# Do I understand correctly that simulation volume influences relative permittivity calculations? If so, how is the result useful?

I am having a lot of trouble understanding the influence of volume on relative permittivity/static dielectric constant. I'm not a chemist, but am using molecular dynamics to calculate relative permittivity for use in other work.

A predecessor used this equation for relative permittivity $$ε_\mathrm{r}$$:

$$ε_\mathrm{r} = 1 + \frac{4π\sigma_M}{3k_\mathrm{B}TV}$$

where $$V$$ is the volume of the simulation, $$T$$ is temperature, $$\sigma_M$$ is the variance of the total dipole moment, and $$k_\mathrm{B}$$ is the Boltzmann constant.

Here is what I do not understand: If I use a different $$V$$(i.e., a different number of atoms) for my simulation compared to my predecessor, then my calculated relative permittivity will be different from his. I have done this to make sure, and indeed, the difference is quite significant.

Is there something silly I am misunderstanding here? Like "$$V$$" is not the average box/cell volume of the simulation, but something that is not size-dependent?

• Your equation looks right. It is the same as can be found in this paper, arxiv.org/ftp/arxiv/papers/1009/1009.5958.pdf. I am not certain of this, but I would guess that the variance of the total dipole moment would change when you change the total volume, and thus your result should be more or less volume independent. Oct 20 '19 at 9:05
• Thanks very much, Erik. I wondered if that were meant to be the case. So then does it seem likely to you that when two calculations (each with a different number of atoms) lead to different calculated values, that the system volume is too small for one of the calculations? Oct 20 '19 at 9:07
• And a follow-up question to that for anyone who is interested: would it make sense to run the smaller system for longer to achieve the same relative permittivity value as the larger system? Or is there something more fundamental that's a problem about using a smaller system? Oct 20 '19 at 9:20
• It's not quite clear how you determine the variance of total dipole moment. Is it just a mean square? Could you probably elaborate on this a bit further and add the exact math expression you are using in the equation? Oct 20 '19 at 10:31
• Thank you very much for asking, andselisk, and thank you for your edits. The molecular dynamics software I'm using outputs the total dipole moment and I'm using Python/numpty's "variance" function (e.g., "np.var") to calculate the variance. Please feel free to ask if you would like me to add any more detail about this. Oct 20 '19 at 10:47