Molecular field theory

The mean-field theory (engl. Mean field theory ) is an approximation , the systems of interacting particles and systems of free particles in an external field viewed. The external field is regarded as constant and therefore does not take into account that each particle changes the field locally due to its behavior (i.e. fluctuations are neglected).

Although this approximation results in quantitatively inaccurate values for many quantities , it gives numerous qualitative information on the scale behavior , i.e. on the critical exponents at phase transitions . The molecular field theory is closely related to the Landau theory of phase transitions.

In formal terms, molecular field theory considers the state with the greatest contribution to the sum of states , which is why it is also referred to as classical approximation or molecular field approximation .

Applications

The molecular field theory is often used in statistical physics or statistical thermodynamics , etc. a. in determining the permittivity of polarizable media, in the Ising model (grid of N  spins ) and in the Van der Waals theory (liquids). The relationship between the Ising model and the fluid theory results from the lattice gas interpretation of the Ising model ( spin up 'lattice site is occupied', spin down 'lattice site is empty'). ${\displaystyle {\hat {=}}}$ ${\displaystyle {\hat {=}}}$

Example: N-spin system

A system of spins is characterized by its Hamilton operator : ${\displaystyle N}$

${\displaystyle {\begin{aligned}{\hat {H}}&=-\sum _{j}g\mu _{\mathrm {B} }{\hat {{\vec {S}}_{j}}}{\vec {B}}-\sum _{i,j}J_{ji}{\hat {{\vec {S}}_{j}}}{\hat {{\vec {S}}_{i}}}\\&=-\sum _{j}g\mu _{\mathrm {B} }{\hat {{\vec {S}}_{j}}}\left({\vec {B}}+{\frac {1}{g\mu _{\mathrm {B} }}}\sum _{i}J_{ji}{\hat {{\vec {S}}_{i}}}\right)\ ,\end{aligned}}}$

in which

• the first term the energy contribution through the interaction of the spins with the magnetic flux density of an external magnetic field ${\displaystyle {\hat {\vec {S}}}}$ ${\displaystyle {\vec {B}}}$
• the second term is the interaction of the spins with each other, the entry of which in the interaction matrix is ​​different from zero, ${\displaystyle J_{ji}}$
• ${\displaystyle g}$the gyromagnetic factor
• ${\displaystyle \mu _{\mathrm {B} }}$the Bohr magneton

describes.

In the sense of molecular field theory, the interaction term is now estimated by replacing the spins with their mean value over the entire system:

${\displaystyle \langle {\hat {\vec {S}}}\rangle ={\frac {1}{N}}\sum _{i=0}^{N}{\hat {{\vec {S}}_{i}}}\ .}$

The expectation value of a single spin is then in the molecular field approximation . ${\displaystyle S_{i}}$ ${\displaystyle \langle {\hat {{\vec {S}}_{i}}}\rangle =\langle {\hat {\vec {S}}}\rangle }$

The Hamilton operator then becomes:

${\displaystyle {\hat {H}}=-\sum _{j}g\mu _{\mathrm {B} }{\hat {{\vec {S}}_{j}}}\left({\vec {B}}+{\frac {1}{g\mu _{\mathrm {B} }}}J_{j}\langle {\hat {\vec {S}}}\rangle \right)\ ,}$

whereby . ${\displaystyle J_{j}={\sum }_{i}J_{ji}}$

In a further estimate, the following is assumed to be the same for all : ${\displaystyle J_{j}}$ ${\displaystyle j}$

${\displaystyle J_{j}=J}$

The term in brackets is now independent of the individual interactions in the system and can be understood as an effective external magnetic field. This can be used instead of the magnetic field in the solutions to the problem of free spins ( ). ${\displaystyle {\vec {B}}_{\text{eff}}=\left({\vec {B}}+{\frac {1}{g\mu _{\mathrm {B} }}}J\langle {\hat {\vec {S}}}\rangle \right)}$ ${\displaystyle J_{ji}=0}$

In the case of a magnetic field aligned along the z-axis, the expected value of the -component of the spin sum results : ${\displaystyle {\vec {B}}=B\,{\vec {e}}_{z}}$ ${\displaystyle z}$ ${\displaystyle S}$

${\displaystyle \langle {\hat {S_{z}}}\rangle =S\ \operatorname {B} _{S}\left({\frac {Sg\mu _{\mathrm {B} }B}{k_{\mathrm {B} }T}}\right)\ ,}$

With

• the Brillouin function for the spin S. ${\displaystyle \operatorname {B} _{S}(\cdot )}$
• the Boltzmann constant ${\displaystyle k_{\mathrm {B} }}$
• the absolute temperature ${\displaystyle T}$

the expectation for interacting spins is:

${\displaystyle \langle {\hat {S_{z}}}\rangle =S\ \operatorname {B} _{S}\left[{\frac {Sg\mu _{\mathrm {B} }}{k_{\mathrm {B} }T}}\left({\vec {B}}+{\frac {1}{g\mu _{\mathrm {B} }}}J\langle {\hat {\vec {S}}}\rangle \right)\right]\ .}$

restrictions

The molecular field theory neglects correlations of the physical quantities, i. H. it is believed that . From this it follows that the molecular field theory breaks down at the critical point of a phase transition and in its vicinity. ${\displaystyle \langle S_{1}S_{2}\rangle =\langle S_{1}\rangle \langle S_{2}\rangle }$

Generalizations

The core of the theory is that for a more complicated operator, a linear approximation, i.e. H. a one-particle approximation is made. Similarly, you can z. For example, in quantum theory, a complicated many-particle theory can be traced back to an optimally adapted one-particle theory, for example by approximating the Hamilton operator using the associated Hartree-Fock approximation or by introducing suitable quasiparticles .

literature

• Molecular field theory of ferromagnetism (University of Bayreuth), accessed December 24, 2009, 4:57 pm
• Mean Field Theory for Ferromagnetism, CAU Kiel, Hyperscript Prof. H. Föll, June 24, 2012

Individual evidence

1. ^ DJ Amit: Field Theory, the Renormalization Group, and Critical Phenomena , World Scientific, 1978, ISBN 9971-966-10-7 .
2. ↑ C. Itzykson, JM Drouffe: Statistical Field Theory , Cambridge University Press, 1989, ISBN 0-521-40805-9 .