The Maxwell Speed Distribution in 2D

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SUMMARY

The discussion focuses on the Maxwell Speed Distribution in two dimensions, specifically addressing the normalization of the integral involving the expression v*exp(-αv²). The key equation mentioned is the number of states with speed between u and u+du, which is given by 2π*u du. Participants highlight the challenges of integrating by parts and using the Gaussian integral for normalization. A suggested approach involves substituting α*v² to simplify the integral.

PREREQUISITES
  • Understanding of Maxwell Speed Distribution
  • Familiarity with Gaussian integrals
  • Knowledge of integration techniques, specifically integration by parts
  • Basic concepts of probability density functions
NEXT STEPS
  • Study the derivation of the Maxwell Speed Distribution in 2D
  • Learn advanced integration techniques, focusing on substitution methods
  • Explore the properties and applications of Gaussian integrals
  • Investigate normalization of probability distributions in statistical mechanics
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Students and researchers in physics, particularly those studying statistical mechanics and thermodynamics, as well as anyone interested in the mathematical foundations of speed distributions.

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Homework Statement


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It seemed much easier to screencap than to write out.

Homework Equations



It helps to know that the number of states with speed between u and u+du is 2pi*u du

The Attempt at a Solution



I've tried quite a few things but every time I get to trying to normalise I either get stuck integrating by parts over and over, or using http://en.wikipedia.org/wiki/Gaussian_integral" <that, which doesn't arrive at the answer wanted.

If someone could give me a push in the right direction it would be much appreciated.
 
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You need some substitution to make the integral doable.
 
Ok so I have the integral of v*exp(-\alphav2) dv between 0 and infinity to normalise. I don't know how a substitution would help because you would still have two functions multiplied by one another.

Edit: I think I'm being stupid. I've substituted for alpha*v^2. Hopefully it will work.
 
Last edited:

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