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 $$r_{m}^{(i)}$$ is the $$\mathrm{m^{th}}$$ Cartesian coordinate of the position vector $$\mathbf{r}^{(i)}$$ of the $$\mathrm{i^{th}}$$ 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 $$r_{CM}$$. Where



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

In the continuum limit,



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

where $$\rho(\mathbf{r})$$ represents the number density of particles at position $$\mathbf{r}$$.

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 proprotional to the inertial tensor if all the particle masses were identical.

Diagonalization
Since the gyration tensor is a symmetric 3x3 matrix, a Cartesian coordinate system can be found in which it is diagonal



\mathbf{S} = \begin{bmatrix} \lambda_{x}^{2} & 0 & 0 \\ 0 & \lambda_{y}^{2} & 0 \\ 0 & 0 & \lambda_{z}^{2} \end{bmatrix} $$

where the axes are chosen such that the diagonal elements are ordered $$\lambda_{x}^{2} \leq \lambda_{y}^{2} \leq \lambda_{z}^{2}$$. These diagonal elements are called the principal moments of the gyration tensor.

Shape descriptors
The principal moments can be combined to give several parameters that describe the distribution of particles. The squared radius of gyration is the sum of the principal moments



R_{g}^{2} = \lambda_{x}^{2} + \lambda_{y}^{2} + \lambda_{z}^{2} $$

The asphericity $$b$$ is defined by



b \ \stackrel{\mathrm{def}}{=}\ \lambda_{z}^{2} - \frac{1}{2} \left( \lambda_{x}^{2} + \lambda_{y}^{2} \right) $$

which is always non-negative and zero only for a spherically symmetric distribution of particles. Similarly, the acylindricity $$c$$ is defined by



c \ \stackrel{\mathrm{def}}{=}\ \lambda_{y}^{2} - \lambda_{x}^{2} $$

which is always non-negative and zero only for a cylindrically symmetric distribution of particles. Finally, the relative shape anisotropy $$\kappa^{2}$$ is defined



\kappa^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{b^{2} + (3/4) c^{2}}{R_{g}^{4}} $$

which is bounded between zero and one.