Diffusion smoothing

Question

What does it means that diffusion has a smoothing effect of spatial heterogeneity?

I read the sentence on a paper and I was wondering about a mathematical explanation of that.

Answer 1

The diffusion equation models heat flow so I will use heat as an example. The idea is that heat flows from high concentration areas to low concentration areas. This flow has a sort of averaging affect wherein the heat at each point is replaced by the average heat in the surrounding area (this is a rough heuristic, it's not exactly what's happening but it helps to demonstrate). So if there is a strong heterogeneity (i.e., large differences in value at nearby points) in the initial data, the diffusion will work to homogenize this (i.e., make the values more similar at nearby points). I've included a picture to demonstrate this. The blue line in the picture is the original heat profile; it is equal to $1$ for $-1 \le x \le 1$ and $0$ otherwise. This function is non-differentiable since it has jumps at $x = 1$ and $x = -1$; this is a large heterogeneity. Consider what is happening near $x = 1$. The value of the data there is $1$, but nearby there are values that are either $1$ or $0$, so these should be 'averaged', and the value should be replaced by $1/2$ which, in some sense, makes the function smoother because now there won't be such a large jump anymore. In fact, if you do something similar at all points and instantaneously in time, the function goes from being non-differentiable at the initial time to being differentiable at later times. That is, while $u(x,0)$ is non-differentiable in $x$, the function $u(x,t)$ is infinitely differentiable in $x$ for any positive value of $t$. This gain in regularity is what is meant by 'diffusive smoothing.' You can see this in the picture; the red line is the heat profile at time $t = 0.1$ and it is much smoother than the initial data.

Answer 2

Let me approach this phenomenon from the "random walk" angle, which is a good model for an actual diffusion process.

Say, you have a particle on a line which can move by "jumps" of length $1$ to the left or to the right each second (or any other unit of time). The direction of the jump is determined by a fair coin toss, i.e. is a random number which can take values $-1$ or $+1$ with equal probability.

Two samples of such a walk can be seen below for a red ball and a blue ball (the numbers on the left count the time elapsed):

Now what happens if we put many particles on the line in position $0$ and let them "jump" as stated above? It's quite obvious, they, while moving absolutely independently of one another, will spread out on the line.

Experiment with $50$ balls below confirms this fact.

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Mathematical description of the random walk is quite simple and in the limit of many particles and small jumps (continuum limit in other words) gives the usual diffusion equation.

Let $N(j,m)$ be the expected number of particles (remember, we are describing a random process) at an integer point $m=0, \pm 1, \pm 2, \dots$ at time $j$.

Obviously, $N(j,m)$ directly depends on $N(j-1,m-1)$ and $N(j-1,m+1)$, i.e. the number of particles on the left and on the right during the previous time step. How does it depend? By the rules stated above, the probabilities of jumping to the left and to the right are equal to $1/2$.

It doesn't depend on $N(j-1,m)$, because all the particles that were here before have to leave according to the rules (the probability of staying put is $0$).

Thus:

$$N(j,m)=\frac{1}{2} N(j-1,m-1)+\frac{1}{2} N(j-1,m+1)$$

$$N(j,m)=\frac{N(j-1,m-1)+N(j-1,m+1)}{2}$$

But look at this: the above states that, as time progresses, the number of the particles averages over all positions (the expression for the number of particles at each position is just the arithmetic mean over numbers of particles that were on the left and on the right).

If, for example, we had:

$$N(1,1)=10$$

$$N(1,2)=0$$

$$N(1,3)=0$$

Then for the middle point at the next step we have: $$N(2,2)=\frac{10+0}{2}=5$$

In other words, a simple random process, where each particles moves independently, automatically leads to the "smoothing" of any inhomogeneities.