Diffusion

Model

The example model single-compartment-diffusion is a single compartment that contains two species: ‘fast’ and ‘slow’, each with the same analytic initial distribution

\[c_s(t=0) = e^{-((x-48)^2+(y-48)^2)/36}\]

The two species have different diffusion coefficients: \(D=1cm^2/s\) for species ‘slow’, and \(D=3cm^2/s\) for species ‘fast’, and the model contains no reactions.

Analytic solution

For this system without reactions, we are simulating the two-dimensional diffusion equation,

\[\frac{\partial c_s}{\partial t} = D_s \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) c_s\]

where

  • \(c_s\) is the concentration of species \(s\) at position \((x, y)\) and time \(t\)
  • \(D_s\) is the diffusion constant for species \(s\)

For the initial condition \(c_s(t=0) = \delta(x)\delta(y)\), the analytic solution at time t of this equation is the heat kernel:

\[c_s(t) = \frac{1}{4 \pi D_s t}e^{-(x^2+y^2)/(4 D_s t)}\]

and a solution for our model can then be found by convolving this expression with our initial condition, to give

\[c_s(t) = \frac{t_0}{t+t_0}e^{-((x-48)^2+(y-48)^2)/(4 D_s (t+t_0))}\]

where \(t_0 = 9/D_s\). Note that this solution ignores boundary effects, so will not be valid at late times or close to the compartment boundary.

The total amount of species in the compartment is a conserved quantity,

\[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} c_s(t) dx dy = 36 \pi\]

and this is also valid at late times, since our zero flux Neumann boundary conditions also conserve the amount of species in the compartment.