssim\hbar{c}/(k_BT)$ quadruple points are possible on the phase
diagram where four different phases coexist. Due to the smallness of solid
helium sample, temperature of Bose-Einstein condensation (BEC) of vacancies is
relatively high, $10-100$ mK. This allowed us to experimentally discover the
BEC in a system of zero-point vacancies, predicted more than 50 years ago.
