Mixed Bose-Fermi statistics: kinetic equation and navigation through a network.
The conventional master equation is generalized to include Bose and Fermi moves. The obtained equation describes the time evolution of a system with mixed Bose-Fermi statistics which are controlled by the fraction of Fermi moves (α) ; the latter plays a role of an exploratory tendency (ET) factor. The theoretical consideration is illustrated with numerical results for navigation through a model scale-free network. Depending on the ET factor and the number of participants involved in the navigation (N(tot)) , a broad variety of the behavior scenarios has been observed. The cases of N(tot)=1 and N(tot)>>1 have been found drastically different. Of particular interest is the case of N(tot)>>1 , when the participants interfere with each other's motion, which is characteristic of many real systems (urban traffic, Internet, epidemic spreading, etc.). It has been found that at α<1 all participants reach a preselected (target) state but an excessive intention to try detour routes (α→1) leads to a critical slowdown in their passage to this state. At the same time, if α=1 (solely Fermi moves, no possibility to return to a previously visited state), only one of the participants attains the target state but it does it much faster than when a single copy navigates through the network (N(tot)=1) , i.e., this one optimizes the way to the target state at the expense of other participants. The procedure used to include Bose and Fermi moves in the master equation offers a framework for derivation of kinetic equations for more complex and general statistics.