Nonergodic dynamics of a system of nuclear spins $1/2$ with identical spin-spin coupling constants

The exact solution for the evolution of nuclear spin polarizations in a system with spin-spin coupling constant $g$ identical for all spin pairs is obtained on the condition that only one (first) spin is polarized at zero time. It is shown that the polarization $P_1(t)$ of the first spin has the form of periodic time pulsations with the period $4\pi/g$. In every period, the function $P_1(t)$ changes from its initial value $P(0)=1$ to $1/3$ during the time on the order of $t\approx4\pi/Ng$, if the number of spins $N\gg1$, and remains in the state $P_1(t)=1/3$ virtually during the entire period. A simple classical model within the framework of mean-field theory explains the physical nature of the nonergodic dynamics of the system.