The evolution of a 2-dimensional ideal gas toward the equilibrium state. The Boltzmann H-theorem

Kostas Papamichalis Dr of Theoretical Physics

The theoretical model on which the simulation is based:

The evolution of a 2D ideal gas toward the state of thermodynamic equilibrium (pdf-file)

The purpose of the application is to test "experimentally" the Boltzmann model for the case of a 2D virtual gas. According to the Boltzmann model, the velocity-distribution of the gas changes with time until it takes the form of the Maxwell-Boltzmann (M-B) distribution, which is stable. In addition, the value of the Boltzmann H-functional at an arbitrary time, depends on the actual velocity distribution of the gas at this same moment. H converges to its minimum value obtained when the gas reaches M-B distribution, irrespectively of the initial velocity distribution of the gas. This is a theoretical prediction of the famous Boltzmann's H-theorem.

In the sim-window you can see the motion and the interactions of the gas-particles. One of the particles has been colored red and its trajectory is visible to the user. At time t=0, you can see the vectors of the particles' velocities corresponding to the initial distribution. The initial distribution is chosen by the user among three alternatives (see the theoretical model file).

In the upper graph-window you can see how the velocity distribution of the gas changes with time because of the particles' interactions, and converges to the M-B distribution (the red theoretical curve), irrespectively of the chosen initial distribution. The lower graph-window shows the variation of the Boltzmann H-functional with time: it converges to its value that is obtained at the M-B distribution. It must be noticed that the graphs are completely "experimental" in the environment of the application: they are composed by measurements of the particles-velocities at a sequence of specified time-moments. The motion of the gas-particles is determined by the Newton laws, their interactions and their initial states. The number of the particles is N=300. The user can choose the mean energy of the particles in the interval [100,400]. The energy is conserved in the interaction between particles and between the particles and the walls of the virtual container. The user controls one more parameter: the "strength" of the interaction. The choice zero means that there is no p-p interaction, and the initial velocity distribution do not change with time. (See "Activities" in the theoretical model file)

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