Determination of specific heat of finite systems in micro-canonical Monte Carlo simulations

The 2d extended XY model has been simulated using micro-canonical Monte Carlo simulation. Simulations have been carried out for q=1 and 50 on 15 x 15 and 30 x 30 spin system with periodic boundary conditions. The vortex unbinding transition and the first order transition have been studied. The specific heat (C) is determined from C=dE/dT. C is also determined from the fluctuation of energy C=(<E^2>-<E>^2)/T^2 using the finite system as a model of the thermodynamic limit. In the later case an individual spin is assumed to be in contact with the spin system which ``appears'' as a ``heat bath.'' The present Monte Carlo estimation of C are compared with the existing spin-wave and vortex unbinding theoretical results and explained in terms of finite and thermodynamic systems. For q=1 we obtain the low temperature C to be 0.50 at T=0.11 for the 15 x 15 spin system.