Locally ordered regions and the first-order phase transition in the disordered systems

The sixth-order Landau-Ginzburg Hamiltonian with random temperature and the locally ordered regions (LOR) are studied to investigate the effect of disorder on the first-order (discontinuous) phase transition. The disordered system is modelled as a lattice, on which each cell has a local reduced temperature. The random part of the local reduced temperature is distributed in the Gaussian form. The one-dimensional saddle point equation of the Landau-Ginzburg Hamiltonian with such a random temperature is solved numerically. The average, the fluctuation and the correlation length of the solution are calculated. The scaling relations for these quantities with the temperature and the disorder strength are derived. The scaling relations agree with the numerical data well. The implications of this model on the experiments are discussed.