Question

Mean Field Theory neglects what flucutations?

Answer 1

This is a topic that has being confusing me for a while. A general phrase that is used in the literature is that:

Mean Field theories neglect fluctuations

My questions is what is meant by fluctuations; spatial or thermal?

To illustrate my confusion let me give two (very predominant) examples of where mean field theory is used.

Example 1: The Ising Model

The Hamiltonian is given by:H=−J∑⟨i,j⟩σiσjwhere σi=±1.

We write σi=(σi−M)+M and neglect termsquadratic in (σi−M).

The Partition function is then given by:

z=e−βNzJM2/2∑{σi}eβJzM∑iσi

This can then be solved to find a consistency relation form M which takes the form:M=tanh(βJzM)

You can proceed to find things like the free energy specific heat etc.

Here we have clearly not neglected thermal fluctuations and have (at least on the surface) neglected spatial fluctuations by ignoring the quadratic term.

Example 2: Saddle point approximation

Here we have a partition function:Z=∫DM(→r)e−β∫d3→rf[M(→r)]

We find the M∗(→r) that minimizes f[M(→r)](which may include derivatives) and then just consider f[M∗(→r)] to be the total free energy.

It should be apparent that this ignores thermal fluctuations (we are ignoring the integral over M(→r)) but allows for spatial fluctuations - as M∗ need not be independent of →r.

Answer 2

Hey mate here is your answer in attachment