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BEC question about number of excited particles

  1. Jul 9, 2015 #1
    Assuming no interactions, and the energy levels ##\epsilon_j## for a single particle state, the occupation number for this state for a system of particles is given by:
    à
    ##n_j=\frac{1}{exp(\frac{\epsilon_j-µ}{kT}) - 1}## (1)

    (Here comes argument 1)

    We conclude that the maximal value that µ can take is equal to ##µ=\epsilon_0##. This means that the distribution

    ##n_j=\frac{1}{exp(\frac{\epsilon_j-\epsilon_0}{kT})- 1}## (2)

    Is an upper limit on any real distribution.

    So using (2) we can calculate a statistical upper limit on the expected amount of particles in the excited states ##N_{exc}##.

    Questions:

    1) What explanation should come in the placeholder for 'argument 1' to conclude that µ is between 0 and ##\epsilon_0##? I've seen one on the net that already uses the result of having a macroscopic amount of particles in the groundstate to conclude this, which is kind of a circular reasoning.

    2) Am I correct in thinking about this in terms of (2) being an upper limit? The problem that I then have is this - the upper limit on the state ##\epsilon_0## is infinite right? How to conclude that this state will only start getting a macroscopic occupation after the number of particles has exceede ##N_{exc}## - if we see an upper limit of infinity here how can we know that we won't get a big occupation in the groundstate from the very beginning?
     
  2. jcsd
  3. Jul 13, 2015 #2
    1) If µ were > than ϵ0, the occupation number would be negative for ground states.

    2) Your question here is unclear to me. As µ -> ϵ0, n_j -> infinity for j = 0, if that's what you mean by "macroscopic number of particles."
     
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