Relativistic Hydrodynamics 2026, 03: Newtonian Kinetic Theory
An ensemble of N free particles interacting through some coupling can be described in a variety of ways, distinguished by the dimensionless ratio of the deBroglie wavelength and the typical inter-particle separation. If this ratio is about 1 or larger, the waveforms of the various particles overlap and thus requires a QM approach, likely resulting in a N-particle wavefunction. If it's significantly smaller than 1, the wavefunctions of the particles are widely separated, and the QM interference is negligible. The individual wave packets evolve like isolated Schroedinger equation solutions, i.e. classical particles (Ehrenfest Theorem). We of course keep around the Boltzmann equation, and in plasmas, only subject to Coulomb forces, it transitions into the Vlasov-Maxwell equation, and if the long-range forces are gravitational, then general relativity holds and the Boltzmann equation transitions into the Einstein-Vlasov equation. Further, if the system itself is so large in particle number and it takes up so much space that the typical inter-particle separation can't be meaningfully pinned down statistically, it gains a continuous description, which is when one would refer to it as a "fluid".
We clearly want to stick to statistical descriptions, so we need a distribution function for the probability that a particle shows up in some spacetime volume. For the fluid description, we'll assume that all particles are of the same species and indistinguishable. The sanity test for this is of course an integral over the entire 6D spacetime volume of the plasma, with the expectation that it returns our total particle number. For the fluid description to make sense, all fluid elements should include a large number of particles, so to minimize statistical variance between different chunks of the integral. As far as it exists, the equilibrium distribution function is very interesting for a system.
Systems in which inter-particle interactions are possible, and those in which they're not (or rather those, in which they're ignored) is the last delineation to make in the description of the system. If no collisions happen, and an external force is applied to the particles with mass m, then all particles would travel into a thin subset of volume elements. Eventually, the number of particles in each cell is invariant. Every time a particle moves from one cell of phase space to the neighbouring one, it will be replaced by another from nearby cells. The collisionless Boltzmann equation is
If instead one includes "binary elastic collisions" produced by short range forces, which only influence the advection from one cell into a neighbouring one (or countering collisions preventing the advection), one gains the classical Boltzmann equation
It requires a numerical approach in application. Exceptions include the Fokker-Planck equation, which describes the case for a gravitational potential, which can be solved analytically. Solutions to the Boltzmann equations are equilibrium distribution functions, representing an asymptotic (time-independent) state of the system. For a monatomic fluid the solution is found in the H-theorem. It follows from the mathematically constructed equilibrium distribution
Assuming that mass linear momentum and energy are collisionally invariant, the conservation / transport equation is otherwise mainly defined by a density term. The transport flux of any quantity follows from this definition by entering the relevant collisional invariants. From this approach follow the moment equations.
By the results of the H-theorem, the coefficients have a uniform integral expression for some particle number density n. With this, and a description of the fluid, the classical Maxwell-Boltzmann distribution emerges, which governs a number of ensemble quantities, such as temperature and drift velocity. The zero-order approximation through the Maxwellian is a perfect fluid, the first-order approximation models non-perfect fluids via the stress tensor and Navier-Stokes equations.