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I've been reading about kinetic theory of gases, which only deals with gases. I know that the lattice Boltzmann method, which is commonly used to simulate fluid flows, finds its origin in the kinetic theory of gases. After reading a brief chronological history, I wonder who extended the theory from gases to fluids in general and what arguments were used. I wonder if it was done along with the creation of statistical physics.

I've read some posts on this website, however I've not seen a clear answer to my question.

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  • $\begingroup$ Does this help: HPP model in Wikipedia ? $\endgroup$
    – Mauricio
    Aug 23, 2022 at 16:22
  • $\begingroup$ Sadly, I already know how HPP and FHP. I don't think they explain why the kinetic theory of gases were extended to fluid simulations ... I've read somewhere it might be Gibbs but I'm not sure $\endgroup$
    – Atmos
    Aug 23, 2022 at 18:40

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Considering that "fluid" is a joint name for gases and liquids, I will interpret the question as about extending the Maxwell-Boltzmann theory of dilute gases to dense gases and liquids. Although there were major developments in this direction since 1920s, they are rather fragmented, and a unified kinetic theory of dense fluids does not exist.

One can see the 1905-06 work of Einstein and Smoluchowski on the Brownian motion as a step in this direction, and the Chapman–Enskog theory (1916-17) that related the Boltzmann equation and the $H$-theorem to Euler and Navier-Stokes equations as another step.

The main difficulty with extension beyond dilute gases was that one could no longer assume molecular chaos, i.e. ignore position and/or velocity correlations between molecules. The simplest version was developed by Enskog in 1920s, some historical details can be found in Cohen's survey The kinetic theory of fluids:

"The simplest such correlation occurs when the presence of a third particle, 3, prevents particles 1 and 2, to be at the positions $r_1$ and $r_2$... If one takes only those into account, and also the difference in position of the two colliding particles, one obtains a kinetic equation for a dense gas of hard spheres that Enskog had already found in the 1920s.

This equation is, of course, only a first step in the generalization of the Boltzmann equation to higher densities. So far it has not been extended to realistic intermolecular potentials. Nevertheless, it has been very useful for describing the density dependence of transport coefficients of dense fluids in practice."

Writing in 1946, Born and Green described a rather skeptical attitude concerning kinetic theory of dense fluids even at equilibrium, while proposing a new approach in a multi-part General kinetic theory of liquids:

"It has been said that there exists no general theory of liquids because it is impossible to utilize the simplifying conditions either of the kinetic theory of gases (Chapman & Cowling 1939) where the density is small, or of the theory of solids (Born 1923) where a high degree of spatial order may be assumed. Yet a mathematical formulation of the problem should be possible, without making such an assumption, since only the general laws of mechanics and statistics are involved; though the solution itself may be extremely difficult."

However, velocity correlations for non-equilibrium fluids proved much more challenging, as Cohen describes:

"Nicolai Bogolubov, Melville Green, I and many others in the years after 1945 tried to continue in this way and considered velocity correlations between the particles 1 and 2 at $r_1$ and $r_2$ due to previous collisions with two particles... However, as was discovered around 1965... in contrast with the case of a fluid in thermal equilibrium, velocity correlations in a nonequilibrium fluid introduce long-range correlations between the particles that prevent a systematic generalization of the Boltzmann equation to higher densities".

Even more complex velocity correlations in high density fluids led to more negative results. They generate long-time tails, discovered by Alder and Wainwright (1967) in computer simulations: hydrodynamic equations beyond the Navier-Stokes equations do not seem to exist, as Burnett and higher-order hydrodynamic equations do not have finite transport coefficients.

The lattice Bolztmann theory is more recent, and was developed, in part, in response to the above negative results:

"The lattice Boltzmann equation (LBE) is a minimal form of Boltzmann kinetic equation which is meant to simulate the dynamic behaviour of fluid flows without directly solving the equations of continuum fluid mechanics. Instead, macroscopic fluid dynamics emerges from the underlying dynamics of a fictitious ensemble of particles, whose motion and interactions are confined to a regular space-time lattice...

The LBE can be shown to quantitatively reproduce the equations of motion (Chapman and Cowling, 1952) of continuum fluid mechanics, in the limit of long wavelengths as compared to the lattice scale. The idea of simplified Boltzmann equations with discrete speeds can be traced to the pioneering work of Broadwell, back in the 60's (Broadwell, 1964). However, to the best of our knowledge, these discrete-velocity Boltzmann equations were mostly intended to provide simpler, sometimes even analytically tractable, model equations for rarefied gas dynamics, but never meant as a computational alternative for the numerical solution of the Navier-Stokes equations of continuum fluid dynamics.

The major conceptual twist in this direction had to wait another 20 years, with the advent of the celebrated Frisch-Hasslacher-Pomeau (FHP) lattice gas automaton (Frisch et al., 1986; Wolfram, 1986, Wolf-Gladrow, 2000). Originally developed in response to the major pitfalls of the Lattice Gas Cellular Automaton approach (Frisch et al., 1986; Wolf-Gladrow, 2000), the LBE rapidly developed into a vigorous self-standing research subject (Mc Namara and Zanetti, 1988; Higuera and Jimenez, 1989; Higuera et al., 1989; Chen et al., 1992)."

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