Generalized approach to description of energy distribution of spin system

Abstract

Abstarct. We examined energy spectrums of some particular systems of N binary spins. It is shown that the configuration space can be divided into N classes, and in the limit N   the energy distributions in these classes can be approximated by the normal distributions. For each class we obtained the expressions for the first three moments of the energy distribution, including the case of presence of a nonzero inhomogeneous magnetic field. We also derived the expression for the variance of the quasienergy distribution in the local minimum. We present the results of computer simulations for the standard Ising model and the Sherrington-Kirkpatrick and Edwards-Anderson models of spin glass. Basing on these results, we justified the new method of the partition function calculation.

Applicability of n-vicinity method for calculation of free energy of Ising model

“…In a general case, the distribution of the energies of the states from the n-vicinity is unknown. However, in [12,13] we found the exact expressions for the mean energy n E and the variance 2 n  in terms of the characteristics of the connection matrix. Numerous computer simulations show that the distribution of energies from n  is a single-mode one and the Gaussian density with the calculated parameters n E and 2 n  approximates it rather accurately (see [13]).…”

“…In [11][12][13] we proposed the n-vicinity method as a way of approximate calculation of the partition function for an arbitrary connection matrix. The main idea of our method is as follows.…”

“…Then in the expression for the partition function F x E depends on the inverse temperature  , the external magnetic field H (when it is present) and the characteristics of the connection matrix. In the general form the expression for ( , ) F x E was obtained in [12,13].…”

n-vicinity Method and 1D Ising Model

“…We denote by () n DE a true energy distribution of states from n  . As a rule, we do not know the function () n DE but we can obtain exact expressions for the mean n E and the variance 2 n  of this unknown distribution [5]. In the framework of the n-vicinity method, we replace the unknown distribution () n DE by a normal distribution with the mean n E and the variance 2 n  :…”

“…The justification of this method we published in [5] and the boundaries of its applicability for Ising models were examined in [1].…”