note: diffusion in 2D and 3D is also automatically net away from any point, but not in 1D, that’s another fun derivation
What? The Green’s function for the 1D heat equation has the same dependance on distance as the 2D and 3D case.
“It’s easy to show that [something.]” A student asks, “Um… is that easy?” The professor starts writing and checking his notes. He leaves, comes back an hour later, says “Yes, it’s easy,” and then continues on with the lecture.
The version I heard used ‘obvious’ instead of ‘easy’. I wonder ‘trivial’ would be funnier yet.
Ah, you’re right, sorry. Yes, diffusion rate works the same in 1D for concentration gradient over time. The difference I was thinking of is that in 1D for an individual molecule a random walk returns to the origin with probability 1, even though avg distance rises over time, while in higher dimensions that isn’t true.
What? The Green’s function for the 1D heat equation has the same dependance on distance as the 2D and 3D case.
The version I heard used ‘obvious’ instead of ‘easy’. I wonder ‘trivial’ would be funnier yet.
Ah, you’re right, sorry. Yes, diffusion rate works the same in 1D for concentration gradient over time. The difference I was thinking of is that in 1D for an individual molecule a random walk returns to the origin with probability 1, even though avg distance rises over time, while in higher dimensions that isn’t true.
Simple random walks return to the origin in 2D as well, but not 3D or higher. I don’t know if the continuous case is different, but I suspect not.
Then I’m probably just misremembering, it’s been about 15 years since I looked at that one. Thanks!