Diffusion of a 2D gas

Kostas Papamichalis, Dr. Theoretical Physics

Diffusion in two dimensions: The theoretical model and activities

A 2D virtual gas is confined in a container thermally isolated. The container is divided into two similar chambers (left and right) which communicate through a gap of length h, initially closed by a barrier. In the virtual environment of the simulation, the user can watch the evolution of the system during the diffusion of the gas through the gap of the container.
At time t=0 the particles are confined in the left chamber of the container, and they have been placed at random positions. The directions of their velocities are also random. The velocity magnitudes are random numbers picked in specific real intervals, according to the chosen by the user energy level. The initial velocity distribution of the gas is not identical to the Maxwell-Boltzmann (M-B) distribution, but it is very 'near' to this (see the description of the theoretical model in the attached file). Nevertheless, because of the p-p interactions, fast enough the gas is converging into the M-B distribution, as predicted by the Boltzmann's H-theorem.
The particles interact with each-other in such a way that the linear momentum and the kinetic energy of the interacting couple are conserved. The interactions with the walls of the container and the barrier are elastic collisions.

The program imposes a delay of 2 time-units for the barrier to be removed, after the user starts the simulation. Aiming at a better perception on the formation of the particles' trajectories under the successive p-p interactions, one specific particle and its path has been colored red.

The user controls three parameters: the energy level of the gas, the length h of the gap and the probability P_pass of passing a particle from D1 to D2 or back, if all the other conditions for that, predicted by the theoretical model, are satisfied. P_pass is a phenomenological parameter used to obtain the best agreement of the theoretical graphs with the experimental.
In the energy graphs window, it is depicted the actual gas's energy in each chamber and the total gas's energy in a sequence of time moments.
In addition, at each moment of this time sequence the program counts in real-time, the actual numbers n₁ and n₂ of particles in each chamber and draws the corresponding experimental graphs in the number graphs window (reddish curves). In the same window, the theoretical graphs are also depicted for the chosen by the user parameters (bluish curves).

The user is prompted to adjust the value of the parameter P_pass so that to succeed the best agreement of the theoretical graphs with the experimental.
Remark: A change of P_pass implies a change of time constant only; it does not affect the other characteristics of the theoretically derived functions n₁(t), n₂(t) (see the attached file: "The theoretical model").

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