- #1
Philip Koeck
- 673
- 182
- TL;DR Summary
- I come up with an expression for the probability of finding R out of a total of N bosons in one half of a volume which depends on the number of available states g in half the volume as long as g is not very large compared to N and 1. I want to know whether this can be understood and explained in a common-sense way.
For the probability of finding R out of N (indistinguishable) bosons in one half of a volume with a total of 2g states (g in each half) I get the following expression:
PR = WR / WT
where WT is the number of ways of distributing N particles in the total volume:
WT = (N+2g-1)! / (N! (2g-1)!)
and WR is the number of ways of distributing R particles in one half of the volume and the remaining N-R in the other half:
WR = ((R+g-1)! (N-R+g-1)!) / (R! (g-1)! (N-R)! (g-1)!)
As long as I don't assume a large number of states g and low occupancy (g >> N), PR depends on the value of g.
My questions:
Does this seem to be correct?
Is there a common-sense way of explaining this dependence on g?
Note that at low occupancy and large g the dependence on g disappears and that for (hypothetical) distinguishable "bosons" PR doesn't depend on g in the first place, whatever the value of g is.
For fermions I can understand that PR depends on g since each fermion blocks a state and alters the situation for all other fermions so that PR must depend on g.
Can one come up with a similar explanation for indistinguishable bosons?
In case my equation is unclear I have a complete derivation here: https://www.researchgate.net/publication/336375268_Probability_of_finding_R_of_N_particles_in_one_half_of_a_volume
PR = WR / WT
where WT is the number of ways of distributing N particles in the total volume:
WT = (N+2g-1)! / (N! (2g-1)!)
and WR is the number of ways of distributing R particles in one half of the volume and the remaining N-R in the other half:
WR = ((R+g-1)! (N-R+g-1)!) / (R! (g-1)! (N-R)! (g-1)!)
As long as I don't assume a large number of states g and low occupancy (g >> N), PR depends on the value of g.
My questions:
Does this seem to be correct?
Is there a common-sense way of explaining this dependence on g?
Note that at low occupancy and large g the dependence on g disappears and that for (hypothetical) distinguishable "bosons" PR doesn't depend on g in the first place, whatever the value of g is.
For fermions I can understand that PR depends on g since each fermion blocks a state and alters the situation for all other fermions so that PR must depend on g.
Can one come up with a similar explanation for indistinguishable bosons?
In case my equation is unclear I have a complete derivation here: https://www.researchgate.net/publication/336375268_Probability_of_finding_R_of_N_particles_in_one_half_of_a_volume