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Nature of lack-of-ergodicity in finite systems of two-dimensional Potts model Physics & Astronomy Key Words

Smita Ota, S Ota, S Ota, D Microcanonical, Monte Carlo, Potts

unpublished

The Potts model has been used to gain an understanding of phase transition in some well defined systems [1]. Moreover first-order phase transitions play an important role in the statistical mechanics of many physical phenomena of finite systems. In this context microcanonical Monte Carlo simulations play an important role [2]. The energy as a function of temperature for such systems shows a 'S' shape at the first-order transition. Therefore, understanding the 1 st order phase transition in this
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... microcanonical MC simulation technique has gained interest. In this communication we report the origin of the lack of ergodicity in 2D q-state Potts model [3]. The Hamiltonian of the q-state Potts model is given by H=-J ( i j) (1) where is the Kronecker delta. J is the interaction strength (>0 for the ferromagnetic case) and the sum is over all the nearest neighbors. The spin at the ith site i can take any one of the q different values. The Potts model has a first-order transition for q>4 and a higher order transition for q4. The thermodynamic first-order transition temperature is given by [4] k B T C /J=[ln(1+q)]-1 (2) We consider a 2d square lattice having N×N spins with periodic boundary condition and simulated the system with N=15 and q=10. For q=10 k B T C /J=0.70. Although periodic boundary condition is used in the simulations the system is 'finite' in the sense that the 'configurations' are defined due to the updating procedure. Initially all the spins are aligned in one state (i.e., state 1). This corresponds to the lowest energy state of the system. An extra degree of freedom called the 'demon' is allowed to move from one spin site to the another sequentially on the lattice as it exchanges energy with spins changing the microstate. The simulation starts with the demon having a fixed amount of energy (E D). This demon energy when added to the system energy (E S) corresponds to the total energy of the system. A random number in the interval [1,q] is generated which corresponds to a possible new state of the spin. The change in energy is calculated corresponding to this change in the spin state. A positive change in energy is allowed if the demon has sufficient energy. Otherwise the old spin state is retained. A negative or zero change in energy is always accepted and PACS: 05.50.+q; 05.70.Fh Two dimensional (2D) q-state Potts model has been studied in microcanonical ensemble using Monte Carlo simulations. For the microcanonical Monte Carlo simulation 15x15 spin system and q=10 state Potts model was chosen with periodic boundary condition. For large system energy the demon energy distribution is found to deviate from linearity and has the form exp(-E D +E D 2). is found to be zero near the first-order transition. We suggest a possible origin of localized two level system observed in glasses. Abstract Full Paper

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