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 , magnetic moments orient themselves along a certain direction, and a non zero total magnetic moment is observedOnes says that a phase transition occurs.\index{phase transition} Historically, two sorts of phase transitions are distinguished ph:physt:Diu89 phase transition of first order (like liquid--vapor transition) whose characteristics are: <li>Coexistence of the various phases. <li>Transition corresponds to a variation of entropy. <li>existence of metastable states. second order phase transition (for instance the ferromagnetic--paramagnetic transition) whose characteristics are: <li>symmetry breaking <li>the entropy S is a continuous function of temperature and of the order parameter. . Ising model has been proposed to describe this phenomenom. It consists in describing each microscopic domain by a moment (that can be considered as a spin)\index{spin}, the interaction between spins being described by the following hamiltonian (in the one dimensional case): partition function of the system is: which can be written as: It is assumed that 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. represents the sum over all possible values of , 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. Partition function can be written as: with We have: Indeed: Thus, integrating successively over each variable, one obtains: eqZisi This result can be obtained a powerful calculation method: the renormalization group methodph:physt:Diu89, ma:equad:Schuster88\index{renormalisation group} proposed by K. WilsonKenneth Geddes ilson received the physics Nobel prize in 1982 for the method of analysis introduced here.. Consider again the partition function: where Grouping terms by two yields to: where This grouping is illustrated in figure figrenorm. center | frame |Sum over all possible spin values . The product is the sum over all possible values of spins and of a function deduced from by a simple change of the value of the parameter associated to function .} figrenorm Calculation of sum over all possible values of yields to: Function can thus be written as a second function with Iterating the process, one obtains a sequence converging towards the partition function defined by equation eqZisi.