Discretization

Assume that ${\cal C}$ is discretized by using the resolutions $k_1$, $k_2$,$\ldots$, and $k_n$, in which each $k_i$ is a positive integer. This allows the resolution to be different for each C-space coordinate. Either a standard grid or a Sukharev grid can be used. Let

$$\Delta q_i = [ 0 \;\; \cdots \;\; 0 \;\; 1/k_i \;\; 0 \;\; \cdots \;\; 0 ],$$ (5.35)

in which the first $i-1$ components and the last $n-i$ components are 0. A grid point is a configuration $q \in {\cal C}$ that can be expressed in the form^5.10

$$\sum_{i=1}^n j_i \Delta q_i ,$$ (5.36)

in which each $j_i \in \{ 0,1,\ldots,k_i\}$. The integers $j_1$, $\ldots$, $j_n$ can be imagined as array indices for the grid. Let the term boundary grid point refer to a grid point for which $j_i = 0$ or $j_i = k_i$ for some $i$. Due to identifications, boundary grid points might have more than one representation using (5.36).