What is (local) pressure within a gas on the microscopic level?

Question

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 of 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.

Answer 1

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.

Answer 2

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.