Diffusion-limited annihilation (part 1) (Rev #2)

This page is a blog article in progress, written by Blake Stacey.

(After Vollmayr-Lee, 2009)

Feynman diagrams, the comic books of physics!

The **propagator** tells us how particles get from one point to another if nothing happens in between. We’re saying that these particles move by diffusion, so the propagator in this case is the Green’s function for the diffusion equation.

$G_D(k, \omega) = \frac{1}{-i\omega + D k^2}.$

Transforming from frequency back into the time domain,

$G_D(k, t) = \int \frac{d \omega}{2\pi} \frac{e^{-i \omega t}}{-i \omega + D k^2}.$

For $t \gt 0$, this is

$G_D (k, t) = \exp(-D k^2 t).$

In position space,

$G_D(x, t \gt 0) = \frac{e^{-x^2 / (4 D t)}}{(4 \pi D t)^{d / 2}}.$

We can think of this as saying that the response to a delta-function spike at $t = 0$ is a Gaussian curve which spreads out as time passes, its standard deviation growing as the square root of the elapsed time.

To each trivalent vertex, we associate a factor $-2\lambda_0$, and each initial vertex gets a $n_0$. Wave-vector (or “momentum”) conservation applies at each vertex. We can read off the self-consistency condition for the tree-level contributions directly from the diagrams:

$a_{tree}(t) = n_0 + \int_0^t d t_1 G_D(0, t - t_1) (-2 \lambda_0) a_{tree}(t_1)^2.$

The propagator with $k = 0$ is just 1. Differentiating both sides of the self-consistency equation yields that the time derivative of $a_{tree}$ is the integrand evaluated at $t$.

$\frac{d a_{tree}}{d t} = -2\lambda_0 a_{tree}^2$

This is just a rate equation for $a_{tree}$. With the initial condition $a_{tree}(0) = n_0,$ this has the solution

$a_{tree}(t) = \frac{n_0}{1 + 2\lambda_0 n_0 t}.$

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