  It was recently shown that entanglement in quantum systems being in a
non-equilibrium state can appear at much higher temperatures than in an
equilibrium state. However, any system is subject to the natural relaxation
process establishing equilibrium. The work deals with the numerical study of
entanglement dynamics in a dipolar coupled spin-1/2 system under the transition
from a non-equilibrium state to an equilibrium state. The spin system is
characterized by a two-temperature density matrix, and the process of the
establishment of equilibrium is in the equalization of these temperatures. The
method of the non-equilibrium statistical operator is used to describe the
evolution of the system. The process of establishing an equilibrium state in
the homonuclear spin systems at low temperature was first considered. It was
shown that the time dependences of the inverse temperatures of the spin
subsystems are given by a solution of non-linear equations in contrast to the
linear equations in the well-known high temperature approximation. It was first
studied the entanglement dynamics during the equilibrium state establishment
and found that, during establishing equilibrium, the concurrence changes
non-monotonically with time and temperatures. Entanglement fades long before
equilibrium is established in the system. It was shown that the entanglement
dynamics depends strongly on the ratio of the Zeeman energy to the dipolar
energy. At a high ratio, the concurrence in the system decreases quickly for
time about 100 {\mu}s while, at a low ratio, establishment of equilibrium and
fading entanglement take prolonged time up to 1 ms.
