Tham khảo tài liệu 'computational physics - m. jensen episode 2 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . PHASE TRANSITIONS IN MAGNETIC SYSTEMS 189 where the vector spm contains the spin value sk 1. For the specific state Ex we have chosen all spins up. The energy of this configuration becomes then . The other configurations give and E -E44 J- 2. We can also choose so-called periodic boundary conditions. This means that if ỉ N we set the spin number to i 1. In this case the energy for the one-dimensional lattice reads .117 . and we obtain the following expression for the two-spin case E J siS2 S2S1 - . In this case the energy for Ex is different we obtain namely E1-En-U The other cases do also differ and we have E3 E - 2 J and Eấ Eu 2J. If we choose to use periodic boundary conditions we can code the above expression as jm N for . j 1 j N j energy spin j spin jm jm j 190 CHAPTER 11. MONTE CARLO METHODS IN STATISTICAL PHYSICS Table Energy and magnetization for the one-dimensional Ising model with N 2 spins with free ends FE and periodic boundary conditions PBC . State Energy FE Energy PBC Magnetization I n -2J 2 ĩ n 2J 0 3 ịt 2J 0 4 ịị -J 2J -2 Table Degeneracy energy and magnetization for the one-dimensional Ising model with N 2 spins with free ends FE and periodic boundary conditions PBC . Number spins up Degeneracy Energy FE Energy PBC Magnetization 2 1 -J 2J 2 1 2 J 2J 0 0 1 -J -2 The magnetization is however the same defined as N M . . where we sum over all spins for a given configuration i. Table lists the energy and magnetization for both free ends and periodic boundary conditions. We can reorganize Table according to the number of spins pointing up as shown in Table . It is worth noting that for small dimensions of the lattice the energy differs depending on whether we use periodic boundary conditions or fri ends. This means also that the partition functions will be different as discussed below. In the thermodynamic limit however N oo the final results do not depend on the kind of boundary conditions we choose. For
