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This question stems from a problem I was doing on the Debye-Hückel theory. It says that the continuum approximation which underlies the Debye-Hückel theory is valid provided that $\lambda_D \gg r_{ij}$ where $\lambda_D$ is the Debye screening length and $r_{ij}$ is the interparticle spacing for some $i$ and $j$.

I am just wondering what exactly this continuum approximation is? I am thinking it is the approximation of smearing out the background charge density to produce, what I have heard of in condensed matter, jellium. But I am not sure how this relates to the given inequality in the above.

Many thanks.

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The continuum approximation that they are referring to, also called the Knudsen approximation, is based on assuming that the molecular mean free path is smaller than the smallest representative length scale of the physical problem.

This essentially ensures that the mathematics of continuum mechanics, rather than that of statistical mechanics, is appropriate for solving the problem; in other words, you don't need to consider anything stochastic. In the case of electrohydrodynamics, this smallest length scale is almost always the Debye screening length $\lambda_D$: hence the requirement that $\lambda_D \gg r_{ij}$, since $r_{ij}$ is a good estimate for the mean free path.

Hope this helps!

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