Conditional average of a field in physics: $\langle \Psi \rangle_{ij}^i = \langle \Psi \rangle_i^i$

Question

I was just reading this article on the quasicrystalline approximation (QCA). The article abstract says the following:

The quasicrystalline approximation (QCA) was first introduced by Lax to break the infinite heirarchy of equations that results in studies of the coherent field in discrete random media. It simply states that the conditional average of a field with the position of one scatterer held fixed is equal to the conditional average with two scatterers held fixed, i.e., $\langle \Psi \rangle_{ij}^i = \langle \Psi \rangle_i^i$.

Can someone please explain the mathematics $\langle \Psi \rangle_{ij}^i = \langle \Psi \rangle_i^i$? In the context of quantum mechanics, I usually interpret the brackets $\langle$ and $\rangle$ to relate to bra-ket notation (inner products in Hilbert space), but it seems that they are used differently in this context (perhaps for some purpose of conditional average?). Furthermore, I would usually interpret the subscripts and superscripts as referring to tensor notation (such as Einstein summation notation), but it isn't clear to me what it means in this context (I wonder if the subscripts refer to conditioning with respect to some values, or something similar?).

I would greatly appreciate it if people would please take the time to explain this.

Answer 1

In statistical mechanics and other parts of physics the symbol $\langle X\rangle$ is used to denote the average of the random variable $X$ respect to its probability distribution. Probabilists usually write $E(X)$ or $\mu_X$. The physics notation probably has its origin in the quantum mechanical expectation $\langle \psi|X|\psi\rangle$, but there need be no quantum aspects to the system being considered.

The appended indices $i$, $j$ and so on have no standard meaning and will be specific to your paper, and so should be defined there.