Exact representation of crossover of transitions from first order to second order in the Potts model for rumor transmission.
The L-state Potts model for rumor is the N-spin chain describing how a simple rumor transmitted by N recreant rumormongers is aggrandized. The studied rumor is represented mathematically by a simple proposition with the universal quantifier, which again is represented geometrically by a point in a proposition space. During the transmission, such a proposition is changed with the change of the rumor, which has individual number N0 at the beginning of the transmission. Correspondingly, the point expressing the proposition may start from an arbitrary site at the proposition space, and then it shifts in the space. Thus, a spin sum of the Potts model corresponds to a walk of a point in the proposition space and spin configuration's numbers is given by enumerating the corresponding walks. The concept of the lattice path in combinatorial mathematics is introduced and the exact series representation of the configuration's numbers is given. The partition function exhibits the transition of the chain and critical equivalent inverse temperature beta(c) is determined. Moreover, there is a crossover value of the individual number, N00. The model has a first-order transition when N0<N00 and a second-order one when N0>N00.
Shao
C G
CG
Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China.
Liu
Z Z
ZZ
Wang
J F
JF
Luo
J
J