Calculating number of microstates to find entropy

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    Entropy Microstates
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Discussion Overview

The discussion revolves around the calculation of the number of microstates in the context of entropy, specifically using the Boltzmann entropy formula and its application to different statistical mechanics frameworks, including Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics. Participants explore the implications of particle distinguishability and the conditions under which various statistical methods can be applied.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions why the Boltzmann entropy formula, which uses Maxwell-Boltzmann statistics, cannot be applied using other statistical methods like Bose-Einstein or Fermi-Dirac statistics.
  • Another participant asserts that the formula for the number of microstates is a general combinatoric formula and asks how the distinguishability of particles would affect its application.
  • A participant provides an example involving the placement of balls in bins, illustrating the difference in counting methods based on whether the balls are treated as identical or distinguishable.
  • One participant agrees with the previous point and explains that the Boltzmann formula can be adapted to calculate the number of ways a collection of distinguishable systems can be arranged, even when the individual particles are indistinguishable.
  • A follow-up request for clarification on the adaptation of the formula indicates that the explanation provided may not have been fully understood by all participants.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the Boltzmann entropy formula to systems of indistinguishable particles and how this relates to the use of different statistical methods. The discussion remains unresolved regarding the best approach to calculating microstates in various contexts.

Contextual Notes

There are limitations in the assumptions made about particle distinguishability and the interpretations of the statistical methods, which may affect the conclusions drawn from the discussion.

UnderLaplacian
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In the Boltzmann entropy formula , the number of microstates is calculated according to Maxwell-Boltzmann statistics , i.e. , W = n!/Πki! , Σki = n . Why cannot we use some other method , such as Bose-Einstein or Fermi-Dirac statistics ?
 
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I am fairly sure that the formula you cite for the number of microstates is a general combinatoric formula. What do you think would make it different if the particles were distinguishable, fermionic, or bosonic?
 
Say , for example , we consider the problem of placing 2 balls in 2 bins . If we treat the balls as identical , we have 3 ways , if not , we have 4 ways . Please point out if I am making some mistake in my interpretation .
 
You are correct. You cannot use the formula $$W = \frac{n!}{k_{1}! ... k_{r}!}$$ to calculate the number of possible states that a system of n identical particles can be distributed among r different energy levels. However, if you change your interpretation, you can use this formula to calculate the number of ways that a virtual collection of N distinguishable systems each of n identical particles can be distributed among the combined energy levels (possible energy levels of the imaginary collection). Then, the Boltzmann entropy formula applies even to systems in which the individual particles are indistinguishable. This is how the core formulas of statistical mechanics are justified in the quantum domain. Does that answer the question?
 
Twigg said:
However, if you change your interpretation, you can use this formula to calculate the number of ways that a virtual collection of N distinguishable systems each of n identical particles can be distributed among the combined energy levels (possible energy levels of the imaginary collection).
Could you please explain your above statement in some more detail ? I did not really get what you meant .
 

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