You can gain a lot of intuition for how heat flows from place to place by imagining it as a bunch of “heat beads”, randomly skittering through matter.
Each bead rolls independently from place to place, continuously changing direction, and the more beads there are in a given place, the hotter it is.
In one of his terribly clever 1905 papers, Einstein described how the random motion of individual atoms gives rise to diffusion. Adding up the probabilities from every possible starting position is the sort of thing integrals were made for: So far this is standard probability fare.
Einstein’s cute trick was to say “Listen, I don’t know what ϕ() is a probability distribution (and the sum of probabilities over all possibilities is 1).
The second derivative, , is a way to describe how a function is curving.