Research Of The Phase Transition In Potts Model For Rumor

As an important social problem, rumor problem has been paid much attention for many years by physical researchers except for sociologists. On the one hand, the rumor problem is studied in order to realize the prediction for the transmission results and the control for the transmission process in factual life, which can provide the decision-making reference for the national security departments or enterprise relevant to secret. On the other hand, the researches on the phase transitions and ensembles equivalence in the physical system corresponding to the social system redound to understand the complicated system property. We study a Potts model which describes a simple rumor transmitting along with a chain constituted with N rumormongers. This transmitting chain is corresponded to a physical spin chain. We calculate the canonical and micro-canonical thermodynamic quantities of the spin system respectively according to the partition function and spin configuration number. Based on the analysis of the continuity of these thermodynamic functions, we obtain the transition results in the spin systems in the canonical and micro-canonical ensemble respectively. In the canonical ensemble, the free energy of the spin system is continuous all the time, but its first-order derivative, namely the entropy, has a discontinuous point in the effective temperature area; in the micro-canonical ensemble, the entropy is continuous in the whole energy area. This shows that there is a first-order transition in the spin system in a canonical ensemble but no in a micro-canonical ensemble. According to the transition results we draw a conclusion that the canonical and micro-canonical descriptions of the transitions' order in Potts model for rumor system are not equivalent. Because the Potts model for rumor is a new non-extensive energy spin model, this result provides a new example for the researches of ensembles equivalence in the frame of non-extensive statistical mechanics.