Conversions between Positions and Connectors

In terms of the struct BONDS_GROUP which has $({\tt np}, {\tt ip[np]}, {\tt bonds[np-1]})$, we can construct the conversions $\{\bm{x}_{\mu}\}\mapsto\{\overline{\bm{Q}}_{i}\}$ and $\{\overline{\bm{Q}}_{i}\}\mapsto\{\bm{x}_{\mu}\}$.

First, we see the system with a single group. Because we only take the independent bonds, the bond index $i$ runs from 0 to $N-2$ (where the number of independent bonds is $N-1$), while the particle index $\mu$ runs from 0 to $N-1$ (where the number of particles is $N$). Adding the $(N-1)$-th element of the connector vector $\{\overline{\bm{Q}}_{i}\}$ by the center of mass $\bm{x}_{\text{COM}}$ as

$$\overline{\bm{Q}}_{N-1} := \bm{x}_{\text{COM}} = \frac{1}{N} \sum_{i=0}^{N-1} \bm{x}_{{\tt ip}[i]} ,$$ (9.11)

(therefore, precisely speaking, $\bm{x}_{\text{COM}}$ is not the center of mass though), the numbers of elements of both $\{\bm{x}_{\mu}\}$ and $\{\overline{\bm{Q}}_{i}\}$ become the same.

The transformations between them can be constructed as follows. By definition of connector vectors,

$$\overline{\bm{Q}}_{i} = \bm{x}_{\alpha_{{\tt bonds}[i]}} - \bm{x}_{\beta_{{\tt bonds}[i]}} ,$$ (9.12)

for $i = 0, \cdots, N-2$. The last element of the connector vector $\overline{\bm{Q}}_{N-1}$ is given by Eq. (9.11). This is implemented in the routine BONDS_pos_to_conn_1. Note that, for the isolated particle, we define the (generalized) connector vector by the position itself as

$$\overline{\bm{Q}}_{0} = \bm{x}_{0}.$$ (9.13)

As we see, the relation is linear and the transformation can be expressed by

$$\overline{\bm{Q}} = \mathscr{B}\cdot\bm{x},$$ (9.14)

where $\overline{\bm{Q}}$ and $\bm{x}$ are the generalized vector with $3N$ dimension for $\{\overline{\bm{Q}}_{i}\}$ and $\{\bm{x}_{\mu}\}$, respectively, and $\mathscr{B}$ is $3N\times 3N$ matrix based on the $N\times N$ matrix $\overline{\mathsf{B}}$ in Eq. (9.4).

The opposite transformation from $\overline{\bm{Q}}_{i}$ to $\bm{x}_{\mu}$ can be obtained from the inversion matrix $\mathscr{B}^{-1}$. Alternatively (and directly), we can calculate the position vectors from the connector vectors (and the center-of-mass vector) as follows: For the isolated particle,

$$\bm{x}_{0} = \overline{\bm{Q}}_{0}.$$ (9.15)

For the group with $N(> 1)$ particles, we take the two steps:

1. at the first step, $\bm{x}_{\mu}$ is formed with respect to the origin at the $\bm{x}_{\alpha_{{\tt bonds}[0]}}$,
2. then, adjust the origin with respect to $\bm{x}_{\text{COM}} = \overline{\bm{Q}}_{N-1}$.

At the first step, therefore, we define

$$\bm{x}_{\alpha_{{\tt bonds}[0]}} = \bm{0},$$ (9.16)

as an initial condition. After it, for the independent bond $i$ from 0 to $N-2$,

$$\bm{x}_{\beta_{{\tt bonds}[i]}} = \bm{x}_{\alpha_{{\tt bonds}[i]}} - \overline{\bm{Q}}_{i} .$$ (9.17)

To make this procedure work, we should assure that $\bm{x}_{\alpha_{{\tt bonds}[i]}}$ has been defined by the $(i-1)$-th step. This condition is satisfied from the fact that

• bonds[] don't have redundant bonds,
• the number of the independent bond is np-1,
• the particle indices $\alpha_{i}$ and $\beta_{i}$ are sorted as in Eq. (9.10),
• bonds[] itself is sorted as ${\tt bonds}[i] < {\tt bonds}[i+1]$.

After the first step, we calculate $\widetilde{\bm{x}}_{\text{COM}}$ as

$$\widetilde{\bm{x}}_{\text{COM}} = \frac{1}{N} \sum_{i=0}^{N-1} \b... ...\left( = \frac{1}{N} \sum_{i=0}^{N-2} \bm{x}_{\beta_{{\tt bonds}[i]}} \right) ,$$ (9.18)

and adjust the center of mass to $\bm{x}_{\text{COM}}$ as

$$\bm{x}_{{\tt ip}[i]} = \bm{x}_{{\tt ip}[i]} - \widetilde{\bm{x}}_{\text{COM}} + \bm{x}_{\text{COM}} ,$$ (9.19)

for $i = 0, \cdots, N-1$.

It should be noted that the calculation of this rather tedeous procedure is $O(N)$, while the inversion $\mathsf{A}^{-1}$ usually costs $O(N^3)$.

For the system containing many groups at the same time, we can apply the above procedure for each group separately, and the overall calculation is still $O(N)$. This system-wide transformations are implimented in the routines BONDS_pos_to_conn and BONDS_conn_to_pos.