The main idea of the mean field theory is to focus on one particle and
assume that the most important contribution to the interactions of such
particle with its neighboring particles is determined by the *mean field*
due to the neighboring particles. In the 1-dimensional Ising model, for
instance, the average force
exerted on spin
is

(300) |

where the index
includes all the nearest neighbors of spin .
Therefore, the average magnetic field
acting on spin
is

(301) |

where

(302) |

is the contribution to the mean field due to the nearest neighbors.
Note that
when all spins are identical. Eq. (301) defines the *self consistent*
aspect of the theory, since according to such equation the mean field
acting on spin
is determined by its own mean value .
The assumption that the interactions of a spin with its neighboring spins
can be approximately described by the *mean field*, introduced by
Eq. (302), introduces an enormous simplification. Such mean field approximation
simplifies the many body statistical mechanics problem to a *one-body*
problem (i.e., Eq. (301) transforms the problem of N interacting spins
influenced by an external magnetic field
to a problem of N non-interacting spins influenced by the mean field ).

The partition function, under the mean field approximation, is

cosh | (303) |

and the average value of
is

(304) |

where
is the probability of state .
The average value of spin is

(305) |

Note that Eq. (305) involves a transcendental equation. Its solution
corresponds to the value of
for which the function on the left hand side of Eq.(305) (i.e., )
equals the function on the right hand side of Eq. (305) (i.e., tanh).
In the absence of an external magnetic field (i.e., when ),
Eq. (305) always has the trivial solution
and a non-trivial solution
only when .
Such solution is represented by the following diagram:

The diagram shows that the mean field theory predicts *spontaneous
magnetization* (i.e., magnetization in the absence of an external magnetic
field) for the 1-dimensional Ising model at any temperature ,
since there is a non-trivial solution
for which Eq. (305) is satisfied. Unfortunately, however, this result is
erroneous! The 1-dimensional Ising model does not undergo spontaneous magnetization
at any finite temperature, since each spin has only two nearest neighbors
and the stabilization energy due to two nearest neighbors is not enough
to overcome the randomization process due to thermal fluctuations. This
simple example, however, illustrates the theory including the fact that
it is sometimes inaccurate near critical points. The theory works better
in higher dimensionality, e.g., in the 2-dimensional
Ising model where the theory predicts spontaneous magnetization at
a critical temperature
that is close to the experimental value 2.3 J/K.

**Exercise:**

Show that there is no spontaneous magnetization in the 1-dimensional Ising model at finite temperature by computing the average magnetization from the exact canonical partition function. Hint: Compute the average magnetization in the presence of an external magnetic field and show that in the limit when such magnetization becomes negligible.