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Suppose I have a gas of particles that is initially uniformly distributed so that the number density is $n_0$ (number of particles per unit volume), and then I displace the particles by the vector field $\vec{d}(\vec{x})$ (i.e. the particle initially at position $\vec{x}$ is displaced by the vector $\vec{d}$). How is the resulting number density $n(\vec{x})$ related to the displacement vector $\vec{d}(\vec{x})$? I'm sure this must be done somewhere in a standard textbook but I can't find where. |
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A simple 1D calculation gives, at first order, something like $$ \frac{1}{n} = \frac{1}{n_0}\left(1+\vec{\nabla}\cdot\vec{d}\right) $$ but only if $\vec{d}$ is small enough. Otherwise, for calculating $n(\vec{x})$, you need to evaluate the divergence at a point $\vec{x'}$ such that $\vec{x'}+\vec{d}(\vec{x'}) = \vec{x}$. |
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Take a point A. Find where it's been translated to, say A'. Add to the density at A' the original density at A divided by the Jacobian of the transformation at A. Integrate this over all A. |
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