# Correlation function $\langle s_1(x, t)s_2(x', t')\rangle$ vs $\langle s_1(x, t)s_2(x', t')\rangle-\langle s_1(x, t)\rangle\langle s_2(x', t')\rangle$

The correlation function in statistical mechanics is defined in either of two ways

1. $$g(\mathbf{x}-\mathbf{x}', t-t') = \left\langle s_1(\mathbf{x}, t)s_2(\mathbf{x}', t') \right\rangle$$
2. $$g(\mathbf{x}-\mathbf{x}', t-t') = \left\langle s_1(\mathbf{x}, t)s_2(\mathbf{x}', t') \right\rangle - \left\langle s_1(\mathbf{x}, t)\right\rangle\left\langle s_2(\mathbf{x}', t')\right\rangle,$$

where in both cases $s_1, s_2$ are random variables and $\left\langle \dots \right\rangle$ is the thermal average.

It seems like a matter of taste, which definition is used. I've even seen that authors decide chapter-wise which one to apply.

How is the additional term in the second variant motivated?

In statistical mechanics and field theory, the second type is referred to as a "connected" correlation function. You sometimes see the notation $$g_{\text{c}}(\mathbf{x}-\mathbf{x'},t-t') = \langle s_1(\mathbf{x},t) s_2(\mathbf{x}',t') \rangle_{\text{c}}\,\text{,}$$ where $\langle\ldots\rangle_{\text{c}}$ indicates that the product of the averages should be subtracted. The connected correlation function corresponds to what is called the covariance in statistics.

The reason for subtracting the second term is that the expectation value of a product $\langle A B \rangle$ factorizes, i.e., $\langle AB \rangle = \langle A \rangle\langle B \rangle$, if $A$ and $B$ are statistically independent. So if the degrees of freedom at $\mathbf{x}$ and $\mathbf{x}'$ are completely uncorrelated, then the connected correlation function is zero. For example, one almost always finds that connected correlation functions vanish in the limit $\lvert \mathbf{x} - \mathbf{x}' \rvert \rightarrow \infty$.

As Gennaro points out, the "one-point functions" $\langle s_1(\mathbf{x},t)\rangle$ and $\langle s_2(\mathbf{x}',t') \rangle$ often vanish, and so the two types of correlation function become equivalent. This usually happens for reasons of symmetry; for example, in the paramagnetic phase of the Ising model, the expectation value of the spin vanishes. In the ferromagnetic phase, on the other hand, $\langle s(\mathbf{x})\rangle = m \neq 0$ (where $m$ is the magnetization). The connected correlation function goes to zero as $\lvert \mathbf{x} - \mathbf{x}' \rvert \rightarrow \infty$, but the other ("disconnected") correlation function does not; in fact, $$\langle s(\mathbf{x}) s(\mathbf{x}') \rangle \rightarrow \langle s(\mathbf{x}) \rangle \langle s(\mathbf{x}') \rangle = m^2\,\text{.}$$

• "....if and only if A and B are statistically independent.", to be precise, it is just "if" and not "if and only if" – ophelia Jan 22 '16 at 11:55

The general Pearson correlation between two variables is defined as $$\textrm{cor}(X,Y) = E[XY] - E[X]E[Y]$$ up to a denominator containing the standard deviations of the distributions of the two variables.

In some field theories the expectation values of the variable itself (one-point function) vanishes, therefore oftentimes the above definition reduces to the first term on the right hand side only.

The second version is the first one but with the observables shifted by their respective averages:

$$\langle \tilde A\tilde B\rangle = \langle (A-\langle A\rangle)(B-\langle B\rangle)\rangle = \langle AB\rangle - \langle A\rangle\langle B\rangle$$

Oftentimes in e.g. numerical studies, it is easier to just sample the average product of the original observables $\langle AB\rangle$ and subtract the (known) product of the averages. The correlation function $\langle \tilde A\tilde B\rangle$ will typically be more interesting physically.