Transient temperature and mixing times of quantum walks on cycles
  The definition of entanglement temperature for the quantum walk on the line is extended to $N$-cycles, which are more amenable to a physical implementation. We show that, for these systems, there is a linear connection between the thermalization time and the mixing time, and also that these characteristic times become insensitive to the system size when $N$ is larger than a few units 
Nicol\'as D\'iaz, Raul Donangelo, Renato Portugal, and Alejandro   Romanelli
quant-ph