Boltzmann factor for the energy change. The Monte Carlo method has been used to verify, with various methods, the critical temperature of the system, and is estimated to be . The Monte Carlo method can also compute average values that are used to compute thermodynamic quantities like magnetization, spin-spin correlation, correlation lengths, and specific heat. These are important ways to characterize the behavior of the system near the critical temperature. The magnetization and squared magnetization, for example, can be computed as where are the number of spins. The mean magnetization characterizes the magnitude of the net magnetic moment of the system; in many magnetic systems this is zero above a critical temperature and becomes non-zero spontaneously at low temperatures. Similarly the mean-squared magnetization characterizes the average of the square of net components of the spins across the lattice. Either of these are commonly used to characterize the order parameter of a system. Rigorous analysis of the XY model shows the magnetization in the thermodynamic limit is zero, and that the square magnetization approximately follows , which vanishes in the thermodynamic limit. Indeed, at high temperatures this quantity approaches zero since the components of the spins will tend to be randomized and thus sum to zero. However at low temperatures for a finite system, the mean-square magnetization increases, suggesting there are regions of the spin space that are aligned to contribute to a non-zero contribution. The magnetization shown (for a 25x25 lattice) is one example of this, that appears to suggest a phase transition, while no such transition exists in the thermodynamic limit. Furthermore, using statistical mechanics one can relate thermodynamic averages to quantities like specific heat by calculating The specific heat is shown at low temperatures near the critical temperature . There is no feature in the specific heat consistent with critical behavior (like a divergence) at this predicted temperature. Indeed, estimating the critical temperature comes from other methods, like from the helicity modulus, or the temperature dependence of the divergence of susceptibility. However, there is a feature in the specific heat in the form of a peak at . This peak position and height have been shown not to depend on system size, for lattices of linear size greater than 256; indeed, the specific heat anomaly remains rounded and finite for increasing lattice size, with no divergent peak. The nature of the critical transitions and vortex formation can be