I'm trying to follow the proof given in Box 5.3 of MCP (Thorne/Blandford)

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Discussion Overview

The discussion revolves around understanding the proof presented in Box 5.3 of the MCP book by Thorne and Blandford, specifically concerning the Van der Waals grand potential. Participants are examining the mathematical details and potential errors in the equations provided in the text.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the presence of a missing factor in the last term of Equation (8), suggesting it should be (2l−1)!/(l−1)!.
  • Another participant requests a high-quality scan of the relevant page to assist in the discussion.
  • A participant provides a link to a high-quality image of the page in question, indicating the previous page may not be necessary for understanding.
  • Some participants assert that the circled l! terms in the equation should actually be (2l)!, claiming that the final line of Equation (8) as presented in the book is correct.
  • One participant expresses appreciation for the clarification regarding the terms, indicating that they initially expected the line to be correct but were unable to identify the error.

Areas of Agreement / Disagreement

There is disagreement regarding the correctness of the terms in Equation (8). Some participants believe there is an error, while others maintain that the equation as given in the book is correct.

Contextual Notes

The discussion reflects uncertainty about specific mathematical details and the interpretation of the proof, with participants relying on the text and each other's insights to clarify their understanding.

jouvelot
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I'm trying to follow the proof given in Box 5.3, page 235, of the MCP book regarding the Van der Waals grand potential. It seems to me that there is a missing factor (2l−1)!/(l−1)! in the last term of Equation (8). What am I doing wrong?
 
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Can you post a high-quality scan image of that box? Use the "Attach files" link below the edit window to upload the image. Thanks.
 
Thanks @berkeman

Here is the page you requested (the attached version seems to be of rather poor quality). I don't think the previous page is needed, but if you think it could be useful, please let me know.
The original, good quality image is available at https://www.dropbox.com/scl/fi/ep2xqpo5ko9ybdancd33f/20240401_185542.jpg?rlkey=gnsw4ftd8ou2ea8konhgh6mn9&dl=0.

20240401_185542.jpg
 
1711998694491.png


The circled ##l!## terms should each be ##(2l)!##. The final line of (8), as given in the book, is correct.
 
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TSny said:
View attachment 342684

The circled ##l!## terms should each be ##(2l)!##. The final line of (8), as given in the book, is correct.
Ah, this makes a lot of sense, indeed. I was expecting Line 8 to be correct, since the rest of the proof follows through with this value, but couldn't find the error. Thanks a lot, @TSny, for spotting the error before that.
 
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