Reaction coordinate as a function of atomic positions

Question

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...

Answer 1

As @Kitchi writes $\delta$ is the Dirac Delta function. Your question is basically why is this true: $P(\xi) = \langle\delta[\xi - \hat{\xi}(x)]\rangle$.

As already noted (by Alex), when $ ξ=\hat ξ (x)$, we have $ \delta(ξ-\hat ξ (x)) = 1$. So the average that needs to be computed has this form: a configuration which has the reaction coordinate value $\xi$ contributes 1 and other configurations contribute 0.

So we have: $\langle\delta[\xi - \hat{\xi}(x)]\rangle = { \frac{ \text {# of conf. with }\xi}{\text{total # configs}}}$ . Clealy the RHS is the $P(\xi)$ So that this computes the ENSEMBLE average.