Taylor expansion of dispersion relation - plasma physics

In summary, the conversation revolves around a person struggling with the derivation of an equation in a research paper and requesting help. They mention having no background in maths and it has been a while since they have studied it. The person offering help suggests looking at the references in the paper and clarifies the meaning of the "A" level label. Ultimately, it is mentioned that an understanding of graduate level content is necessary to discuss and solve the problem.
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  • #3
vanhees71 said:
What's your problem?
I have no idea how to derive the equation
 
  • #4
SMannion said:
I have no idea how to derive the equation

We need more information than that. What have you tried? Have you looked at the references given in the paper?
 
  • #5
PeterDonis said:
We need more information than that. What have you tried? Have you looked at the references given in the paper?
I need help with the maths, I am unsure how or where to begin with this?
 
  • #6
SMannion said:
I need help with the maths, I am unsure how or where to begin with this?

You labeled this thread as "A" for advanced. That implies that you have a graduate level understanding of the subject matter. What background do you have in this field? The paper you referenced is definitely at the "A" level, and you would need a graduate level background to be able to understand it or to reproduce any derivations it discusses.
 
  • #7
My apologies, I thought the "A" was for the content level. I am a graduate but not in maths, it has been about 20 years since I studied anything like this.
Any help people could provide would be greatly appreciated
 
  • #8
SMannion said:
I thought the "A" was for the content level

It can reflect both. Sometimes there are various levels at which content can be discussed, so the thread level gives readers an idea of your background so we know what level the discussion should be at. But truly "A" level content very often (as in this case) requires an "A" level background to have a useful discussion, because pretty much any questions you could ask are only answerable at the "A" level.
 
  • #9
The content of this thread is clearly A level, but it is impossible to help, if we don't know, where the problems are. For a very good introduction to transport theory and a brillant treatment of plasma physics, see the cited textbook by Landau and Lifshitz, vol. X (Kinetics).
 

1. What is the purpose of using a Taylor expansion for the dispersion relation in plasma physics?

The Taylor expansion of the dispersion relation is used to approximate the behavior of waves in a plasma. It allows us to study the properties of waves at different frequencies and wavelengths, which is crucial for understanding the behavior of plasmas.

2. How is the Taylor expansion of the dispersion relation derived?

The Taylor expansion is derived by taking a series of derivatives of the dispersion relation with respect to frequency. This results in an infinite series of terms, but typically only the first few terms are used in practical applications.

3. What is the significance of the first term in the Taylor expansion of the dispersion relation?

The first term in the Taylor expansion, also known as the "zeroth-order term," represents the behavior of the wave at the plasma frequency. This term is particularly important as it describes the resonant behavior of waves in a plasma.

4. How does the Taylor expansion of the dispersion relation relate to plasma instabilities?

The Taylor expansion can be used to study the stability of plasmas by analyzing the higher-order terms in the series. These terms can reveal the presence of instabilities, such as the growth of waves or the formation of particle beams, which can have important implications for plasma behavior.

5. Are there any limitations to using the Taylor expansion for the dispersion relation?

While the Taylor expansion is a useful tool for studying plasma waves, it does have some limitations. For example, it assumes that the plasma is homogeneous and does not account for any nonlinear effects. Additionally, the accuracy of the approximation decreases as the frequency of the wave deviates from the plasma frequency.

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