Consider the simple linear transport equation

$\displaystyle \partial_t f + v\partial_x f = 0,\quad f(x,0) = \phi(x)$

with a velocity ${v}$. Of course the solution is

$\displaystyle f(x,t) = \phi(x-tv),$

i.e. the initial datum is just transported in direction of ${v}$, as the name of the equation suggests. We may also view the solution ${f}$ as not only depending on space ${x}$ and time ${t}$ but also dependent on the velocity ${v}$, i.e. we write ${f(x,t,v) =\phi(x-tv)}$.

Now consider that the velocity is not really known but somehow uncertain (while the initial datum ${\phi}$ is still known exactly). Hence, it does not make too much sense to look at the exact solution ${f}$, because the effect of a wrong velocity will get linearly amplified in time. It seems more sensible to assume a distribution ${\rho}$ of velocities and look at the averaged solutions that correspond to the different velocities ${v}$. Hence, the quantity to look at would be

$\displaystyle g(x,t) = \int_{-\infty}^\infty f(x,t,v)\rho(v) dv.$

Let’s have a closer look at the averaged solution ${g}$. We write out ${f}$, perform a change of variables and end up with

$\displaystyle \begin{array}{rcl} g(x,t) & = &\int_{-\infty}^\infty f(x,t,v)\rho(v)dv\\ & =& \int_{-\infty}^\infty \phi(x-tv)\rho(v)dv\\ & =& \int_{-\infty}^\infty \phi(x-w)\tfrac1t\rho(w/t)dw. \end{array}$

In the case of a Gaussian distribution ${\rho}$, i.e.

$\displaystyle \rho(v) = \frac{1}{\sqrt{4\pi}}\exp\Big(-\frac{v^2}{4}\Big)$

we get

$\displaystyle g(x,t) =\int_{-\infty}^\infty \phi(x-w)G(t,w)dw$

with

$\displaystyle G(t,w) = \frac{1}{\sqrt{4\pi}\,t}\exp\Big(-\frac{w^2}{4t^2}\Big).$

Now we make a time rescaling ${\tau = t^2}$, denote ${h(x,\tau) = h(x,t^2) = g(x,t)}$ and see that

$\displaystyle h(x,\tau) = \int_{-\infty}^\infty \phi(x-w)\frac{1}{\sqrt{4\pi \tau}}\exp\Big(-\frac{w^2}{4\tau}\Big)dw.$

So what’s the point of all this? It turns out that the averaged and time rescaled solution ${h}$ of the transport equation indeed solves the heat equation

$\displaystyle \partial_t h - \partial_{xx} h = 0,\quad h(x,0) = \phi(x).$

In other words, velocity averaging and time rescaling turn a transport equation (a hyperbolic PDE) into a diffusion equation (a parabolic PDE).

I’ve seen this derivation in a talk by Enrique Zuazua in his talk at SciCADE 2015.

To end this blog post, consider the slight generalization of the transport equation

$\displaystyle \partial_t f + \psi(x,v)\partial_x f = 0$

where the velocity depends on ${x}$ and ${v}$. According to Enrique Zuazua it’s open what happens here when you average over velocities…