Decoherence, wave function collapses and non-ordinary statistical mechanics

We consider a toy model of pointer interacting with a 1/2-spin system, whose x variable is measured by the environment, according to the prescription of decoherence theory. If the environment measuring the variable x yields ordinary statistical mechanics, the pointer sensitive to the 1/2-spin system undergoes the same, exponential, decoherence regardless of whether real collapses or an entanglement with the environment, mimicking the effect of real collapses, occur. In the case of non-ordinary statistical mechanics the occurrence of real collapses make the pointer still relax exponentially in time, while the equivalent picture in terms of reduced density matrix generates an inverse power law decoherence