Consider the diffusion equation for settled systems in the free-streaming limit (which I think should be appropriate)

\frac{\partial \rho}{\partial t} = k \frac{\partial ^2 \rho}{\partial x^2} \rightarrow \frac{\partial \rho}{\partial t} = v \frac{\partial \rho}{\partial x} where v is the free streaming velocity roughly corresponding to the thermal velocity. (For flux-limited radiation diffusion, this is just the speed of light).

We can then combine that with a logistic style source term that accounts for local probe activity

S=\frac{1}{T_p}\rho(1 - \frac{\rho}{\rho_{max}})

We then normalize the density of settled systems by the density of total systems

\eta=\frac{\rho}{\rho_{max}}

which gives us

\frac{\partial \eta}{\partial t} = -v \frac{\partial \eta}{\partial x} + \frac{1}{T_p} \eta (1-\eta)

We then look for self-similar solutions of the form

\eta(\xi) where \xi = x - \mathcal{V} t

Upon substituting that in to the continuity equation we arrive at

-\mathcal{V}\frac{\partial \eta}{\partial \xi} = -v\frac{\partial \eta}{\partial \xi} + \frac{1}{T_p} \eta \left ( 1 - \eta \right )

or

\frac{\partial \eta}{\partial \xi} = -\frac{\eta (1-\eta)}{T_p (\mathcal{V}-v)}

which is just a logistic curve with length scale L=T_p \left (\mathcal{V}-v \right) where v is the thermal speed and \mathcal{V} is the speed of the front (maximum speed from maxwellian).

Ran a simulation with periodic BC's, but particles enter from the right unsettled. Also shifted the random velocities by the expected front velocity 3 v_rms.

Then fit the number of settled systems to a logistic curve

\rho = \frac{\rho_{\max}}{1+\exp^{(x-x0)/L}}