Introduction to Mathematical Physics/N body problems and statistical equilibrium/Ising Model

In this section, an example of the calculation of a partition function is presented. The Ising model [ma:equad:Schuster88], [ph:physt:Diu89] \index{Ising} is a model describing ferromagnetism\index{ferromagnetic}. A ferromagnetic material is constituted by small microscopic domains having a small magnetic moment. The orientation of those moments being random, the total magnetic moment is zero. However, below a certain critical temperature T_c, magnetic moments orient themselves along a certain direction, and a non zero total magnetic moment is observed[1] . Ising model has been proposed to describe this phenomenom. It consists in describing each microscopic domain by a moment S_i (that can be considered as a spin)\index{spin}, the interaction between spins being described by the following hamiltonian (in the one dimensional case):

H=-K\sum S_lS_{l+1}

partition function of the system is:

Z=\sum_{(S_l)}\Pi_{l=0}^{N-1}e^{-KS_lS_{l+1}},

which can be written as:

Z=\sum_{(S_l)}\Pi_{l=0}^{N-1}\mbox{ ch } K +S_lS_{l+1}\mbox{ sh } K.

It is assumed that S_l can take only two values. Even if the one dimensional Ising model does not exhibit a phase transition, we present here the calculation of the partition function in two ways. \sum_{(S_l)} represents the sum over all possible values of S_l, it is thus, in the same way an integral over a volume is the successive integral over each variable, the successive sum over the S_l's. Partition function Z can be written as:

Z=\sum_{S_1}\dots\sum_{S_n}f(S_1,S_2)f(S_2,S_3)\dots

with

f_{K}(S_i,S_{i+1})=\mbox{ ch } K +S_iS_{i+1}\mbox{ sh } K

We have:

\sum_{S_1}f(S_1,S_2)=2 \mbox{ ch } K.

Indeed:

\sum_{S_1}f(S_1,S_2)=\mbox{ ch } K +S_2 (+1)\mbox{ sh } K + \mbox{ ch } K +S_2 (-1) \mbox{ sh } K.

Thus, integrating successively over each variable, one obtains:

eqZisi

Z=2^{n-1} (\mbox{ ch } K)^{n-1}

This result can be obtained a powerful calculation method: the renormalization group method[ph:physt:Diu89], [ma:equad:Schuster88]\index{renormalisation group} proposed by K. Wilson[2]. Consider again the partition function:

Z=\sum_{S_1}\dots\sum_{S_n}f_K(S_1,S_2)f_K(S_2,S_3)\dots

where

f_{K}(S_i,S_{i+1})=\mbox{ ch } K +S_iS_{i+1}\mbox{ sh }K

Grouping terms by two yields to:

Z=\sum_{S_1}\dots\sum_{S_n}g(S_1,S_2,S_3).g(S_3,S_4,S_5)\dots

where


g(S_i,S_{i+1},S_{i+2})=(\mbox{ ch } K +S_iS_{i+1}\mbox{ sh }K)(\mbox{ ch } K +S_{i+1}S_{i+2}\mbox{ sh }K)

This grouping is illustrated in figure figrenorm.

Sum over all possible spin values math>S_{i},S_{i+1},S_{i+2}</math>. The productf_K(S_i,S_{i+1})f_K(S_{i+1},S_{i+2}) is the sum over all possible values of spins S_{i} and S_{i+2} of a function f_{K'}(S_{i},S_{i+2}) deduced from f_K by a simple change of the value of the parameter K associated to function f_K.}
figrenorm

Calculation of sum over all possible values of S_{i+1} yields to:

\sum_{S_{i+1}}g(S_i,S_{i+1},S_{i+2})=2(\mbox{ ch }^2K+S_{i}S_{i+2}\mbox{ sh }^2K)

Function \sum_{S_{i+1}}g(S_i,S_{i+1},S_{i+2}) can thus be written as a second function f_{K'}(S_i,S_{i+2}) with

K'=\mbox{ Arcth }(\mbox{ th }^2K).

Iterating the process, one obtains a sequence converging towards the partition function Z defined by equation eqZisi.

  1. Ones says that a phase transition occurs.\index{phase transition} Historically, two sorts of phase transitions are distinguished [ph:physt:Diu89]
    1. phase transition of first order (like liquid--vapor transition) whose characteristics are:
      • Coexistence of the various phases.
      • Transition corresponds to a variation of entropy.
      • existence of metastable states.
    2. second order phase transition (for instance the ferromagnetic--paramagnetic transition) whose characteristics are:
      • symmetry breaking
      • the entropy S is a continuous function of temperature and of the order parameter.
  2. Kenneth Geddes ilson received the physics Nobel price in 1982 for the method of analysis introduced here.
Last modified on 31 May 2009, at 08:02