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Suppose we have a large set of particles, and we know, that some parameter $x$ of each particle satisfies the following equation:

$\dfrac{dx_i}{dt}=f(x_i)$.

Our set of particles can be described with particle distribution function $\Psi(x, t)=dN/dx$, which gives the number of particles with parameter in range ($x, x+dx)$, so total number of particles $N$ is

$N=\int\Psi(x,t)dx$

Suppose for certainty the distribution is similar to normal distribution. How one can derive the corresponding equation for the distribution function:

$\dfrac{\partial\Psi(x, t)}{\partial t}=?$

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2 Answers 2

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This question seems like it may be , so I will only provide a hint here for now. (Please read the tag wiki and clarify, if this is insufficient.)

A reasonable approach would be to use a (Lagrangian) control volume over an interval $[x,x+\Delta{x}]$ to define an integral equation for the number of particles $n$ it contains. Then you can apply the Leibniz rule for differentiating an integral to compute $\frac{dn}{dt}$, and explore the behavior as the length $\Delta{x}$ goes to zero.

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Here is another hint that you should think about: The number of particles in a certain region of spaces changes become some particles move in and other particles move out. For a one (spatial) dimensional, discrete example, imagine that the number of particles at lattice points labeled by integers $k$ and $t$, and that a particle at $k$ at time $t$ winds up at $k+n$ at time $t+1$, for some constant integer $n$. Then, the number of particles at position $k$ at time $t+1$ is equal to the number of particles at position $k-n$ at time $t$. Using similar reasoning, try to set up a differential equation that expresses how particles move into and out of a small region of space.

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