A Collision integral approximation in boltzmann equation

[mertcan]

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?

 

[PeterDonis]

@mertcan what reference (book/article/web page/whatever) is this taken from?

 

[mertcan]

@mertcan what reference (book/article/web page/whatever) is this taken from?

@PeterDonis the link is http://www.am.ub.edu/~blai/plasma_physics/Lecture_3.pdf and page 3

 

[thephystudent]

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.

 

[mertcan]

thanks for response @thephystudent ,@PeterDonis you have not given any answers, what is your response about my question?

 

[PeterDonis]

@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]

@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?

 

[thephystudent]

@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.

 

[mertcan]

thanks for responses but, I need some mathematical demonstration to learn better, could you help me about where relative velocity comes from that?