I am trying to understand why smoothing an image with a gaussian kernel of different sigma values and then computing the gradients of the smoothed image leads to "thicker" trails. In the image below, the left image is produced by convolving the image with the derivative of a gaussian kernel where $\sigma = 1$ while the right picture shows image gradients when the image is convolved with a gaussian kernel where a $\sigma = 2$.
From what I understand, the gaussian kernel with a higher $\sigma$ value tends to make pixels look "more" like it's neighbors, because it places higher weights on pixels that are further away from the middle.