Diffusion
2d 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
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,
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:
and a solution for our initial condition can then be found with an overall rescaling and a shift in t:
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,
and this is also valid at late times, since our zero flux Neumann boundary conditions also conserve the amount of species in the compartment.
3d Model
The example model single-compartment-diffusion-3d is a single compartment that contains two species: ‘fast’ and ‘slow’, each with the same analytic initial distribution.
Key differences from the 2d example above are that the origin of the geometry lies in the centre of a 100cm^3 cube, with initial species concentration given by:
and the heat kernel in 3d is given by:
which gives the solution at time t (again ignoring boundary effects):
where \(t_0 = 9/D_s\), with total amount of species in the compartment a conserved quantity: