I'm going over some (molecular dynamics) related literature - specifically the derivation of the Weighted Histogram Analysis Method (WHAM).
As a quick backdrop WHAM is a method for stitching together conceptually overlapping and independent experiments done in Monte Carlo or Molecular Dynamic simulations. This isn't actually super important, as this is more a notation question.
Before the paper gets to the WHAM derivation, there's a discussion on the actual problem, where two variables are defined
$ x = \text{coordinates of atoms}$
$ \xi = \text{reaction coordinate - is a function of x} $
There's also a discussion on the notation used, where a caret above a value indicates it's a function,
"Thus, $\hat{V_i}(x)$ denotes the function and $V_i$ a particular value the function takes"
Having defined this and after some more work, the paper defines the probability density obtained from a simulation, which can be written in the following manner.
$P(\xi) = \langle\delta[\xi - \hat{\xi}(x)]\rangle$
For the record
$P(\xi) = e^{\big(-\beta W(\xi)\big)}$
Firstly, there's no explanation of what $\delta$ is at all, which makes me think it's probably some common operator. I don't think a Kronecker delta because that doesn't make sense in this context (and it lacks subscripts) but maybe?
Secondly, is there some common way to interpret a variable which can also be a function? When I initially read it I assumed that $\xi(x)$ was looking at deviation from the reaction coordinate the $x$ variable showed, but in hind-site I don't know why I thought that.
The reaction coordinate, traditionally, is some physical component of the system. It could be the distance between two residues, a specific orientation of all the atoms in the system, the distance of a specif residue to some arbitrary point etc. I'm not sure if that's the same here, or if it has some other, more specific meaning in this context.
EDIT: I'm currently reading another paper in the literature ("Use of the Weighted Histogram Analysis Method for the "Analysis of Simulated and Parallel Tempering Simulations" by Chodera et al.) which a colleague recommended and uses very different notation - hopefully I can use this to reconcile what's going on here...
