Why can we not reduce the size of a system below the correlation length without qualitatively changing its properties?

This question is posed in the context of thermodynamics/statistical mechanics. Suppose we define the correlation length as the $\xi$ in the exponential factor $e^{-r/\xi}$ that appears in the correlation function $$C(r) = \langle m(0)m(r) \rangle - \langle m \rangle^2$$ (where $m$ is some relevant order parameter).

Then my question is: how can we, from this mathematical definition, argue the following two physical properties.

Suppose we have our thermodynamic system and we take a chunk out of it with side-length L.

1. If $L > \xi$, the sample has the same qualitative behavior as the original sample;
2. If $L < \xi$, the qualitative behavior is different.

If one can add extra conditions for these statements to be true, that would be nice/welcome.

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Just so I understand -- you have a correlation function $C(r) \sim e^{-r/\xi}$? And you want to know why large-enough subsets of the system behave the same as the whole while small-enough subsets don't? –  tpg2114 Dec 8 '12 at 10:03

Mathematically, the expression

$$C(r) = \langle m(0)m(r) \rangle - \langle m \rangle^2$$

does not support the two relations. Without knowing how $C(r)$ changes with $r$, all this expression does is define a correlation. Without knowing that the correlation decreases with $r$ and without defining a cut-off for what we consider to be small, this relationship alone is just a definition. So, using the rest of the information, namely:

$$C(r) \sim e^{-r/\xi}$$

we can now proceed.

You defined $\xi$ to be a correlation length, which based on your relation for the correlation, implies that when $r > \xi$, $C(r)$ becomes small enough that $m(0)$ and $m(r)$ are sufficiently uncorrelated.

In light of that concept, let's look again at your two cases:

1. $L > \xi$ implies the same qualitative behavior:
If we take a chunk of the domain such that $L > \xi$, then our two edge-points are sufficiently uncorrelated because we exceeded the correlation length. So a change in the value at $m(0)$ does not influence $m(L)$ in a statistical sense. If $m(0)$ doesn't affect $m(L)$ then it likewise doesn't affect $m(2L)$ or anything greater. Nothing is lost, statistically, from taking the smaller domain because the largest statistical relationships are contained within it.
2. $L < \xi$ implies different qualitative behavior:
Here we can make the reverse argument. The value at $m(0)$ is influence by values at $m(\xi)$ but we don't have that point in our domain we are considering. So we do not have all the information needed and the behavior of the system will be different because we can't include the influence of these larger scales that we know have an influence.

It's really important to note two things here. First, we are looking at the statistical behavior of the systems. In no way are we saying that the point-wise values are qualitatively the same in the system when considering smaller subsets. But the statistical relationships are preserved (or not preserved depending on $L$).

Second, we are only saying that the qualitative behavior is the same, not the quantitative behavior. So taking a subset may not give you the same value but the trend will be the same.

An example may make this easier to understand, so let's consider one from the area I know (fluid dynamics and turbulence):

Assume we are interested in the flow through a wind tunnel with a wing in the test section. The test section is 10m long and we determine (somehow, not important) that $\xi$ is 1m. This is a great thing because we can take our wind tunnel and move it to another room in another building and get qualitatively the same answer as in the original room because the room/building effects don't change the statistics because changes more than 1m away are uncorrelated!

Let's look at the counter example. Consider a hurricane formed over the Atlantic Ocean. Here, the correlation length may be proportional to the storm radius, so let's say it's on the order of 100 miles for sake of argument. If we were to consider only the storm within a radius of 10 miles, the behavior is not the same as the entire storm, not even qualitatively. If the energy were to double in that 10 mile radius part, it wouldn't look anything like if the energy were to double over the entire 100 mile radius storm.

This is extremely important when it comes to numerical simulation of turbulence. We know that turbulence is correlated within certain distances and uncorrelated outside of those distances. So if we are designing a numerical simulation, we know our domain size doesn't need to be much bigger than the correlation length. This is great -- if we want to simulate the flow through a channel, we don't need to include the room the channel is in, or the building the room is in, or the city the building is in, etc. The exact values we get in our channel may not identically match an experiment, but the trends in values should.

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