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…