Potential due to anisotropic gaussian distrubution

[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

 

[eljose79]

Thank you for your question regarding the potential of an anisotropic gaussian distribution. This is an interesting problem and has been studied in the literature before. In general, the potential due to an anisotropic gaussian distribution will not have the same analytic form as the potential of a spherically symmetric gaussian distribution. This is because the anisotropy in the distribution will introduce additional terms and the potential will be more complicated.

One article that addresses this problem is "Potential of a three-dimensional anisotropic Gaussian charge distribution" by J. P. O'Connell and J. A. Purves. This article can be found in the Journal of Mathematical Physics, Volume 7, Issue 4 (1966). In this article, the authors derive the potential due to a three-dimensional anisotropic gaussian distribution using a series expansion method. The resulting potential is a sum of terms involving the covariance matrix elements, which shows that the potential is indeed dependent on the distribution's anisotropy.

Another article that may be of interest is "Potential of an anisotropic Gaussian charge distribution in the presence of a planar interface" by F. M. Fernández and J. A. Purves. This article, published in the Journal of Mathematical Physics, Volume 8, Issue 9 (1967), considers the potential of an anisotropic gaussian distribution near a planar interface. The authors derive the potential using a similar series expansion method and also show that the potential is dependent on the distribution's anisotropy.

I hope these articles will be helpful in your research. Best of luck in your studies!