I Do equipotential lines fall on the equiprobability contours?

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Equipotential lines and equiprobability contours may align closely near the core of a 2D charge distribution, particularly when the standard deviations σx and σy are equal. As the distance from the core increases, the equipotential surfaces tend to become circular for equal standard deviations. The discussion raises questions about the relationship between these contours in general terms, not limited to specific particle distribution functions. Clarification is sought on whether the normalized distribution function (NDF) is Gaussian. Overall, the relationship between equipotential and equiprobability contours remains a nuanced topic in the context of charge distributions.
Mikheal
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Are equipotential lines fall on the equiprobability contours of charge distribution?
For 2D charge distribution ρ(x,y)=Ne PDF(x,y), where PDF is the normalized probability density function with its peak on (0,0) and has standard deviations σ x. and σ y. Are the contours with the equal probability "PDF(x,y)=const" the same as the equipotiential contours?, I tend to think that near the core of the distribution, they will be similar, and as the distance from the core increases, the equipotential surfaces will be circles for σx=σy.

Edit 1: I am speaking in general, not about certain particle distribution functions, such as 2D Gaussian with different σ x and σ y, 2D bi-Gaussian, 2D super-Gaussian, Flat-top, ....

Edit 2: I know that for 2D Gaussian with σ x = σ y, they fall on each other.
 
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Is NDF a Gaussian? Your question needs to be a little bit more definitive.
 

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