What is a correlation length?

Question

What is a correlation length?

I encountered this term in my space physics lecture, in the context of the "correlation length of the magnetic field magnitude," but I am not sure what does it mean.

Answer 1

Loosely speaking: say you have a current in a plasma that generates a magnetic field; the field that current generates will be important in a region around the current of a size given by $\sim \ell$ before it gets swamped out by the fields generated by other currents elsewhere in the plasma. $\ell$ is the correlation length.

More formally, in a plasma the magnetic field $B$ will be a random field depending on the position in the plasma. If the two-point correlation function for the magnetic field $B$ at $x$ and $y$ falls off exponentially with $|x-y|$, \begin{equation} \langle B(x) B(y) \rangle \sim e^{-|x-y|/\ell}, \end{equation} then the length scale $\ell$ is the correlation length. This behavior is very common in thermal states like a hot plasma may be in. Physically, this happens because there are lots of random, uncorrelated currents that are each giving rise to their own magnetic field, and so any given current only has a finite "sphere of influence" before it gets swamped by other currents.