- #1

Philip Koeck

- 712

- 198

- 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:

P

where W

W

and W

W

As long as I don't assume a large number of states g and low occupancy (g >> N), P

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" P

For fermions I can understand that P

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

P

_{R}= W_{R}/ W_{T}where W

_{T}is the number of ways of distributing N particles in the total volume:W

_{T}= (N+2g-1)! / (N! (2g-1)!)and W

_{R}is the number of ways of distributing R particles in one half of the volume and the remaining N-R in the other half:W

_{R}= ((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), P

_{R}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" P

_{R}doesn't depend on g in the first place, whatever the value of g is.For fermions I can understand that P

_{R}depends on g since each fermion blocks a state and alters the situation for all other fermions so that P_{R}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