- #1

terra

- 27

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First, two definitions: let ## \varrho (M)## be the probability density of macro states ##M ## (which correspond to a subgroup of the phase space) and ## \mathrm{d} \Gamma ## be the volume element of a phase space.

In my lecture notes, the derivation for continuity equation of probability density starts with:

$$ \frac{\mathrm{d}}{\mathrm{d}t} \int \mathrm{d} \Gamma \, \varrho= \int \frac{ \bar{n} \cdot \bar{v} \mathrm{d}A \, \mathrm{d}t}{\mathrm{d}t} \varrho + \int \mathrm{d} \Gamma \, \partial_t \varrho ,$$

where the first term on right hand side is supposed to be the time-derivative of the volume element in phase space.

This step bothers me.

In my lecture notes, the derivation for continuity equation of probability density starts with:

$$ \frac{\mathrm{d}}{\mathrm{d}t} \int \mathrm{d} \Gamma \, \varrho= \int \frac{ \bar{n} \cdot \bar{v} \mathrm{d}A \, \mathrm{d}t}{\mathrm{d}t} \varrho + \int \mathrm{d} \Gamma \, \partial_t \varrho ,$$

where the first term on right hand side is supposed to be the time-derivative of the volume element in phase space.

This step bothers me.

**Why do we also take the derivative of**## \boldsymbol{ \mathrm{d} \Gamma} ## - are these not usually ignored (see, for instance, Leibniz integral rule for taking a derivative inside an integral)?**What kind of a mathematical object is**## \boldsymbol{\mathrm{d} \Gamma}##**?**
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