States counting of many particales under a constraint

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Discussion Overview

The discussion revolves around the counting of states for a system of n identical classical non-interacting particles distributed across N sites, with a fixed total energy constraint. Participants explore the implications of this constraint on the number of possible configurations and the mathematical formulation involved.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant presents a formula for the number of possible states, suggesting that the division by n! accounts for the indistinguishability of particles, while the division by (n/2)! is unclear and seeks clarification on its origin.
  • Another participant questions what determines the energy for a given distribution of particles, indicating that understanding this is crucial for addressing the original question.
  • A third participant references the Maxwell-Boltzmann distribution but expresses uncertainty about its relevance to the (n/2)! term in the context of a microcanonical ensemble where energy is fixed.
  • Further clarification is sought regarding whether all configurations yield the same energy, suggesting that this could be a source for the (n/2)! factor.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the role of the (n/2)! term or how energy is determined for different configurations, indicating ongoing debate and uncertainty in the discussion.

Contextual Notes

The discussion highlights limitations in understanding the relationship between particle distribution and energy, particularly in the context of fixed energy constraints and the implications for state counting.

oneshai
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Let’s say i have n identical classical non interacting particles and N sites where i can put them in. BUT the total energy is given.
The number of possible states is (N)^n/n!/(n/2)!
Where N^n is the total possibilities to arrange the particles.
We divide it by n! since they are identical.
and the reason we divide by (n/2)! has some thing to do with the fact that the energy is given but I don’t know how.

i whould be very thankful to anyone who can say why (n/2)!

thanks shai
 
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What determines the energy for a given distribution? This is kinda the input you need for the question you're asking...
 
Maxwell - Boltzmann as usual. I though about it but i fail to see how it leads to the n/2!
 
No, no, that's a distribution function for finite temperature. You're working in a microcanonical ensemble, so you don't have temperature (the energy is fixed).

What I'm asking is: what's the energy for a given configuration of the particles among the lattice sites. Do all configurations give the same energy or not? It's the only possible source of where the (n/2)! is coming from as far as I can tell.
 

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