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Sorry I asked that wrong. I don't mean heat flow in the first case, there are no diffusing particles there. Say concentration of tracer in fluid suspension or something.
I'm not a physicist so my question may be really old hat, but whatever.
I can think of two situations in which one ends up with a diffusion equation but in which the underlying physics is quite different.
First, the flow of heat in a solid. Here there is a continuous 'heat flows down a temperature gradient' picture that is mathematically equivalent to a picture in which individual particles follow Brownian motions. Physically, the former is just a sort of averaged version of the latter - some accounting short cuts - while the latter is some way closer to reality; the particles are really diffusing.
Second, the flow of water in an aquifer. Here the Darcian flow is proportional to the pressure gradient. For the sake of argument, imagine a medium that is a perfectly regular and homogenous 3D network of tiny tubes or something. In this case, there is no 'diffusion' of the fluid particles; they flow in a completely deterministic (indeed reversible) way through the network. But of course, one could presumably 'solve' the aquifer equation with a Monte Carlo similation of a diffusion process, if one really wanted to or if that was handy.
So to my question: Does the Feynman path integral purport to represent what's actually going on in any sense? Or is it more in the nature of a device for solving the problem? Or is this one of those things that is not answerable?