Thermodynamical fluctuations, mean square deviation help

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SUMMARY

The discussion focuses on deriving the density of fluctuations for an ideal gas, specifically demonstrating that the mean square density deviation is given by the formula <(dp)^2> / p^2 = 1 / (N*Na), where p represents density and Na is Avogadro's number. Participants emphasize the importance of understanding expectation values and the relationship between fluctuations and the number of molecules in a subsystem. The discussion also highlights the relevance of the grand canonical ensemble and the grand partition function in solving related problems.

PREREQUISITES
  • Understanding of thermodynamic concepts, particularly ideal gas behavior.
  • Familiarity with statistical mechanics, including expectation values.
  • Knowledge of the grand canonical ensemble and its applications.
  • Basic proficiency in mathematical statistics, specifically in calculating mean square deviations.
NEXT STEPS
  • Study the derivation of the grand canonical ensemble and its implications for fluctuations in thermodynamic systems.
  • Learn how to calculate expectation values in statistical mechanics, focusing on <(dN)^2> and .
  • Explore the grand partition function and its role in understanding thermodynamic fluctuations.
  • Practice problems involving mean square deviations in various thermodynamic contexts.
USEFUL FOR

This discussion is beneficial for students and researchers in physics, particularly those studying thermodynamics and statistical mechanics, as well as anyone tackling problems related to fluctuations in ideal gases.

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Homework Statement



Recalling that k=R/Na (Na is Avogadro's number), show that the density of fluctuations of an ideal gas are given by :

<(dp)^2> / p^2 = 1 / (N*Na) where p is the density (mass/V)

That is, the relative mean square density deviation is the reciprocal of the number of molecules in the subsystem.


Homework Equations



<(dN)^2> = <N^2> - <N>^2 from my book

The Attempt at a Solution



I have no idea where to even begin with this..I have a whole series of problems that ask me to find the "mean square deviation of ____". I understand what an expectation value is, and that i have the given equation for the value.

But HOW do I find <P> or <P^2> ?? Normally I would take the value*probability (sum of X*P(X) right?) but in this case...what is my probability? what is my value??

Where do I even start??

Plz help I have 4 problems like this and am stuck in the exact same place on all 4.
 
Physics news on Phys.org
You may look up "grand canonical ensemble" or "grand partition function" in any thermodynamics textbook. It'll help.
 

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