Long range correlations

We now want to show that one of the characteristic traits of critical transitions, that distinguishes them from other kinds of phase transitions, is the fact that the divergence of response functions in the proximity of the critical point is intimately bound to the existence of long-ranged strongly correlated microscopic fluctuations. We shall do that in a very qualitative way, and to make things more clear we consider a magnetic system (but of course our considerations are absolutely general and extensible to other kinds of systems).

When a magnetic field is present[1], the partition function of a magnet is:

where is the Hamiltonian of the system. The magnetization of the system at equilibrium is:
while its susceptibility is:
However, from a microscopic point of view the magnetization of the system can be written as:
where is the local magnetization in . If we substitute this expression in we get:
If our system is spatially homogeneous, then is constant and , called correlation function, depends only on the relative distance between two points. Defining the connected correlation function as:
sometimes also called , and changing variable to we get:
where in the last step we have renamed the variable to .

The connected correlation function is essentially (as it can be understood from its definition) a measure of how the fluctuations of the magnetization from the mean value in a part of the system influence those in another part of the system. Now, we expect that in general the correlation function will decrease on long distances. We can therefore write[2]:

where is a characteristic length of the system called correlation length. If we call the characteristic value of the connected correlation function for (namely we suppose that it is constant and equal to for lengths smaller than [3]), then from we have (neglecting any proportionality constant):
We thus immediately see that if , since (as we know from the thermodynamics of critical phase transitions) we must also have : in the neighbourhood of a critical point the correlation length of the system diverges. As we have seen in Critical exponents and universality, we describe the divergence of by means of the critical exponent , namely we set:
Furthermore, experiments and exact solutions of simple models show that the correlation function decays near the critical point as a power of , with a critical exponent called~:
where is the dimensionality of the system (see Critical exponents and universality).

  1. We are treating , just like the magnetization , as scalars instead as vectors: in order to make things easier, we are supposing that the real fields and are both directed along the same direction, say the axis of our reference frame, and we are only considering their magnitudes.
  2. This form is also justified, as we will see, from the fact that in many cases turns out to actually decay exponentially with .
  3. In fact, is of the order of 1 when .