Traffic of indistinguishable particles in complex networks

In this paper, we apply a simple walk mechanism to the study of thetraffic of many indistinguishable particles in complex networks. Thenetwork with particles stands for a particle system, and everyvertex in the network stands for a quantum state with thecorresponding energy determined by the vertex degree. Although theparticles are indistinguishable, the quantum states can bedistinguished. When the many indistinguishable particles walkrandomly in the system for a long enough time and the system reachesdynamic equilibrium, we find that under different restrictive conditionsthe particle distributions satisfy different forms, including theBose--Einstein distribution, the Fermi--Dirac distribution and thenon-Fermi distribution (as we temporarily call it). As for theBose--Einstein distribution, we find that only if the particle density islarger than zero, with increasing particle density, do more and moreparticles condense in the lowest energy level. While the particledensity is very low, the particle distribution transforms from thequantum statistical form to the classically statistical form, i.e.,transforms from the Bose distribution or the Fermi distribution tothe Boltzmann distribution. The numerical results fit well with theanalytical predictions.