Considering different universality classes of the QCD phase transitions, we perform the Monte Carlo simulations of the 3-dimensional $O(1, 2, 4)$ models at vanishing and non-vanishing external field, respectively. Interesting high cumulants of the order parameter and energy from O(1) (Ising) spin model, and the cumulants of the energy from O(2) and O(4) spin models are presented. The critical features of the cumulants are discussed. They are instructive to the high cumulants of the net baryon number in the QCD phase transitions.
The phase transition of hard-sphere Heisenberg and Neutral Hard spheres mixture fluids has been investigated with the density functional theory in mean-field approximation (MF). The matrix of second derivatives of the grand canonical potential $Omega$ with respect to the total density, concentration, and the magnetization fluctuations has been investigated and diagonalized. The zero of the smallest eigenvalue $lambda_s$ signalizes the phase instability and the related eigenvector $textbf{x}_s$ characterizes this phase transition. We find a Curie line where the order parameter is pure magnetization and a mixed spinodal where the order parameter is a mixture of total density, concentration, and magnetization. Although in the fixed total number density or temperature sections the obtained spinodal diagrams are quite similar topology, the predominant phase instabilities are considerable different by analyzing $textbf{x}_s$ in density-concentration-magnetization fluctuations space. Furthermore the spinodal diagrams in the different fixed concentration are topologically different.
