### abstract

- It was recently shown that entanglement in quantum systems being in a nonequilibrium 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 nonequilibrium 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 nonequilibrium 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 dependencies of the inverse temperatures of the spin subsystems are given by a solution of nonlinear equations in contrast to the linear equations in the well-known high-temperature approximation. We have first studied the entanglement dynamics during the equilibrium state establishment and have found that the concurrence changes nonmonotonically with time and temperatures while equilibrium is being established. 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 a time of about 100 $\ensuremath{\mu}\text{s}$ whereas, at a low ratio, establishment of equilibrium and fading entanglement takes a prolonged time up to 1 $\text{ms}$.