Phase transition in d-dimensional long-range interacting systems

Abstract

We present new iterative method for the derivation one-point distribution function in the equilibrium state. For derivation of the distribution function, we must solve Lane-Emden equation. In general case, we solve the equation with iterative method. However the traditional method is not ensured convergence of the algorithm, we cannot often obtain solutions. In order to obtain the stable stationary distribution function, we apply an iterative method, inspired by a previous one used in 2D turbulence. Our method ensures entropy increase and convergence of the algorithm. Furthermore, our method can obtain the distribution function quickly [1]. Here we present the phase transition in longrange interacting systems. The Hamiltonian Mean-Field (HMF) model describes the motion of globally coupled particles on a 1D circle. Nevertheless the interaction is described only one cosine function, both the dynamical and the thermodynamical properties of this mode are quite various and complicated. For the HMF model, the extension for 2D model had been proposed [3]. We have extended the HMF model for 3D and 4D models. Then we analyze the phase-transition for d-dimensional models. The Hamiltonian of the HMF models is written as