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For the quantity $\rho(x,t)$ with the continuum hypothesis am I taking the average value of the density at each point in the small volume surrounding the point $x$ or am I taking the average density of each molecule in the small volume. If it's the latter what does this intuitively mean?

Winther
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usainlightning
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  • It's the former. If we imagine the matter as continuous, there are no molecules. –  Mar 28 '15 at 23:34
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    What is the reason for averaging here though? Why can't we just sum up all the masses at each point of the small volume and divide by the small volume to get the density and if there are no molecules at a specific point say that it has zero mass? – usainlightning Mar 29 '15 at 00:20
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    Because there is no mass at any particular point. It's spread out over the space continuously. No molecules. –  Mar 29 '15 at 00:20

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It's the average of a small enough volume. In order to quantify the validity of continuum hypothesis you can use Knudsen number. $K_n=\lambda/L$ where $\lambda$ is the mean free path of a material particle and $L$ is the characteristic length scale of the problem. if Knudsen is larger than one, you can't use continuum hypothesis.

Arietul
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