Is there a circuit that can create dispersion?
What you say makes me think of a Gaussian filter (for the Gaussian distribution output), or a Bessel (for constant group delay until fc). However, the Bessel filter, given a high enough order, will converge towards a Gaussian, but that will happen only after many stages. If your requirements are not that high, an 8th order Bessel will give you a fairly good temporal response, which can be made with a quad opamp; a 16th order will be even better. Here's an attempt at simulating a "distribution" Dirac input, and the responses of 8th and 16th order Bessel filters (non frequency scaled):



Also, what @ThePhoton says is true, the Dirac function is quite a peculiar one. The way I modeled it is like a distribution, but it's physically impossible.

For reference, here's how the frequency responses of the two filters look like, in terms of magnitude and group delay: