Space Dependent Mean Field Approximation Modelling
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It is shown that the self-consistency condition which is the basic equation for calculating the mean-field order parameter of any mean-field model Hamiltonian can be replaced by the standard Metropolis Monte Carlo scheme. The advantage of this method is its ease of implementation for both the homogeneous mean-field order parameter and the heterogeneous one. To be specific, the mean-field version of the Ising model spin system is discussed in detail and the resulting magnetization is the same as in the case of solving the respective mean-field self-consistency equation. In addition, it is shown that if a high temperature phase of such system is quenched below critical temperature then the mean field experienced by spins develops into a network of domains in analogous way as it happens with the spins in the case of the exact many-body Hamiltonian system and the coarsening processes start to take place. To show that the introduced Metropolis Monte Carlo method works also in case of the continuous variables the order parameter for the Maier-Saupe model for nematic liquid crystals has been calculated.
KeywordsMean field approximation Magnetic domains Coarsening Monte Carlo method
We thank professor Dietrich Stauffer for his comments on the mean-field Metropolis Monte Carlo algorithm as well as his suggestions concerning the paper. C. Zerafa and R. Cauchi acknowledge the support of the Strategic Educational Pathways Scholarship Scheme (Malta). These STEPS scholarships are part-financed by the European Union European Social Fund. B. Zapotoczny thanks for the PhD grant under Sub-Action 8.2.2 Regional Innovation Strategies, Activity 8.2 Know-How Transfer, Priority VIII Regional Business Personnel of the Human Capital Operational Programme, co-funded from the EU resources within the European Social Fund as well as the state budget and the Lubuskie Voivodship.
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