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Suppose I have data that represents a field of positions and velocities. If the distribution function (DF) for the data is $f(x,v,t)$, I know that the DF must obey

$$\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} v + \frac{\partial f}{\partial v}a = 0$$

Is there an equivalent way to write the same condition, but for the trajectory a single particle (rather than "fluid element") must follow?

I.e., for $x$: $$\frac{\partial x }{ \partial t} + v_x + \cdots$$

Note, I am working in the context of collisionless dynamics of gravitating systems.

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A single particle will just follow Newton's force equation, using the acceleration as $a=F/m$, where $F$ is the prescribed force on particles. The Boltzmann equation is built from the assumption that particles do this, which is why you can change $dx/dt$ to $v$ and $dv/dt$ to $a$. You're considering flux of particles through phase space, where each particle follows Newton's force equation, and the particles don't collide with each other.

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I think what you're looking for is called Newton's second law. You know the force acting on a single particle, you calculate its acceleration, and you follow this particle in its motion through space and time. This approach is called Lagrangian and it is not appropriate for the description of certain physical phenomena, for which a so-called Eulerian approach is recommended. Here you look at a small region of the position-velocity space, you count particles in this region and you look at how this number changes in time. This is the fundamental meaning of Boltzmann's equation.

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