# Potential due to anisotropic gaussian distrubution

1. Sep 15, 2011

### vvgobre

Hello All

If there is spherically symmetric gaussian charge density (http://en.wikipedia.org/wiki/Poisson's_equation)
$$\rho(\mathbf{r})=\frac{q_{i}}{(2 \pi)^{3/2} \sigma^3} e^{- \frac{\lvert r \rvert^2}{2 \sigma^2}}$$

then it will have have potential $\phi(r)$ by solving Poisson equation $\bigtriangledown^{2} \phi(r)=-4\pi\rho(\mathbf{r})$
$$\phi(r) = \frac{\mbox{erf} \bigg(\frac{r}{2\sigma}\bigg)}{r}$$

But this is the case if gaussian is spherical symmetric which is diagonal element of convariance matrix are same $\sigma_{x}^{2} = \sigma_{y}^{2} = \sigma_{z}^{2} = \sigma^2$

Since in general covariance matrix for tri-variate case give by
$$\varSigma= \begin{bmatrix} \sigma_{x}^{2} & \sigma_{x}\sigma_{y} & \sigma_{x}\sigma_{z} \\ \sigma_{y}\sigma_{x} & \sigma_{y}^{2} & \sigma_{y}\sigma_{z} \\ \sigma_{z}\sigma_{x} & \sigma_{z}\sigma_{y} & \sigma_{z}^{2} \\ \end{bmatrix}$$

Now i want potential due to anisotropic gaussian distribution (i.e. if i have full covariance matrix). Will it have same analytic form of potential as give by spherically symmetric gaussian?

Is there article/ reference in literature where such problem solved. As i didn't found till now.

Any help will be highly thank full