States counting of many particales under a constraint

In summary, the number of possible states for n identical classical non-interacting particles placed in N sites with a given total energy is calculated by dividing N^n by n! and (n/2)!, with the latter term related to the fact that the energy is fixed in a microcanonical ensemble. The distribution function for finite temperature, Maxwell-Boltzmann, is not applicable in this scenario. The source of the (n/2)! term is the difference in energy for different configurations of the particles among the lattice sites.
  • #1
oneshai
2
0
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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  • #2
What determines the energy for a given distribution? This is kinda the input you need for the question you're asking...
 
  • #3
Maxwell - Boltzman as usual. I though about it but i fail to see how it leads to the n/2!
 
  • #4
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.
 

1. How are states counted under a constraint?

States are typically counted by observing the physical properties or characteristics of the particles. This could include their position, velocity, charge, or other measurable quantities.

2. What is meant by a "constraint" in this context?

A constraint refers to a limitation or restriction that must be taken into account when counting states. This could be a physical constraint, such as the size or shape of the system, or a conceptual constraint, such as the exclusion of certain states due to conservation laws.

3. Why is it important to count states under a constraint?

Counting states under a constraint is important because it allows for a more accurate understanding of the system being studied. By taking into account the limitations or restrictions, we can better predict the behavior and properties of the particles.

4. What factors can affect the accuracy of state counting under a constraint?

The accuracy of state counting can be affected by various factors, such as the complexity of the system, the accuracy of measurement tools, and the understanding of the constraints themselves. Additionally, external influences, such as temperature or pressure, can also impact the accuracy of state counting.

5. How does state counting under a constraint relate to other scientific principles?

State counting under a constraint is closely related to other scientific principles, such as statistical mechanics, thermodynamics, and quantum mechanics. These principles provide the framework for understanding how particles behave under different constraints and how they contribute to the overall properties of a system.

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