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 Template:IMP/cite, Template:IMP/cite \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): Template:IMP/eq partition function of the system is: Template:IMP/eq which can be written as: Template:IMP/eq 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: Template:IMP/eq with Template:IMP/eq We have: Template:IMP/eq Indeed: Template:IMP/eq Thus, integrating successively over each variable, one obtains: Template:IMP/label Template:IMP/eq

This result can be obtained a powerful calculation method: the renormalization group methodTemplate:IMP/cite, Template:IMP/cite\index{renormalisation group} proposed by K. Wilson^[2]. Consider again the partition function: Template:IMP/eq where Template:IMP/eq Grouping terms by two yields to: Template:IMP/eq where Template:IMP/eq 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 $f_{K}$ .} Template:IMP/label

Calculation of sum over all possible values of $S_{i + 1}$ yields to: Template:IMP/eq 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 Template:IMP/eq Iterating the process, one obtains a sequence converging towards the partition function $Z$ defined by equation eqZisi.

↑
Ones says that a phase transition occurs.\index{phase transition} Historically, two sorts of phase transitions are distinguished Template:IMP/cite
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.
↑ Kenneth Geddes ilson received the physics Nobel prize in 1982 for the method of analysis introduced here.

Template:BookCat

[1] Ones says that a phase transition occurs.\index{phase transition} Historically, two sorts of phase transitions are distinguished Template:IMP/cite

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.

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] se 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.

[3] Coexistence of the various phases.

[4] Transition corresponds to a variation of entropy.

[5] xistence of metastable states.

[6] 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.

[7] symmetry breaking

[8] the entropy S is a continuous function of temperature and of the order parameter.

[2] Kenneth Geddes ilson received the physics Nobel prize in 1982 for the method of analysis introduced here.

[10] se 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.

[11] Coexistence of the various phases.

[12] Transition corresponds to a variation of entropy.

[13] xistence of metastable states.

[14] 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.

[15] symmetry breaking

[16] the entropy S is a continuous function of temperature and of the order parameter.

[1]

[2]

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

Navigation menu

Search