Mutual Information And Transfer Entropy In The Potts Model

In information theory,the mutual information is a measure of the mutual dependence between two variables which quantifies the amount of information obtained about one variable through another variable.Transfer entropy shares some of the desired properties of mutual information and also takes dynamical information into account.It can measure the information flow from one variable to another. Mutual information and transfer entropy have been widely applied in the study of phase transition and critical phenomenon in recent years.On the one hand,it provides a new perspective to understand the regulations of phase transition.On the other hand,many systems in the field of chemistry,biography and economics also have phase-transition-like behaviors.For the lack of well-defined order parameters, it is unlikely to study the phase transition of such systems in a conventional way.Therefore,mutual information and transfer entropy provide a method of studying and predicting phase transition for these systems.Study shows that in a ferromagnetic 2D lattice Ising model,pairwise and global mu-tual information based on Shannon entropy both at,tain their maximums at the critical point.Pairwise transfer entropy also peaks at the critical point,while global transfer en-tropy peaks on the disordered side.The results will be different if these two information measures are based on Renyi entropy.In 2D lattice Ising model,the Renyi pairwise and global mutual information still peak at the phase transition,which is the same as the Shannon counterparts. While the peak of Renyi pairwise and global transfer entropy both occur in the disordered phase.The phase transition of Ising model is second order.To explore the situation of first order phase transition,we extended the studies to the case of 2D q-state Potts model which has second order phase transition if q?4 and first order phase transition if q > 4.We first studied the Shannon mutual information and transfer entropy in 2D q-state Potts model,and compared with the results in Ising model. Using Glauber dynamics, we derived the expressions of pairwise and global mutual information and transfer entropy, and then obtained the plots of these physical quantities against temperature. The results show that no matter what value of q is,two kinds of mutual information and pairwise transfer entropy peak at the phase transition,and global transfer entropy peaks in the disordered phase.It is as same as the result in Ising model.Then we studied the two information measures based on Renyi entropy in Potts mod-el,and compared with the results in Ising model. We still used Glauber dynamics to derive the expressions of Renyi mutual information and transfer entropy,and got the same results with Ising model when q = 2.In Ising model,there are only third-order terms of (1/N) in the expressions of transfer entropy,while higher-order terms are not considered.But when q > 2,the expressions of transfer entropy in Potts model have both second-order and third-order terms.Then we obtained the plots of Renyi mutual information and transfer entropy against temperature.We found that both pairwise and global Renyi mutual information still attain their maximums at the critical point,and the two kinds of transfer entropy attain their maximums in the disordered phase,which is also the same as the result in Ising model.In conclusion,no matter based on Shannon entropy or Renyi entropy,the behaviors of mutual information and transfer entropy are irrelevant with the value of q,and are the same as the results in Ising model.Thus, We can conclude that the type of phase transition does not affect the behaviors of mutual information and transfer entropy in Potts model.