As we have seen, the main assumption (and the most important problem) of mean field theories is that the fluctuations of the order parameter are completely neglected in the computation of the partition function; this approximation breaks down in the neighbourhoods of critical points, where as we have seen in Long range correlations the correlation length becomes comparable with the size of the system.
What we would now like to do is to include these fluctuations in a mean field theoretical framework; this will lead to the so called Ginzburg-Landau theory.
As a first approach we can try to estimate how big is the error we make in mean field theories neglecting the fluctuations of the order parameter near a critical point, so that we can understand under which conditions mean field theories are actually a good approximations.
To make things explicit, let us use the Ising model as a base for our considerations.
We have seen in Weiss mean field theory for the Ising model that the Weiss mean field theory for the Ising model is based on the assumption that , i.e. that the spins are statistically independent; therefore, a possible estimate of the error made with this assumption can be:
The numerator of
is, by definition (see also Long range correlations
), the two-point connected correlation function:
If we neglect the fluctuations of the order parameter (which in our case is of course
) we see that
is constantly zero: therefore, in order to have non-null correlation functions we need that the system exhibits some kind of inhomogeneity, not necessarily due to thermal fluctuations. In fact, the connected correlation function describes not only the spatial extension of the fluctuations of the order parameter, but also the way it varies in space in response to an external inhomogeneous field. Let us see this explicitly.
We know that from the partition function of the Ising model in an inhomogeneous external field
can indeed be seen as a response function.
If we now call:
we will have:
If our system is invariant under translations and subject to a uniform field, then:
which is a version of the fluctuation-dissipation theorem
Let us now try to understand when the error done in mean field theories is negligible.
Now, in general terms if we formulate a mean field theory for a system we will make the error in the region where correlations are relevant, namely if is the distance between two points of the system the error is made for , with the correlation length.
Supposing , so that the order parameter is non null, then:
where we have called
the dimensionality of our system.
Therefore, our mean field theory will be a good approximation if
known as Ginzburg criterion
In order to express it in a useful fashion, let us write it in terms of critical exponents; using also the version we have just found of the fluctuation-dissipation theorem we get (supposing our system is continuous)
where we have used the definitions of critical exponents (see Critical exponents and universality
). Therefore, the Ginzburg criterion can be reformulated as:
and in the limit
this is possible only if
This means that Ginzburg criterion allows us to determine the upper critical dimension
of a system, namely the dimension above which mean field theories are good approximations
fluctuations become too relevant and mean field theories don't work.
Let us note that since it depends on the critical exponents, the upper critical dimension
ultimately depends on the universality class of the system considered; furthermore, in order to actually be able to compute
we must generalize Landau theory to systems with spatial inhomogeneities so that we are able to compute the critical exponent
- ↑ For the origin of the first equation, see what we have stated above and also in Long range correlations.
- ↑ This also mean that above the upper critical dimension the critical exponents determined with mean field theories are exact, or at least in good agreement with experiments.