# How to define (and solve) the diffusion equation with a sticky boundary at the origin?

For the diffusion equation $\frac{\partial} {\partial t} P_t(x)=D \frac{\partial^2} {\partial x^2} P_t(x)$, a reflecting boundary at the origin for example, means: $\frac{\partial} {\partial x} P_t(x=0)=0$.

What is the mathematical way of setting the condition that whenever a particle reaches the origin it stays there forever? Note that it does not 'vanish' from the system upon reaching x=0, namely I am not talking about an absorbing boundary.

Also, how do I solve that differential equation in that case?

(Thanks for to all helpers!)

• What is the difference between sticky and absorbing? If particle is absorbed, it doesn't participate in further system evolution ( and certain function, number of the particles, is not invariant). If boundary is sticking, you have at least two possible situations: 1. particle doesn't participate in further dynamics, number of particles is invariant - it gives you probably attractor like solution with all particles glued to boundary surface at the end and 2. The same as above but even glued, particle do participate in evolution, in some way you should describe by another equations – kakaz Jan 2 '18 at 12:54
• Cont. For example particles of finite sizes may cover the whole boundary surface and some kind of saturation of sticking effect may occur. In both situations I will look for simplest generalisation of absorbing boundary problem, counting vanishing particles and connecting solutions in both regimes. I don't expect complicated situation in between saturation and totally gluing, rather exponential decay of particles till saturation occur. But of course you may define more complicated relationship between sticking and dynamics, for example that only certain amount of particles may be glued... – kakaz Jan 2 '18 at 12:59
• Cont. And this gives you other possibilities like periodic dynamic or hysteresis. – kakaz Jan 2 '18 at 13:00

I would just take an absorbing boundary condition and then add the absorbed density as a delta function at the sticking point. For convenience, translate the origin so that the sticking point is $x_a>0$ and the particle starts from $x=0$ at $t=0$. The solution then is

$$P(x,t)=f(x,t)-f(2x_a-x,t)+N(t)\delta(x-x_a)$$ $$f(x,t)=(4\pi Dt)^{1/2}e^{-x^2/4Dt}$$ $$N(t)=1-{\rm erf}\,(x_a/\sqrt{4Dt})$$

see for example these lecture notes.

Diffusions with partially reflected (including sticky) boundary conditions are discussed in detail in

Kushner's proof for weak existence/uniqueness of this class of diffusions is based on the submartingale problem formulation developed in

As a byproduct, Kushner also explains how to numerically solve this problem by using the Markov Chain Approximation Method.

I agree with kakaz and Carlo Beenakker on stitching together the counted absorbed particles and the absorbing boundary condition solution. Carlo Beenakker quoted the solution for diffusion in 1D starting from a Dirac delta function initial condition. I'd like to add that this problem was also solved in 3D for a uniform initial density in a landmark paper by Smoluchowski in 1917 (http://link.springer.com/10.1007/BF01427232) and was reviewed in English by Chandrasekhar in 1943 (https://link.aps.org/doi/10.1103/RevModPhys.15.1). The result has been used to provide diffusion-limited reaction rates for chemical reaction rates and annihilation reactions between singlet excitons. Worth a read!