# Averaging over random potential of impurities

## Main Question or Discussion Point

Hey,

In doing calculations on superconductors, I often hear that people say "averaging over the random potential of impurities make the theory translationally invariant both in time and space".

I do not exactly understand it? Could you please explain it through a simple example or by citing a readable reference?

Thanks a lot!

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Hey,

In doing calculations on superconductors, I often hear that people say "averaging over the random potential of impurities make the theory translationally invariant both in time and space".

I do not exactly understand it? Could you please explain it through a simple example or by citing a readable reference?

Thanks a lot!
I can't address specific issues with superconductors, but in general the statement means location in space and time does not affect the statistical expected value of the "potential" for impurities. That is, the expected value is invariant with respect to translation in space and independent of time (at least within some understood context). Specifically, the expected value would be the mean of a suitably large sample (or multiple samples) of measurements of concentrations of impurities in specific semiconductors.

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I can't address specific issues with superconductors, but in general the statement means location in space and time does not affect the statistical expected value of the "potential" for impurities. That is, the expected value is invariant with respect to translation in space and independent of time (at least within some understood context).
Thanks, SW VandeCarr! Intuitively, I understand that because of huge number of scatterings from impurities in all directions, preferred directions in space disappear; however I do not see how it can happen in mathematics through a concrete example.

Thanks, SW VandeCarr! Intuitively, I understand that because of huge number of scatterings from impurities in all directions, preferred directions in space disappear; however I do not see how it can happen in mathematics through a concrete example.
I anticipated your response and edited my previous post. Just speaking generally,an average is based on data, not theory. You also need to define as precisely as possible the "population" being sampled in terms of homogeneity so that you can make useful inferences. Are you talking about all superconductors? I would think the distribution of concentrations of impurities might vary by type, processing etc.

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