Gas Energy Distribution Vs generation in 1, 2, and 3 dimensions Please click link below for a full discussion of this subject. This animation shows the evolution of the energy distributions of a 2D gas that initially has a mono-energetic distribution (MED where all atoms have energy E0). That means that the energies of all generations of the outside atoms will be random values in the range between 0 and E1n+ E2n or En+1,1=(En,1+En,2)*r En+1,2=(En,1+En,2)*(1-r) r=random(0,1) where n is the generation number and random(0,x) provides a randomly selected floating point number between 0 and x. Note that the generation n+1 energy values are chosen so that neither is negative. This non-negativity is the most important property driving the final distribution since it results in a compaction at lower energies and a spreading at higher energies which are unrestricted in value except that they must obey energy conservation on each interaction. The distribution after many generations always fits an exponential N(E)=N0*exp(-E/<E>) where N0 is a normalizing constant and the symbol <x> means the average value of x. This would seem to explain energy distributions for one dimension (1D) and three dimensions (3D) as well. Unfortunately this is not true for the other dimensions. To explain 3D energy distributions, one has to use the full conservation of momentum equations as well as the energy conservation equations. The result for 3D is a distribution of form sqrt(E/<E>)exp[E/(2/3<E>)]. The individual momentum probabilities are all like f(px)dpx=exp(-px*px/<px*px>)dpx and the 3D momentum probability is F(px,py,pz)=f(px)f(py)f(pz)dpxdpydpz we can change the differential to energy and rewrite this as G(E)=exp(-p^2/<p^2>)p*p*dp=exp(-E/<E>)sqrt(E/<E>)dE thus getting the expression used to fit the 3D energy here. Using the same collision equations elevated to higher dimensions, we could model particles with access to 4 or more dimensions. Of course, these higher dimensions usually refer to rotation and vibration modes. For 1D the situation is even more complicated for 2 reasons: First, the particles can interact only with their nearest initial neighbors. Second, there is no orthogonal dimension (like x,y) for the momentum to be transferred to during a collision. As a result, the momentum distribution does not approximate a gaussian exp[-p*p/(<p*p>)] which is a correct distribution for, say, the y component of the momentum in 2D. So I have used a function involving the square root of the energy to describe the 1D energy distribution. Converting the momentum differential, dp, to energy we would have dp=dE/sqrt(E) which is what I have used for 1 dimension energy distributions. It should be noted that, while the energy distributions for 2D and 3D are mono-energetic, the initial distribution for 1D is flat. with the same average energy as for 2D and 3D.
2D Energy Distributions Information
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