tailieunhanh - Báo cáo hóa học: " Research Article The Stochastic Ising Model with the Mixed Boundary Conditions"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article The Stochastic Ising Model with the Mixed Boundary Conditions | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 571950 17 pages doi 2009 571950 Research Article The Stochastic Ising Model with the Mixed Boundary Conditions Jun Wang Department of Mathematics College of Science Beijing Jiaotong University Beijing 100044 China Correspondence should be addressed to Jun Wang wangjun@ Received 16 December 2008 Revised 12 April 2009 Accepted 19 June 2009 Recommended by Veli Shakhmurov We estimate the spectral gap of the two-dimensional stochastic Ising model for four classes of mixed boundary conditions. On a finite square in the absence of an external field two-sided estimates on the spectral gap for the first class of weak positive boundary conditions are given. Further at inverse temperatures p pc we will show lower bounds of the spectral gap of the Ising model for the other three classes mixed boundary conditions. Copyright 2009 Jun Wang. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction and Definitions We consider the most popular ferromagnetic model of statistical physics which is the Ising model see 1-5 . The property of ferromagnetism comes from the quantum mechanical spinning of electrons. Because a small magnetic dipole moment is associated with the spin the electron acts like a magnet with one north pole and one south pole. Both the spin and the magnetic moment can be represented by an arrow which defines the direction of the electron s magnetic field. The spin can point up spin value 1 or down spin value -1 and it flips between the two orientations. Ferromagnetic models were invented in order to describe the ferromagnetic phase transition via a simple model. Considering the Ising model on the two-dimensional integer lattice Z2 at sufficiently low temperatures we know that the model exhibits a phase .