Gyration tensor information

In physics, the gyration tensor is a tensor that describes the second moments of position of a collection of particles

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

where ${\displaystyle r_{m}^{(i)}}$ is the ${\displaystyle \mathrm {m^{th}} }$ Cartesian coordinate of the position vector ${\displaystyle \mathbf {r} ^{(i)}}$ of the ${\displaystyle \mathrm {i^{th}} }$ particle. The origin of the coordinate system has been chosen such that

${\displaystyle \sum _{i=1}^{N}\mathbf {r} ^{(i)}=0}$

i.e. in the system of the center of mass ${\displaystyle r_{CM}}$. Where

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

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

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

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

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

In the continuum limit,

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

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

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.