Decoherence in the Heisenberg model

We study a simplified Heisenberg spin model in order to clarify the idea of decoherence in closed quantum systems. For this purpose, we define a new concept: the decoherence function \Xi(t), which describes the dynamics of decoherence in the whole system, and which is linked with the total (von Neumann) entropy of all particles. As expected, decoherence is understood both as a statistical process that is caused by the dynamics of the system, and also as a matter of entropy. Moreover, the concept of decoherence time is applicable in closed systems and we have solved its behaviour in the Heisenberg model with respect to particle number N, density \rho and spatial dimension D in a 1/r -type of potential. We have also studied the Poincare recurrences occurring in these types of systems: in an N=1000 particle system the recurrence time is close to the order of the age of the universe. This encourages us to conclude that decoherence is the solution for quantum-classical problems not only in practice, but also in principle.