# Diffusion tensor as a covariance matrix

TLDR: In nuclear magnetic resonance (NMR), to study molecular diffusion we assume that molecules displace in 3D space according to a trivariate gaussian distribution. The variables are then the displacements along the 3 directions and the "diffusion tensor" I get from NMR experiments is definitely a covariance matrix..BUT the physical meaning of its off-diagonal elements is by no means the covariance of displacement along different directions, but rather the covariance of displacement along a direction and the STRENGTH OF THE MAGNETIC FIELD along another direction. Why?

I do chemistry and my topic of work is a physical quantity: diffusion, or Brownian motion of molecules in a medium. When the molecules move in a restricted space and can experience boundaries, their ability to move (their "diffusion") has a different extent along the three axes x,y,z, depending on the geometry of that environment. In this case, the features of molecular diffusion are expressed with a 3x3 symmetric tensor that brings information about the restriction and therefore about the geometry.

If we assume that the molecular displacement can be modelled with a trivariate Gaussian distribution, the 3x3 tensor should be in fact the covariance matrix.

Now, the way we derive this tensor is by running Nuclear Magnetic Resonance experiments, namely measurements in which you place your sample (containing the diffusing molecules) in a magnetic field that has a certain direction itself. Things are such that if we change the orientation of the sample in respect with this external magnetic field, the outcome of the measurement changes and we get a different diffusion tensor. In short: the diffusion tensor contains information not only on the geometry in which diffusion occurs, but also on the "relationship" between this geometry and the external magnetic field.

The diffusion tensor that we want needs to be specific for the geometry of the sample in which molecules move, and therefore CANNOT be different depending on the mutual orientation between the sample and the external magnetic field. This is the reason why we need to diagonalise the tensor to get an object which is invariant under rotations and only contains information about the sample, free from any contribution coming from its orientation in respect with the magnetic field. With this procedure, we make the off-diagonal elements equal to zero, and we obtain rotational invariant eigenvalues that give us information solely about the geometry.

Now, as far as I know, the off diagonal elements of the covariance matrix are non zero if there is a correlation between the variables. In my case, the variables are the diffusion along x, the diffusion along y and the diffusion along z.

But in the tensor that I get, these off diagonal elements do not represent the correlation between displacements along the 3 directions, but rather the correlation between the displacements along the 3 directions and the strength of the magnetic field along the 3 directions.

The point is that the assumption on which all this treatment is based on, is that MOLECULES DIFFUSE IN 3D SPACE FOLLOWING A TRIVARIATE GAUSSIAN DISTRIBUTION, and so my variables are the displacements along the 3 directions and the diffusion tensor that I get is definitely a covariance matrix..but its off diagonal elements are by no means covariance between displacement along the different directions!

What am I missing?