Number of microstate for an N-Particle system is not equal to Number of microstate for 1 Particle system to the power N. Here's why !!
A small misconception I had while studying Statistical Mechanics:
Can we always say:
Number of microstates of N particles = (Number of microstates of 1 particle)^N ?
At first it feels natural because if one particle has Ω₁ possible states, N independent particles should simply give (Ω₁)ᴺ.
But in the microcanonical ensemble, the system has a fixed total energy. The N particles are not represented by N separate momentum spheres. Instead, the entire system forms one hypersphere in the full phase space.
For one particle in D dimensions → momentum space is D-dimensional.
For N particles → momentum space becomes DN-dimensional.
Which is basically dof (degrees of freedom dimensional)
That changes the geometry. The Gamma function term changes from:
[ Γ(D/2 + 1) ]ᴺ
to:
Γ(DN/2 + 1)
and these two are not the same.
So the N-particle microstate is not obtained by blindly raising the one-particle answer to the power N. You have to count states in the complete phase space of the system.
For identical particles, we also divide by N! to remove the overcounting due to particle exchange (Gibbs correction).
A small detail mathematically, but a very important idea physically. Let me know if any error.