Let us consider a 3D box of volume $V$, containing $N$ identical Brownian particles. The diffusion coefficient of the particles is noted $D$. Inside this box there is a square surface of area $L^2$.
To keep everything as simple as possible, the box dimensions are much larger than that of the surface, and the surface dimensions are much larger than that of the Brownian particles. Moreover, the Brownian particles do not interact with each other and do not interact with the surface either.
What is the rate $j$ at which particles cross the surface?
Attempt 1: I have tried to combined intuition and dimensional analysis. $j$ physical dimensions are number/time. If the surface area is twice larger, $j$ is twice larger as well, and $j \propto L^2$. If there are twice more particles, $j$ will be twice larger, and $j \propto N$. If the box volume is twice smaller, it is equivalent than the number of particles being twice larger, and $j \propto 1/V$. The last two considerations amount to $j$ being proportional to the particle concentration $N/V$. To match the correct physical dimensions of the quantity $j$, I also need a quantity that involves time. $D$ is the only time-dependent quantity in my problem, and is expressed as length^2/time. Therefore I must have $j \propto D$. In summary I have
$$ j \propto \frac{N}{V}L^2D $$
This is wrong since the dimensions are not matching: left hand side dimension is 1/time and right hand side is length/time.
I think the correct answer is $j \propto \frac{N}{V}L D$ but it does not make sense to me, since in my opinion $j$ must scale with $L^2$.
What am I missing? Maybe the mean free path is also relevant?
Attempt 2. I also tried to solve the problem from a simulation point of view and discretized both time and space. So now the $N$ particles are randomly distributed on a 3D lattice which sites are cubes of dimension $a$. The surface is aligned with the lattice axis for simplicity. Every time step $\Delta t$, particles jump on a randomly selected neighbour site. The concentration of particles is small enough so that the probability that two particles occupy the same site is negligible.
During one time step $\Delta t$, the only particles that have a chance to cross the surface are the one located on sites that are adjacent to the surface. The total number of such sites is $$ 2 \frac{L^2}{a^2} \ . $$ The probability that any given site is occupied by a particle is $$ \frac{N a^3}{V} \ . $$ For any given particle located on a site adjacent to the surface, the probability to cross the surface during $\Delta t$ is $1/6$ (since the particles randomly choose one direction out of the six available directions).
Therefore the average number of particles crossing the surface during $\Delta t$ is $$ \frac{N a L^2}{3 V} \ . $$ Since every time step is equivalent, the number of particles crossing the surface, per unit time $\Delta t$, is $$ \frac{N a L^2}{3 V \Delta t} \ . $$
Now we need to relate the diffusion coefficient $D$ to the lattice dimension $a$ and the time step $\Delta t$. This a discrete random walk problem and the solution is $$ D = { \frac{a^2}{2 \Delta t}} \ . $$
We therefore find the rate $j$ of crossing: $$ j=\frac{2}{3} \frac{N}{V} \frac{L^2}{a} D $$
In my opinion the lattice size $a$ can be interpreted as the mean free path.
Now I recover the scaling with $L^2$, but I am uncomfortable with this result, as $D$ is a function of $a$ and $\Delta t$.
