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Collision integral approximation in boltzmann equation
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In summary, the collision integral approximation in the Boltzmann equation is a method used to simplify the complex calculation of particle interactions in a gas. It is based on the assumption that the majority of collisions are elastic, meaning that the total kinetic energy of the particles remains constant. This approximation allows for a more efficient and manageable way of solving the Boltzmann equation, which is essential for understanding the behavior of gases at a molecular level. However, it may lead to some inaccuracies in certain scenarios and is often used in combination with other approximations and numerical techniques.
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PeterDonis
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@mertcan what reference (book/article/web page/whatever) is this taken from?
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mertcan
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@PeterDonis the link is http://www.am.ub.edu/~blai/plasma_physics/Lecture_3.pdf and page 3PeterDonis said:
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thephystudent
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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.
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.
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mertcan
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thanks for response @thephystudent ,@PeterDonis you have not given any answers, what is your response about my question?
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PeterDonis
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mertcan said:
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.
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mertcan
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@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?
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thephystudent
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mertcan said:
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.
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mertcan
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thanks for responses but, I need some mathematical demonstration to learn better, could you help me about where relative velocity comes from that?
1. What is the Boltzmann equation and how is it related to the collision integral approximation?
The Boltzmann equation is a mathematical formula used to describe the behavior of particles in a gas or fluid. It takes into account the interactions between particles, such as collisions, to predict their movement and distribution. The collision integral approximation is a simplification of the Boltzmann equation that is commonly used in calculations and simulations.
2. How does the collision integral approximation work?
The collision integral approximation simplifies the Boltzmann equation by assuming that the collisions between particles are elastic and that the interactions between particles can be represented by a single parameter, known as the collision cross-section. This allows for a more efficient calculation of particle behavior and distribution.
3. What are the limitations of the collision integral approximation?
While the collision integral approximation is a useful tool for predicting particle behavior, it has some limitations. It assumes that all collisions are elastic, which may not always be the case in real systems. It also does not take into account higher-order interactions between particles, which may be important in certain scenarios.
4. How is the collision integral approximation used in practical applications?
The collision integral approximation is used in a variety of fields, including fluid dynamics, plasma physics, and molecular dynamics simulations. It plays a crucial role in understanding and predicting the behavior of particles in these systems and is often used in conjunction with other techniques and approximations.
5. Are there any alternative methods to the collision integral approximation?
Yes, there are alternative methods to the collision integral approximation, such as the Chapman-Enskog method and the Grad method. These methods also simplify the Boltzmann equation and have different assumptions and limitations compared to the collision integral approximation. The choice of method depends on the specific application and the desired level of accuracy.
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