We investigate the statistical mechanics of the periodic one-dimensional Ising chain when the number of positive spins is constrained to be either an even or an odd number. We calculate the partition function using a generalization of the transfer ma
trix method. On this basis, we derive the exact magnetization, susceptibility, internal energy, heat capacity and correlation function. We show that in general the constraints substantially slow down convergence to the thermodynamic limit. By taking the thermodynamic limit together with the limit of zero temperature and zero magnetic field, the constraints lead to new scaling functions and different probability distributions for the magnetization. We demonstrate how these results solve a stochastic version of the one-dimensional voter model.
Uncoordinated individuals in human society pursuing their personally optimal strategies do not always achieve the social optimum, the most beneficial state to the society as a whole. Instead, strategies form Nash equilibria which are often socially s
uboptimal. Society, therefore, has to pay a price of anarchy for the lack of coordination among its members. Here we assess this price of anarchy by analyzing the travel times in road networks of several major cities. Our simulation shows that uncoordinated drivers possibly waste a considerable amount of their travel time. Counterintuitively,simply blocking certain streets can partially improve the traffic conditions. We analyze various complex networks and discuss the possibility of similar paradoxes in physics.
