Collision integral approximation in boltzmann equation

Click For Summary

Discussion Overview

The discussion revolves around the approximation of the collision integral in the Boltzmann equation, focusing on how this approximation is derived and the underlying assumptions. Participants seek clarification on the mathematical formulation and its intuitive basis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant requests a derivation of the approximation of the collision integral as a discrete sum.
  • Multiple participants inquire about the source of the material being discussed, indicating a need for references.
  • Another participant suggests that the approximation is more about intuitive reasoning than strict mathematical derivation, proposing that the velocity change depends linearly on the differences in velocities of other components.
  • A participant shares a link to a lecture that may contain relevant equations and questions whether the provided equation is appropriate for the collision integral.
  • There is a discussion about the meaning of the difference in velocities in the context of the collision integral, with one participant explaining it as the relative velocity of one particle with respect to another.
  • Another participant expresses a desire for a mathematical demonstration to better understand the origin of relative velocity in the context of the collision integral.

Areas of Agreement / Disagreement

Participants express varying views on the nature of the approximation, with some emphasizing intuitive reasoning while others seek mathematical clarity. There is no consensus on the derivation or the interpretation of the collision integral.

Contextual Notes

Participants reference multiple sources and equations, indicating a reliance on specific definitions and contexts that may not be universally agreed upon. The discussion reflects a range of assumptions about the mathematical treatment of the collision integral.

mertcan
Messages
343
Reaction score
6
upload_2017-12-11_10-58-10.png


Hi, as you can see at the end of the picture/attached file collision integral is approximated to a discrete sum. Could you express how this approximation is derived?
 

Attachments

  • upload_2017-12-11_10-56-11.png
    upload_2017-12-11_10-56-11.png
    49.8 KB · Views: 1,871
  • upload_2017-12-11_10-58-10.png
    upload_2017-12-11_10-58-10.png
    49.8 KB · Views: 1,733
Physics news on Phys.org
It is not really an approximation of the integral from a purely mathematical point-of-view I think, it rather seems a mather of intuitive common sense.

It is reasonable to suppose that the velocity change of component alpha depends linearly on the differences with the velocities of the other components beta. I don't think there is much more in it than that.
 
thanks for response @thephystudent ,@PeterDonis you have not given any answers, what is your response about my question?
 
mertcan said:
@PeterDonis you have not given any answers, what is your response about my question?

I don't have one, at least not at the moment. I asked you for a reference because that's part of moderating the forum, and to hopefully help other possible responders.
 
mertcan said:
@thephystudent , I cut my attachment off that link https://courses.physics.ucsd.edu/2015/Fall/physics210b/LECTURES/CH05.pdf and page 10, so is it the right equation for collision integral for you? by the way why difference of velocity takes place in that equation?

Seems related indeed, where the $\langle \nu_{\alpha\beta}$ coefficient captures the average of some remaining coefficients. Difference of velocities= the relative velocity of particle beta when traveling along particle alpha.
 
thanks for responses but, I need some mathematical demonstration to learn better, could you help me about where relative velocity comes from that?
 

Similar threads

Graduate References: continuum approximation of discrete sums?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Born rule in classical Ising model: Feynman -> Boltzmann ensemble?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad What does the Lattice Boltzmann Equation actually say?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Is there a Boltzmann distribution for a system with continuous energy?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Are Boltzmann brains predicted by our current cosmological theories?
  • · Replies 12 ·
Replies
12
Views
1K
Graduate Understanding Harmonic oscillator conventions
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Wigner Weisskopf method for time varying Perturbation
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Converting momentum sums to integrals in curved spacetime
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Lattice Boltzmann simulation with arbitrary equilibrium func
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Strange feature of BE, FD and Boltzmann distributions
  • · Replies 6 ·
Replies
6
Views
2K