Non-Gaussian velocity distribution

KudroDi A, Hetuy J Non-Gaussian velocity distributions in excited granular matter in the absence of clustering, Phys Rev E 62 R1489—R1492, 2000. 

Unfair, I hear you mutter under your breath, velocities have a lower limit of zero but lack an upper limit, so it is no wonder that they exhibit an asymmetrical distribution. True enough, but this also applies to many other quantities, such as absolute temperature, mass, concentration, and absorbance. The point here is not why a number of distributions are decidedly non-Gaussian, but that they are. 

For the discussion of the chaotic behavior statistical methods will be used. The relative cumulative frequency or the probability, respectively, is shown in Figure 7 for four velocity classes, see Table 3. From these data the relative frequency or probability density, respectively, is obtained, see Figure 8. It turns out that the frequency distribution is completely non-Gaussian, and the range characterizing the statistical dispersion is increasing with the relative velocity of the impact, while midrange point and mean value coincide fairly well, see Table 4. 

Specifically, at least three typical features are related with nonequilibrium granular flow systems (i) feasibility of one granular temperature (ii) non-Gaussian behavior of velocity distribution, and (iii) strong correlated density fluctuations in forms of bubbling or clustering. In the following section, we detail the above three points. 

Exercise. A cannon projects a ball with initial velocity v at angle 9 with the horizontal. Both v and 9 are subject to uncertainties that can be described by a Gaussian distribution for each. The distributions are centered at v0 and 60, respectively, and so sharp that non-physical values, such as negative v or 9, may be ignored. What is the probability distribution of the distance covered by the cannon ball ... 

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