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Fluid dynamics view
Fluid dynamics describes liquids and gases in terms of local pressure (among other variables), which varies from a point to point. While mathematically the concept is well-defined (see, e.g., internal pressure), it is not fully obvious (to me at this moment) how the parts of the media interact with each other. To be more precise:

  • in a solid there are forces between the constituent atoms/molecules that resist deformation of any chosen volume, which immediately translate into normal and shear stresses, which are essentially pressures;
  • in a liquid the atoms/molecules can be easily displaced in respect to each other, but they are tightly packed and interact, so one can talk about a force resisting compression or extension fo a volume, hence the pressure;
  • in a gas however the atoms/molecules interact only via collisions, which are rare - the more so the more the gas is diluted. Can we meaningfully talk about pressure within a gas?

Statistical physics view
Another take on it: in statistical physics the gas pressure is calculated as the result of the atoms/molecules collisions with the walls of the container. However, in the bulk of the gas the molecules do not interact - apart from the collisions that are typically neglected (although they are responsible for the establishment of the thermodynamic equilibrium). If we were to place a continuous surface anywhere within the container, tehre would be pressure exterted on this surface by the molecules scattered from it. However, in the absence of such a surface the molecules continue to move freely. One could thus define the local pressure within a gas as a pressure that would be exerted on a surface rather than an actually existing force, but this undermines the fluid dynamics, where local forces are assumed to exist. The two specific examples that interest me are: (a) pressure on a volume of hot air (see How does hot air rise?), and (b) pressure waves in air.

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    $\begingroup$ Isn't pressure a macroscopic concept? Think about the statistical mechanics view you described: the "pressure" is the cumulative result of many atoms striking a wall. If you are interested in a molecule-by-molecule analysis, you shouldn't be talking about a "pressure", but should be analyzing the forces directly. $\endgroup$
    – Michael M
    Nov 17 '21 at 11:10
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    $\begingroup$ @MichaelM Pressure is a force pur unit area, so it doesn't make much difference whether we talk about one or the other. The question is more of how a macroscopic force and motion arise... from atomic collisions? $\endgroup$ Nov 17 '21 at 11:19
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    $\begingroup$ The forces can arise from many sources, as you describe in your question. I think that fact is evident in the variety of constitutive equations that exist for different media. Gases can often be described by the ideal gas law, while I don't know of any constitutive equation that works for arbitrary amplitudes in liquids or solids (usually you just assume a Taylor expansion for wave motion). $\endgroup$
    – Michael M
    Nov 17 '21 at 11:23
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    $\begingroup$ @MichaelM if we use the treatment from the standard statistical mechanics, the collisions are neglected - one uses the Hamiltonian function for free particles (in partition function, etc.) But there is usually underlying reasoning that the collisions do exist and lead for the establishment of the thermodynamic equilibrium, and that we waited long enought. I suspect the transmission of pressure through gas is considered on microscopic level in physical kinetics (i.e., non-equilibroum stat physics) using Boltzmann equation. $\endgroup$ Nov 17 '21 at 12:43
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    $\begingroup$ @RogerVadim Clearly macroscopic models that work on a larger scale can neglect collisions but models in kinetic theory do not, even though it might not be obvious: E.g. the 1/6th model in kinetic theory assumes molecular chaos, meaning $\overline{v^2} = \overline{v_x^2} = \overline{v_y^2} = \overline{v_z^2}$. This is actually not neglecting internal collisions but instead assuming isotropy of these collisions (no Cartesian direction is preferred) such that their precise effect does not have to be modelled. $\endgroup$
    – 2b-t
    Nov 17 '21 at 22:18
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Yes, generally speaking it makes sense to talk about macroscopic averages such as pressure also for gases as they are sufficiently dense in most common situations on earth ($Kn \ll 1$). In this case the pressure emerges from elastic collisions as well as additional repulsive forces.


Your statements

"in a gas however the atoms/molecules interact only via collisions, which are rare - the more so the more the gas is diluted."

and

However, in the bulk of the gas the molecules do not interact - apart from the collisions that are typically neglected (although they are responsible for the establishment of the thermodynamic equilibrium). If we were to place a continuous surface anywhere within the container, there would be pressure exerted on this surface by the molecules scattered from it. However, in the absence of such a surface the molecules continue to move freely.

are only partially true, gases on earth are actually denser than you would believe: Under norm conditions there are around $2.69 \times 10^{19}$ molecules in a cubic centimeter of air, which makes particle collision fairly common. The underlying continuum assumption (the assumption that suitable averages on a larger scale - macroscopic values such as pressure - can be found) only breaks down in extremely diluted flows such as in atmospheric entries. Under these circumstances the definition of macroscopic values such as pressure indeed does not make sense.

Kinetic theory of of ideal gases

In order to quantify under which conditions this happens (meaning that the continuum hypothesis loses its validity) one has to turn to kinetic theory of gases. There, starting from considerations based on elastic inter-particle collision (this is only a viable assumption for light mono-atomic gases, for larger molecules one would actually have to assume also far-field interactions caused by repulsive forces) in highly diluted gases - by employing statistical mechanics - one can find an evolution equation for a particle distribution, the Boltzmann equation.

From there you can go on and make estimates about the number of collision per time and phase space element, given by the so called Stoßzahl. For a gas in equilibrium one could find the number of collisions by assuming a Maxwell-Boltzmann equilibrium distribution and integrating out.

Finally one can try to find equations that emerge on a larger time scale by performing a so called Chapman-Enskog expansion, a perturbation analysis. There one finds that for the limits of small ratios of the mean free path $\lambda$ and the characteristic length scale of a given problem $L$, the Knudsen number

$$ Kn := \frac{\lambda}{L} \phantom{spacespace} \frac{\text{mean free path}}{\text{representative physical length scale}}$$

the Navier-Stokes equations which were initially derived for dense fluids on a macroscopic level. In this context one can find that the average fluctuations of the relative velocity (the velocity of the individual fluid particles relative to the macroscopic flow)

$$ p = \frac{m_P}{3} \int |\vec \xi - \vec u|^2 f d \vec \xi,$$

correspond to the macroscopic pressure.

As a threshold to continuum based flow generally $Kn \lesssim 0.01$ is taken. In this limit macroscopic averages like pressure and viscosity make sense, they are sufficiently smooth and vary only slowly in time. This is by no means a strict boundary but rather an order of magnitude for which the aforementioned second-order expansion in a perturbation series still makes sense. For larger Knudsen numbers common continuum assumptions like the no-slip condition will not be accurate but instead one will have a certain slip-velocity at a solid boundary and said average will become too noisy to actually have physical meaning on the scale of interest.

For an ideal gas one can correlate the Knudsen number to two other common characteristic numbers, the Reynolds number $Re := \frac{U \, L}{\nu}$ - a measure for turbulence - and the Mach number $Ma := \frac{U}{c_s}$ - a measure for compressibility

$$ Kn \propto \frac{Ma}{Re} $$

For non-ideal gases one actually has to include far field interactions and inner degrees of freedom which makes the evaluation of the integrals quite tricky and leads to the kinetic theory of non-ideal gases.

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  • $\begingroup$ Thank you! I have myself began looking in the direction of how hydrodynamics emerges from the Boltzmann equation, and I am glad to have an input from an expert. Could you comment on my other question How does hot air rise? - what remains unclear to me is how macroscopic and microscopic motions segregate, as both ultimately result from collisions. $\endgroup$ Nov 18 '21 at 8:09
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    $\begingroup$ @RogerVadim You are welcome. I can recommend you "An Introduction to the Theory of the Boltzmann Equation" by Harris Stewart. The topic is very interesting, I hope you will like it. I will have a look at the question and try to answer it. I think you should not think about a segregation of macroscopic and microscopic. In the end both are describing the same thing. The macroscopic equations emerge on a much larger scale from the microscopic description. $\endgroup$
    – 2b-t
    Nov 18 '21 at 19:41
  • $\begingroup$ @RogerVadim You have probably heard about traffic waves, that the behaviour of drivers in some scenarios can be described with a wave equation. Each individual sitting there makes the decision to speed up and slow down, influenced by what the people in front of them do. If you ask the drivers why they reacted the way they did, they might tell you a lot, but I guess nobody was planning for a wave equation. Maybe this wave equation is also not perfect: There is this one old grandma that drives slowly and that one guy overestimating himself crashing his car into the one of the guy in front. $\endgroup$
    – 2b-t
    Nov 18 '21 at 19:52
  • $\begingroup$ @RogerVadim Nonetheless you can describe it that way on a larger scale with an accuracy that is enough to predict this situation. Similarly you might have heard about "Conway's game of life". It is a cellular automaton with very basic rules, you react to your neighbouring cells. But complex life emerges from it. There has been some maniac on Code Golf Stackexchange that created a digital clock by designing an appropriate initial distribution. $\endgroup$
    – 2b-t
    Nov 18 '21 at 19:58
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    $\begingroup$ @RogerVadim The microscopic behaviour causes this macroscopic phenomenon. On the microscopic level you might be interested in the rules that cause it, but all you care on the macroscopic level is this coarse-grained large-scale view. Some things can't be explained on this blurred macroscopic level, some things are not obvious on the microscopic level. The different descriptions coexist. The interesting thing about kinetic theory is that you can bridge the two levels and find connections between the two worlds, e.g. with the aforementioned Chapman-Enskog expansion. $\endgroup$
    – 2b-t
    Nov 18 '21 at 20:02
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You have to bear in mind that there are two kind of pressures: 1) the static pressure due to external forces like gravity (e.g. the pressure exerted on the surface of a table by an object sitting on it), and 2.) the dynamic pressure due to atoms/molecules bouncing back from the surface. Both mechanisms exert a force on the surface but their physical nature is different. For a gas, the pressure is of course solely due to collisions, but here the static part of the pressure shows up indirectly as the density gradient that arises from the gravitational compression, thus cancelling the gravitational force and resulting in a net zero force on each volume element. The volume element has to be much larger than the mean free path for particle collisions though (for air at atmospheric pressure the latter is about $10^{-5} cm$). On a smaller scale it is not possible to apply the concepts of macroscopic physics like density, temperature and pressure other than as probabilistic quantities.

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  • $\begingroup$ Actually there are three contributions that contribute and make up the momentum flux tensor: The dynamic pressure is a consequence of the directed macroscopic motion interacting with its surroundings, it stems from the advective term on a microscopic level. The static pressure $p \delta_{ij}$ emerges from the random motion (relative to the macroscopic velocity) as the isotropic part in the normal direction. In addition you have the viscous stresses $\tau_{ij}$ which are actually the deviatoric part caused by the relative motion. $\endgroup$
    – 2b-t
    Nov 18 '21 at 21:22
  • $\begingroup$ @2b-t One could call all contributions resulting from a momentum change of particles (be it directed or random/thermal) as dynamic pressure.. With my definition, static would only relate to pressure generated by large scale external forces. For instance, objects lying at rest on a scale exert only static pressure on it (due to gravity) and no dynamic pressure (as they are at rest). $\endgroup$
    – Thomas
    Nov 18 '21 at 22:42
  • $\begingroup$ One could indeed introduce that definition as it follows a well-defined logic but static and dynamic pressure in the context of fluid dynamics are well defined terms, generally used in the context of Bernoulli's principle in inviscid flows on a macroscopic level. As pointed out in the comment these terms emerge from said microscopic expressions in the Boltzmann equation. $\endgroup$
    – 2b-t
    Nov 18 '21 at 22:59
  • $\begingroup$ @2b-t The Boltzmann equation is essentially a transport equation describing the change of the particle distribution function in real and velocity space due to external forces and collisions. As such, it is probably a bit of overkill using it to explain the microscopic interpretation of pressure forces in gases. $\endgroup$
    – Thomas
    Nov 19 '21 at 22:50
  • $\begingroup$ I think this really depends on the audience. I can see that the target audience of this forum is very broad and most users are not interested in expertise but rather in somebody that explains them complex things in simple but less rigorous terms. Answering a question in all its complexity for them might be an overkill and instead one tries to reduce it as much as possible resulting in a loss of generality. My former professors of fluid dynamics would not approve most of the answers given here because they are over-simplified. $\endgroup$
    – 2b-t
    Nov 19 '21 at 23:11

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