Mean field theory

The mean field theory (English mean-field theory ) is an approximation, the systems of interacting particles considered as systems of free particles in an external field. The external field is regarded as a constant and thus does not take into account that each particle is locally changed by his conduct the field ( ie fluctuations are neglected).

Although this approximation for many sizes quantitatively imprecise values ​​arise, they will have many qualitative evidence of the scaling behavior, ie the critical exponents at phase transitions. The mean field theory is closely related to the Landau theory of phase transitions.

The molecular field theory finds frequent application in statistical physics or statistical thermodynamics, including in determining the permittivity of polarizable media, the Ising model (lattice of N spins) and the van der Waals theory (liquids ), with the relationship between the Ising model and the fluid theory of the so-called lattice gas interpretation of the former model gives ( ' spin up' ' lattice site is occupied ', ' spin down ' lattice site is empty ').

Formally, consider the mean-field theory the state with the largest contribution to the partition function, which is why it is also referred to as classical approximation or molecular field.

Example: N -spin system

A system of N spins is characterized by its Hamiltonian:

Wherein the first term represents the contribution of energy by the interaction of the spins with an external magnetic field and the second term is the interaction of the pins to one another, the input of which is different in the interaction matrix of zero.

In the sense of the mean-field theory, the interaction term is then estimated by the spins replaced by the average of the whole system. The average spin of the system is:

The expected value of a single spin is then in the molecular field approximation. The Hamiltonian

Is thus

Said.

In a further assessment is assumed to be the same for all. The term in the bracket is now independent of the various interactions in the system and can be understood as an effective external magnetic field. This can now be used in the solutions to the problem of free spins ( ) instead of the magnetic field. In the case of a along the z - axis oriented magnetic field resulting from the expectation value of the z component of the spinning sum S:

With the Brillouin function for spin S, the expected value for interacting spins

The mean-field theory neglects correlations of physical magnitudes, ie, it is assumed that. It follows immediately that the mean-field theory at the critical point of a phase transition, and in its vicinity, collapses.

Generalizations

The core of the theory is that a complicated operator a linear approximation, that a Einteilchennäherung is made. Analogously, one can trace a complicated many-body theory to an optimally adapted Einteilchentheorie eg in quantum theory by approximating the Hamiltonian, for example, by the corresponding Hartree- Fock approximation or fitting so-called quasiparticles introduced.