  Recently, it has been observed that the effective dipolar interactions
between nuclear spins of spin-carrying molecules of a gas in a closed
nano-cavities are independent of the spacing between all spins. We derive exact
time-dependent polarization for all spins in spin-1/2 ensemble with spatially
independent effective dipolar interactions. If the initial polarization is on a
single (first) spin,$P_1(0)= 1$ then the exact spin dynamics of the model is
shown to exhibit a periodical short pulses of the polarization of the first
spin, the effect being typical of the systems having a large number, $N$, of
spins. If $N \gg 1$, then within the period $4\pi/g$ ($2\pi/g$) for odd (even)
$N$-spin clusters, with $g$ standing for spin coupling, the polarization of
spin 1 switches quickly from unity to the time independent value, 1/3, over the
time interval about $(g\sqrt{N})^{-1}$, thus, almost all the time, the spin 1
spends in the time independent condition $P_1(t)= 1/3$. The period and the
width of the pulses determine the volume and the form-factor of the ellipsoidal
cavity. The formalism is adopted to the case of time varying nano-fluctuations
of the volume of the cavitation nano-bubbles. If the volume $V(t)$ is varied by
the Gaussian-in-time random noise then the envelope of the polarization peaks
goes irreversibly to 1/3. The polarization dynamics of the single spin exhibits
the Gaussian (or exponential) time dependence when the correlation time of the
fluctuations of the nano-volume is larger (or smaller) than the $<(\delta g)^2
>^{-1/2} $, where the $<(\delta g)^2>$ is the variance of the $g(V(t))$
coupling. Finally, we report the exact calculations of the NMR line shape for
the $N$-spin gaseous aggregate.
