Gyration Tensor

The gyration tensor is a tensor that describes the second moments of position of a collection of particles

$S_{mn} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{N}\sum_{i=1}^{N} r_{m}^{(i)} r_{n}^{(i)}$

where is the Cartesian coordinate of the position vector of the particle. The origin of the coordinate system has been chosen such that

$\sum_{i=1}^{N} \mathbf{r}^{(i)} = 0$

i.e. in the system of the center of mass . Where

$r_{CM}=\frac{1}{N}\sum_{i=1}^{N} \mathbf{r}^{(i)}$

Another definition, which is mathematically identical but gives an alternative calculation method, is:

$S_{mn} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{2N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N} (r_{m}^{(i)} - r_{m}^{(j)}) (r_{n}^{(i)} - r_{n}^{(j)})$

Therefore, the x-y component of the gyration tensor for particles in Cartesian coordinates would be:

$S_{xy} = \frac{1}{2N^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N} (x_{i} - x_{j}) (y_{i} - y_{j})$

In the continuum limit,

$S_{mn} \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r} \ \rho(\mathbf{r}) \ r_{m} r_{n}$

where represents the number density of particles at position .

Although they have different units, the gyration tensor is related to the moment of inertia tensor. The key difference is that the particle positions are weighted by mass in the inertia tensor, whereas the gyration tensor depends only on the particle positions; mass plays no role in defining the gyration tensor. Thus, the gyration tensor would be proportional to the inertial tensor if all the particle masses were identical.

Read more about Gyration Tensor: Diagonalization, Shape Descriptors