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What is kinetic theory? I am taking a course on fluid dynamics right now, and I have been wondering about one thing for some time now. We have three ways to look at a gas ($N$ particles):

1) Microscopic: The state (position and velocity) of each particle is known at every point of time. Evolution of the state is determined by Newton.

2) Kinetic: We have a distribution function f (as the solution of the Boltzmann-equation) of the particles. We have derived the Boltzmann equation by using Heuristic arguments and Newton.

3) Macroscopic: Looking at macroscopic quantities.

Is this correct? And is the essence of the kinetic view the fact that we have a distribution function and an evolution equation for that? Especially in contrast to the microscopic view? In addition, I wrote down the word "continuum" next to kinetic in my lecture notes but can't make sense of that.

And another question: We derived the macroscopic view from the kinetic view, but i am not so sure about the relation between the microscopic view and the kinetic view.

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  • $\begingroup$ Continuum mechanics: There are an infinite number of infinitesimally small particles, making it impossible to know the state of each and every one of them. Instead one describes state in terms of thermodynamic variables (pressure, temperature, density, ...) as functions of position and time. $\endgroup$ – David Hammen Aug 17 '18 at 15:13
  • $\begingroup$ Please refine this question so (a) you are asking just one question, and (b) the question isn't so broad that we need to write a textbook, or even a chapter of a textbook to answer it. As is, this question will almost certainly be closed. $\endgroup$ – David Hammen Aug 17 '18 at 15:14
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The question is broad, but I'll try to address some of the key inquiries.

The most common definition of a kinetic theory, as you mention, is one that prescribes probability distributions for the states (usually positions and momenta) of particles composing a material system. These probability distributions can be derived from microscopic models fairly rigorously using combinatorial/statistical/physical arguments (see here, for example).

In this sense, you find that you are not dealing with an overwhelming amount of coupled deterministic dynamical equations, and are instead dealing with the temporal evolution (or equilibrium properties) of a probability distribution over state space or some sub-region of it.

Macroscopic or continuum theories of material systems are then almost always derived by integrating over momentum space to generate momentum-averaged "deterministic" equations, which constitute the principal equations of continuum mechanics. This book discusses such things in length.

I should also mention that the "vanilla" Boltzmann equations have very little heuristics associated with them; they can be readily obtained from a transport-theoretical argument over 6-D phase space for an arbitrary probability distribution.

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