Class Mattics The Thermodynamic Properties Of The Hamiltonian Mean Field Model

Long-range interactions are common in nature.Examples include:self gravitating systems,plasmas,dipolar magnets and wave-particle interactions.In this paper we mainly discuss the properties of the long-rage systems in equilibrium.Our main work is following:Firstly,we introduce the basic theory of statistical mechanics,we introduce the concept of Microcanonical ensemble,canonical ensemble and phase transition in the many-body systems;Secondly,we introduce the basic statistical theory of the long-range systems, include the problem of additivity,definition of long range systems,difficulties with the gravitational problem,applications to large systems and small systems,and thermodynamics and dynamical aspects of the systems;Thirdly,as a paradigmatic model for long-range interacting classical many-body systems,we introduce the Hamiltonian Mean Field(HMF)model where all particles interact with the same strength.The model is exactly solved in the canonical ensemble by a Hubbard-Stratonovich transformation.The equilibrium solution of the model in the canonical ensemble predicts a second order phase transition from a high temperature paramagnetic(PA)phase to a low temperature ferromagnetic(FE) one.As we know,HMF model where all particles interact with the same strength, but in the real world,interactions are complex.According to this,in this paper we consider the so called Mattics-like HMF model,a system of N particles rotating on a unit circle which are not fully coupled.The coupling bonds are J_ij.Inspired by Mattics spin-glass model,we ascribes J₀ξ_iξ_j to J_ij,andξ_i has a probability distribution.When c equals to zero,which means all particles are coupled fully,the Mattics-like HMF model turns to the traditional HMF model;when c equals to one, which means all particles are free.By means of a Replica-Symmetry analysis performed on an appropriate effective Hamiltonian,we will show that it is possible to find out a self-consistent equation for an order parameter describing,in the thermo dynamic limit,we show the solution of the equation.We find the critical temperature with the probability c,and it is well consistent with the known results.