Maxwell - Boltzman distribution

[mattT1227]

I made a spreadsheet to plot the 3d speed distribution from the MB probability function. It matches the peak and fall-off of published graphs. I then tried to integrate it by summing over interval widths times probability. I thought the area should be 1. My result is around 5 or 6. I've tried really small intervals, around 0.25 m/s, that didn't help. Must be I don't understand what the area under the probability curve represents.

 

[SteamKing]

Check out this article:
http://en.wikipedia.org/wiki/Maxwell–Boltzmann_distribution

It gives properties of the M-B distribution, including the PDF and corresponding CDF. You must have made an error in your calculations.

 

[UltrafastPED]

You have failed to normalize! For example, see http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/maxspe.html

 

[mattT1227]

Thanks all. Problem solved. Area under curve is 0.98 by numerical integration, close enough.

 

[sardar]

The Maxwell-Boltzmann distribution is a probability distribution that describes the speed distribution of particles in a gas at a given temperature. It is a fundamental concept in statistical mechanics and has various applications in physics and chemistry.

Your approach of using a spreadsheet to plot the 3D speed distribution from the MB probability function is a valid method to visualize the distribution. It is also commendable that your results match the peak and fall-off of published graphs, which indicates that your calculations are accurate.

However, your difficulty in integrating the probability function and obtaining a result of around 5 or 6 instead of 1 is understandable. The area under the probability curve represents the total probability of finding a particle within a certain speed range. In other words, it represents the total likelihood of a particle having a speed within that range. Therefore, the area under the curve should always be equal to 1, as the total probability of finding a particle within the entire speed range is 100%.

Based on your description, it seems that your approach of summing over interval widths times probability may not be accurate. I would suggest consulting with a statistician or a colleague who has expertise in probability distributions to ensure that your calculations are correct.

In conclusion, the Maxwell-Boltzmann distribution is a crucial concept in statistical mechanics, and it is essential to have a thorough understanding of its properties and applications. Keep in mind that the area under the probability curve represents the total probability, and it should always be equal to 1. I wish you all the best in your future experiments and calculations.