Ratio of canonical and microcanonical temperatures of a vibratory antiferromagnetic ising chain

A quite simple statistical principle claims that the measured value of a thermodynamic observable of a system and the ensemble average or the time average of the observable are identical at the thermodynamic limit. Some examples of the principle were given in canonical ensemble, which describes the thermodynamics of a closed system @1#. For an isolated classical Hamiltonian system, the time average of an observable of a system may be replaced by the average of the microcanonical ensemble. In present understanding, there is an equivalent canonical ensemble for the system, which gives the same average values for all observables at the thermodynamic limit @1#. A typical observable is temperature T. In a canonical ensemble, the temperature is a free parameter, which determines the internal energy U5U(Tc) of a system. But in a microcanonical ensemble the energy E of a system is a free parameter, and the temperature is an observable to be calculated as Tm5Tm(E) @1,2#. Here the subscripts c and m represent the canonical and the microcanonical ensemble averages, respectively. The equivalence of the two ensembles means that Tm5Tc for E5U at the thermodynamic limit. Rugh proposed a method calculating the reciprocal temperature in a microcanonical ensemble and deduced an equation of the ratio Tc /Tm for some classical interaction model @3,4#. He gave the result that the ratio Tc /Tm differs from one by a term of order N under the natural assumption on the fluctuations in the kinetic energy @4,5#. However, the assumption, up to now, lacks a rigorous argument for a system with interaction because of the difficulty in the analytical calculation. One-dimension ferromagnetic and antiferromagnetic chains have been widely considered in statistical physics and nonlinear physics @1,2,6,7#. In this paper, we give a descriptive example to calculate the ratio Tc /Tm of a classical Hamiltonian system, which is an antiferromagnetic Ising chain with N spins and its sites vibrate harmonically. Our model is closer to a practical system. The ratio Tc /Tm is given by a rigorous analytical calculation. The result is Tc /Tm511O(N ), which is consistent with the natural assumption @4,5#. The Ising chain has N sites labeled by qi , i51, . . . ,N and there is a spin Si on every site. Moreover, every site vibrates as a three-dimension harmonic oscillator. Then the total Hamiltonian of the system is