# 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 . 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):

$H=-K\sum S_{l}S_{l+1}$ partition function of the system is:

$Z=\sum _{(S_{l})}\Pi _{l=0}^{N-1}e^{-KS_{l}S_{l+1}},$ which can be written as:

$Z=\sum _{(S_{l})}\Pi _{l=0}^{N-1}{\mbox{ ch }}K+S_{l}S_{l+1}{\mbox{ sh }}K.$ 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

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

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

$g(S_{i},S_{i+1},S_{i+2})=({\mbox{ ch }}K+S_{i}S_{i+1}{\mbox{ sh }}K)({\mbox{ ch }}K+S_{i+1}S_{i+2}{\mbox{ sh }}K)$ This grouping is illustrated in figure figrenorm. Sum over all possible spin values $S_{i},S_{i+1},S_{i+2}$ . The product$f_{K}(S_{i},S_{i+1})f_{K}(S_{i+1},S_{i+2})$ 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:

$\sum _{S_{i+1}}g(S_{i},S_{i+1},S_{i+2})=2({\mbox{ ch }}^{2}K+S_{i}S_{i+2}{\mbox{ sh }}^{2}K)$ 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

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