[ENHANCEMENT] Gaussian dynamic incoherent scattering function for systems undergoing diffusion anisotropically.

[ChiCheng45]

Is your feature request related to a problem? Please describe.
The implementation of the Gaussian dynamic incoherent scattering function in MDANSE is only valid for systems undergoing isotropic diffusion; it will not give good results for systems undergoing diffusion anisotropically.

The natural log of the dynamic intermediate scattering function can be expressed as a series of cumulants (see https://spindynamics.org/documents/sd_m5_lecture_01.pdf)

$$\ln \langle \exp(i q_x d_x(t) + i q_y d_y(t) + i q_z d_z(t) ) \rangle = \sum_a^{\infty} \sum_b^{\infty} \sum_c^{\infty} \frac{\langle \langle d_x^{a}(t) d_y^{b}(t) d_z^{c}(t) \rangle \rangle}{ a ! b ! c !} (iq_x)^{a} (iq_y)^{b} (iq_z)^{c}$$

where

$$d_x = r_x(t) - r_x(0)$$

In the Gaussian approximation, the above is truncated to terms of second order in $q$

$$\ln \langle \exp(i q_x d_x(t) + i q_y d_y(t) + i q_z d_z(t) ) \rangle \approx \langle \langle d_x(t) \rangle \rangle i q_x + \cdots - \frac{1}{2}\langle \langle d_x^2(t) \rangle \rangle q_x^{2} - \langle \langle d_x(t) d_y(t) \rangle \rangle q_x q_y - \cdots$$

where

$$\langle\langle d_x(t) \rangle \rangle = \langle d_x(t) \rangle = 0$$

since our system should be in equilibrium and

$$\langle \langle d_x^2(t) \rangle \rangle = \langle d_x^2(t) \rangle - \langle d_x(t) \rangle^2 = \langle d_x^2(t) \rangle$$ $$\langle \langle d_x(t) d_y(t) \rangle \rangle = \langle d_x(t) d_y(t) \rangle - \langle d_x(t) \rangle \langle d_y(t) \rangle = \langle d_x(t) d_y(t) \rangle.$$

Removing all terms that are zero,

$$\ln \langle \exp(i q_x d_x(t) + i q_y d_y(t) + i q_z d_z(t) ) \rangle \approx -\frac{1}{2} \langle d_x^2(t) \rangle q_x^{2} - \langle d_x(t) d_y(t) \rangle q_x q_y - \cdots$$

and then taking the exponential of the series, we obtain the Gaussian approximation of the intermediate scattering function.

$$\langle \exp(i q_x d_x(t) + i q_y d_y(t) + i q_z d_z(t) ) \rangle \approx \exp(- \frac{1}{2}\langle d_x^2(t) \rangle q_x^{2} - \langle d_x(t) d_y(t) \rangle q_x q_y - \cdots)$$

In MDANSE, we simplify the above since for an isotropic system, which has all the off-diagonal terms equal to zero,

$$\langle d_x(t) d_y(t) \rangle = \langle d_x(t) d_z(t) \rangle = \cdots = 0$$

therefore

$$\langle \exp(i q_x d_x(t) + i q_y d_y(t) + i q_z d_z(t) ) \rangle \approx \exp(-\frac{1}{2}\langle d_x^2(t) \rangle q_x^{2} - \frac{1}{2}\langle d_y^2(t) \rangle q_y^{2} - \frac{1}{2}\langle d_z^2(t) \rangle q_z^{2})$$

all diagonal terms are equal since the system is isotropic and are equal to 1/3 of the mean-squared displacement (the sum of them is equal to the MSD)

$$\frac{\mathrm{MSD}}{3}= \langle d_x^2(t) \rangle = \langle d_y^2(t) \rangle = \langle d_z^2(t) \rangle$$

Finally, we get the equation used in MDANSE

$$\langle \exp(i q_x d_x(t) + i q_y d_y(t) + i q_z d_z(t) ) \rangle \approx \exp(- \frac{\mathrm{MSD}}{6}[q_x^{2} + q_y^{2} + q_z^{2}]) = \exp(- \frac{\mathrm{MSD}}{6} \vert \mathbf{q} \vert^2 )$$

Describe the solution you'd like
Generalise the GDISF job so it also works for systems that diffuse anisotropically. Use

$$\langle \exp(i q_x d_x(t) + i q_y d_y(t) + i q_z d_z(t) ) \rangle \approx \exp(- \frac{1}{2}\langle d_x^2(t) \rangle q_x^{2} - \langle d_x(t) d_y(t) \rangle q_x q_y - \cdots)$$