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I have been reading a recent paper. In it, the authors performed molecular dynamics (MD) simulations of parallel-plate supercapacitors, in which liquid resides between the parallel-plate electrodes. To simplify the situation, let us suppose that the liquid between the electrodes is argon liquid.

The system has a "slab" geometry, so the authors are only interested in variations of the liquid structure along the $z$ direction. Thus, the authors compute the particle number densities averaged over $x$ and $y$: $\bar{n}_\alpha(z)$, where $\alpha$ is a solvent species. (That is, in my simplified example, $\alpha$ is argon -- an argon atom.) $\bar{n} _\alpha(z)$ has dimensions of $\frac{\text{number}}{\text{length}^3}$ or simply $\text{length}^{-3}$, I think.

The $xy$-plane is given by the inequalities $-x_0 < x < x_0$ and $-y_0 < y < y_0$. The area $A_0$ of the $xy$-plane is thus given by $A_0 = 4x_0y_0$.

So, the authors define the particle number density averaged over $x$ and $y$ as follows: $$\bar{n}_\alpha(z) = A_0^{-1} \int_{-x_0}^{x_0} \int_{-y_0}^{y_0} dx^\prime dy^\prime n_\alpha(x^\prime, y^\prime, z)$$ where $A_0 = 4x_0y_0$ and $n_\alpha(x, y, z)$ is the local number density of $\alpha$ at $(x, y, z)$.

Thus, $\bar{n}_\alpha(z)$ is simply proportional to $n_\alpha$ integrated over $x$ and $y$. But, my question is, what is $n_\alpha(x, y, z)$? How is $n_\alpha(x, y, z)$ determined in practice?

As far as the computer is concerned, the argon atoms are point particles; they are modeled as having zero volume (although they interact by Lennard-Jones interactions). So how is it possible to define a number density?

Do we simply "cut" the "slab" in "slices" along $z$ and then assign the particles to these slices? There might be 5 particles in the first $z$ slice, 10 in the second, 7 in the third, and so on. If I then divide 5, 10, and 7 by the volume of the respective slice, then I have a sort of number density, with units of $\frac{\text{number}}{\text{length}^3}$ or simply $\text{length}^{-3}$. But how do I now integrate this $n_\alpha(x^\prime, y^\prime, z)$ over $x$ and $y$? Do I have to additionally perform binning in the $x$ and $y$ directions?

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    $\begingroup$ I don't have access to the article. I believe it is common to bin, i.e., discretize the $z$ direction. $\endgroup$ – Deathbreath Jul 16 '12 at 12:40
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As is often the case in simulation papers, the mathematical description of the reported quantities are not literally describing the algorithm used to compute these quantities. (This typically happens when the main author and the compute monkey are not the same person.)

In your case, there is no point to first compute $n_\alpha(x',y',z)$ and then integrate out $x$ and $y$. As you suggest in the question, one may estimate $\bar{n}_\alpha(z)$ directly by only constructing bins along the Z-axis. Just compute (the time-average of) the number of particles in a bin and divide it by the volume of the bin. That will give you an estimate of $\bar{n}_\alpha(z)$ in the bin.

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