Numerical methods and programming
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Num. Meth. Prog.: Year: Volume: Issue: Page: Find

 Num. Meth. Prog., 2012, Volume 13, Issue 3, Pages 452–464 (Mi vmp51)

Âû÷èñëèòåëüíûå ìåòîäû è ïðèëîæåíèÿ

Optimal Gaussian approximation in the Ising model

N. B. Melnikova, Yu. A. Romanenkob

a Central Economics and Mathematics Institute, RAS, Moscow
b M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: The effect of spin fluctuations on the magnetic phase transition in the Ising model is studied. The calculation of basic characteristics is reduced to the integration over configurations of a stochastic (fluctuating) field. To evaluate the integrals, the optimal Gaussian approximation of the fluctuating field is constructed. An explicit expression for the system of nonlinear equations that defines the parameters of the optimal Gaussian approximation at each value of temperature is obtained. It is shown that, for weak interaction of spins, the temperature of the phase transition becomes smaller than that in the mean-field theory, but the phase transition remains second order. With an increase of interaction, the solution becomes nonunique at high temperatures and a jump first-order phase transition is observed.

Keywords: fluctuating-field theory; ferromagnetism; Stratonovich-Hubbard transformation; free energy minimum principle; parameter differentiation method.

Full text: PDF file (243 kB)
UDC: 517.97:51-72:519.677:537.9

Citation: N. B. Melnikov, Yu. A. Romanenko, “Optimal Gaussian approximation in the Ising model”, Num. Meth. Prog., 13:3 (2012), 452–464

Citation in format AMSBIB
\Bibitem{MelRom12} \by N.~B.~Melnikov, Yu. A. Romanenko \paper Optimal Gaussian approximation in the Ising model \jour Num. Meth. Prog. \yr 2012 \vol 13 \issue 3 \pages 452--464 \mathnet{http://mi.mathnet.ru/vmp51}