Free diffusion with a one absorbing boundary and one "re-loading" boundary

Question

If we are considering free diffusion of a particle which is heavily damped then the evolution of its position $x$ can be expressed by a Langevin equation:

$\dot{x} = A\cdot\eta (t)$

where $A$ is constant and $\eta$ is a gaussian noise function with unit variance:

$\left\langle \eta(t) \eta (t')\right\rangle = \delta (t - t')$

I know that the unrestricted diffusive kernel or Green's function for this problem looks like:

$G(x,x';\Delta t) = \dfrac{1}{A\sqrt{2\pi\cdot\Delta t}} \text{exp} \left( -\dfrac{1}{2\cdot A^2 \cdot\Delta t}(x-x')^2\right)$

where $\Delta t = t - t'$ and $x'$ is the position of the particle at the initial time $t'$.

My problem is this. I want to impose two boundary conditions. The first is an exit boundary condition at $x_1$ which would correspond to an absorptive kernel $G_e$. I know how to do this straightforwardly enough: you simply subtract the same free diffusive kernel above and set $x \rightarrow 2x_1 -x$ :

$G_{e}(x,x';\Delta t) = G(x,x';\Delta t) - G(2x_1 -x,x';\Delta t) $

$\Rightarrow G_{e}(x,x';\Delta t)= \dfrac{1}{A\sqrt{2\pi\cdot\Delta t}} \left[\text{exp} \left( -\dfrac{1}{2\cdot A^2 \cdot\Delta t}(x-x')^2\right) - \text{exp} \left( -\dfrac{1}{2\cdot A^2 \cdot\Delta t}(2x_1 -x-x')^2\right)\right]$

The second boundary condition is more complicated. When the particle reaches another point $x_2$ I want the particle to be "sent back" to where it started i.e. $x'$. In this way it is "reloaded" back to the initial point although obviously at a later time. I have no idea how to go about this or even what sort of things to search for. If it all possible I would like to add a fixed time delay between reaching $x_2$ and being "transported" back to $x'$ but if that's too tricky even the instantaneous transport would be useful.

Thanks for any help people can offer.

Answer 1

Edit: After posted I found that you were talking about re-loading. That is too weird a condition. You might want to double check of thie "re-loading" condition. I keep the post here in case you need a reflection boundary condition.

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For bounce-back (reflection) boundary condition at $x_2$, the differential form boundary condition i s $$ - \frac{\partial N(x,t)}{\partial x } |_{x_2} = 0; $$ where $N(x,t)$ is the particle distribution function, a negative sign indicates the right side boundary condition (though it make no difference here).

In term of your diffusion kerner, the reflection boundary would be (assume the boundary at the right hand side $x_2 > x, x'$:

$$ G_r(x, x';t) = G(x, x'; t) + G(2 x_2 -x, x'). $$

This construction keeps the differentiation of $G_r(x, x';t)$ vanishes at $x_2$.

The reason of this boundary condition refers to an earlier article of S. Chandrasekhar, Stochastic Problems in Physics and Astronomy, Rev. Mod. Phys, 15 (1943).