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Monte Carlo simulation of two dimensional 4 state Potts model has been carried out in microcanonical ensemble. The simulations were done on a 30 × 30 system with periodic boundary conditions. The temperature dependence of energy and order parameter has been calculated. The transition in 4-state Potts model is concluded to be first-order in nature. The transition temperature and latent heat of the first-order transition have been found to be 0.92 and 0.18, respectively.

The two dimensional (2D) q-state Potts model undergoes a well studied transition (first-order or higher-order) and provides a system of increasing complexity as q increases [

In the Potts model the spin at the ith site

where J is the interaction strength (>0 for the ferromagnetic case) and the sum is over all the nearest neighbors on a square lattice. It has been suggested that the Potts model has a first-order transition for

We consider a 2D square lattice having 900 spins with periodic boundary conditions and simulated the system for

where

where

The equilibration and the nature of the fluctuation of the order parameter with this algorithm has been studied before and has been found that 1 × 10^{5} Monte Carlo step per spin (MCSS) are sufficient for equilibration and averaging of the physical quantities.

The expression for the exact value of the transition temperature (

For ^{5} MCSS for equilibration and 4 × 10^{5} MCSS for averaging. The simulation constituted of a cooling run followed by a heating run. The physical quantities computed are averages of the heating and cooling runs. ^{st} order transition is characterized by three regions in T vs. E which are high and low temperature regions and the latent heat region with constant temperature. Monte Carlo simulation in microcanonical ensemble, a 1^{st} order transition appears as S'-bend. The 1^{st} order transition is characterized by negative specific heat which is the equilibrium response of a finite isolated system [^{7} in the 30 × 30 spin system. Therefore, we conclude that 4-state Potts model has 1^{st} order transition. The latent heat involved in this transition is about 0.18 and T_{C} agrees with the exact value.

In conclusion we have studied the 2D Potts model for

The transition temperature has been found to be 0.92 and the latent heat involved in the transition is about 0.18.

Ota, S.B., Ota, S. and Ota, A. (2017) Microcanonical Monte Carlo Simulation of 2D 4-State Potts Model. Journal of Modern Physics, 8, 602-606. https://doi.org/10.4236/jmp.2017.84040