Analytical And Numerical Study Of The HMF Model Outfield

As is known to all that long-range interaction system is very complicated,and presents some special properties.In order to have a further research of its properties,Hamilton-Mean field model,which is a representative model of long range interaction system,has been discussed for almost more than a dozen years,and get many important results.In recent years,some physicists study the HMF model with a magnetic field.In the absence of the magnetic field,the Hamiltonian of the HMF model is rotational symmetry,but the external magnetic field broke the property,and made the system appears some new properties.In the recent study about the HMF model with a magnetic field,some physicists adopt the method of phenomenological theory,and get some results.But this method is a kind of semi classical analytic method.The conclusion which is gotten by this method need the verification by experiments.So we hope to present the results by classical analytic method and numerical simulation method.On the one hand, we obtain the partition function of the system in the microcanonical ensemble and in the canonical ensemble respectively,based on the statistical mechanics,and also we get the field equations of systems under the mean-field approximation.we find that the field equation has the solution of the equilibrium state,the metastable state and the unstable state.When the system reaches equilibrium,as the energy or temperature changes.the system will turn out no phase transition,and the canonical ensemble is completely equivalent with the microcanonical ensemble in this state.When the external field strengthens,the metastable state or unstable state will disappear. On the other hand,we use the symplectic algorithm,which is one of the dynamic method,to have a numerical simulation of this system,and get the corresponding results when the system is at the equilibrium.By comparing the results which is gotten by theory analysis and numerical simulation,finally we find our method is feasible,the result is reliable,and also confirms that the results made by the phenomenological theory is reliable.