Stat. Mech. Macrostates for four particles?

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SUMMARY

The discussion focuses on finding macrostates for four particles distributed between two energy levels, one of which is twofold degenerate. The solutions for the indistinguishable particles yielded thermodynamic probabilities of 5, 4, 3, 2, and 1 for each macrostate. For distinguishable particles, the user attempted to apply Fermi-Dirac (FD) statistics but encountered issues with the calculations. The correct thermodynamic probabilities for distinguishable particles were ultimately confirmed as 16, 32, 24, and 48, leading to a total assembly of 84 and 15 for the respective problems.

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  • Understanding of statistical mechanics, specifically macro and microstates.
  • Familiarity with Bose-Einstein (BE) and Fermi-Dirac (FD) statistics.
  • Knowledge of thermodynamic probability calculations.
  • Concept of particle degeneracy in energy levels.
NEXT STEPS
  • Study the application of Bose-Einstein statistics in thermodynamic probability calculations.
  • Learn about Fermi-Dirac statistics and its implications for indistinguishable particles.
  • Explore the concept of particle degeneracy and its effects on statistical distributions.
  • Investigate advanced topics in statistical mechanics, such as the exclusion principle and its applications.
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Students and researchers in physics, particularly those studying statistical mechanics, thermodynamics, and quantum statistics. This discussion is beneficial for anyone looking to deepen their understanding of particle distributions and thermodynamic probabilities.

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Homework Statement


a. Find macro states of 4 particles among two energy level one of which is twofold degenerate. SOLVED
b. Find the thermodynamic probability of each macrostate for indistinguishable particles. SOLVED
c. ------------------distinguishabled problems (need help)
d. the assembly for b and c.
Answers: a. 5. b. 5,4,3,2,1, c. 16,32,24,48,24. d. 84,15

Homework Equations


Concept: MBE statistics applies to classical distinguishable particles not obeying exclusion principle. FD applies to indistinguishable obeying exclusion principle. Bose Einstein applies to indistinguishable without exclusion.

Thermo. Prob. = W = (gj+Nj -1)! / (gj-1)!Nj! this is for BE stats which I used for b and obtained correcting answer
W = N! (Pi)j gj^(Nj) / Nj! (FD statistics I thought to use for c.)

where (Pi) is big pi standing for product, gj is degeneracy (2 here), Nj is particles in Energy state, N is total particle (4) FD statistics I thought to use for c.

The Attempt at a Solution


W = N! (Pi)j gj^(Nj) / Nj!
applying for macro state of 4 where one can have different arrangements of 4 particles in each level.
W4 = 16 CORRECT
W3 = 32 CORRECT
W2 = 24 CORRECT
W1 = 48 INCORRECT
W0 = 24 INCORRECT
Am I using the wrong formula? I know for d you can just sum up each macro state so I mainly need help with c please.
 
SOLVED.
 

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