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The reason for this animation is that a previous hard sphere animation was shown not able to carry an acoustic wave. An acoustic wave is a wave where the pressure variations are periodic and move at a speed `c=sqrt(B/rho)` where `B` is the bulk modulus `and rho` is the density. For this animation I have used an impeller at radian frequency `omega` that moves in such a way as keep the gas volume constant and therefore does not compress the gas. The enclosure and the impeller are implemented with "soft" repelling potentials (see color bar for scale). The soft potentials rise as exponentials so that there are no harsh collisions with the boundaries. Also the particles are kept separated by soft repelling Lennard-Jones potentials. My physicsanimations.org Gas Physics menu items demonstrate agreement of hard sphere gas physics with almost all other aspects of gas physics but acoustic waves do not seem to be demonstrable with hard spheres. Any wave that is started, decays away in just a few gas mean free paths. I thought that this decay could be caused by the lack of connectivity between particles. An inter-particle potential might provide that connectivity so that is what is being tried here.
An acoustic wave is a wave where the pressure and density variations are periodic and move at a speed of approximately `c=sqrt(B/rho)` where `B` is the bulk modulus and `rho` is the density. Since the wave vector `k= omega/c`, where `omega` is the driving frequency, we expect to see spatial wavelengths equal to `lambda=2*pi/k=2*pi*c/omega`. Both the particle density and the particle pressure are expected to have these spatial wavelengths. Of course, since the particle motion is random, we cannot expect to easily see the spatial variations. For that reason, the density and pressure Vs x are separated into bins and to these a Fast Fourier Fransform (FFT) is applied to find the frequency content.
The FFTs show "some" agreement with the expected wavelengths but I'd have to say that this is far from conclusive. Perhaps it will be necessary to complete the Lennard Jones force by adding its attractive component. That would be like saying that the gas has to have some properties of a liquid in order to carry an acoustic wave and would be a major departure from the way we think of gases.
In addition to exploring the acoustic wave aspects of this animation, I am showing at left how the energy distribution behaves as the disc energy increases due to the motion of the enclosure. That is the Accumulated N(E) plot at the left. For a 2D group of particles we expect the N(E) to be a simple Boltzmann exponential `N(E)=N_0exp(-E/E_(avg)) where E_(avg) is the average kinetic energy. Since E_(avg) changes as time goes on, I chose to sum the energy number bins for just 200 frames and plot the distribution as these 200 frames accumulate and then dump the previous bins. The agreement with the Boltzmannn distribution is very good even under these dynamic conditions. To help visualize the actual distribution of energy, I've color coded the discs which can be interpreted by the color bar at the left. Of course some of the discs will have energies greater than 4 times the average so I just make these red.