# probability density distribution: From free diffusion to presence of a barrier

I am a biologist and I am not very comfortable with statistical mechanics. However, I want to learn and I am trying to understand. I just need some clues from people that handle these topics easily.

I would like to model the probability density distribution (PDF) for a particle moving from xo to x in time t, in presence of a barrier. We have the solution for "absence of a barrier" = free diffusion (1st case). We have the solution in presence of a barrier (2nd case). However, I'm interested in knowing how PDF would change from "1st case" to "2nd case" if I model the adding of a barrier at time t1.

Statistical mechanics is used in both cases. I'm interested to know what I should study and look at if I want to model this "phase transition" from case 1 to case 2.

Actually, all the work has been done in the paper that you refer to. The solution of diffusion equations is conveniently handled by calculating the propagator or Green's function, $G(x,y;t)$. This is the probability density of finding a particle at position $x$ at time $t$, given that it started at position $y$ at time $0$. You can imagine it describing how a very narrow initial distribution of density (a Dirac delta function $\delta(x-y)$ ) spreads with time, and becomes a Gaussian function of $x$ at later times: $$G_0(x,y;t) = \frac{1}{\sqrt{4\pi Dt}} \exp[-(x-y)^2/4Dt]$$ The useful feature is that, because the diffusion equation is linear, any initial distribution of density $\rho(x,0)$ can be described as a linear superposition of spreading Gaussians, determined by the initial conditions. So at time $t_1$ $$\rho(x,t_1) = \int_{-\infty}^\infty \, G_0(x,y;t_1) \, \rho(y,0) \, dy$$ depending on your initial density $\rho$ at $t=0$. If your particles really are localised at position $y$ at time zero, $\rho(x,0)=\delta(x-y)$, then this simplifies to $\rho(x,t_1)=G_0(x,y;t_1)$.
Now, the propagator must be constructed to comply with the imposed boundary conditions. The propagator above, $G_0$, is for no barrier: the boundary conditions are simply that the density should vanish as $x\rightarrow\pm\infty$. When the semi-permeable barrier is present, the propagator is more complicated. However, the cited paper gives the propagator for this case, $G(x,y;t)$, in eqn (5), in terms of complementary error functions. It is a combination of slightly different formulae depending on the signs of $x$ and $y$. Nonetheless, it functions in exactly the same way as $G_0$. Therefore, if you insert the barrier at time $t_1$, start with the density $\rho(x,t_1)$ determined by the no-barrier propagator up to that time, and seek solutions at $t>t_1$, you get $$\rho(x,t) = \int_{-\infty}^\infty \, G(x,y;t-t_1) \, \rho(y,t_1) \, dy$$ In the case that $\rho(x,0)=\delta(x-y)$ we can write this as $$\rho(x,t) = \int_{-\infty}^\infty \, G(x,y';t-t_1) \, G_0(y',y;t_1) \, dy'$$ and the quantity on the left can be interpreted as the overall propagator (probability density to be at position $x$ at time $t$, given initial position $y$ at time $0$), incorporating the effects of free diffusion up to time $t_1$, followed by diffusion in the presence of the barrier for later times.
NB in the equations above $\rho(x,t)$ represents the probability density of the diffusing particles, as a function of position and time. In the cited paper, $\rho$ is used for something different.
• That sounds like the starting point for a new question! A lot depends on the details of the measurements. Presumably, if you could measure the density profile accurately as a function of position and time, and if it were a very narrow distribution at $t=0$, you could detect the time at which it started to become non-Gaussian. You might even deduce something from the time evolution of the mean-squared displacement, if that can be measured. But maybe things are not so simple, in which case I doubt that I'm the right person to ask. Best to frame a new question, explaining what can be measured. – user197851 Sep 16 '18 at 23:20