# BEC question about number of excited particles

• Coffee_
In summary, the conversation discusses the occupation number for a single particle state in a system of particles, given by the formula n_j = 1/(exp((ϵ_j-µ)/(kT)) - 1). It is concluded that the maximum value for µ is equal to ϵ0, and this leads to the distribution n_j = 1/(exp((ϵ_j-ϵ0)/(kT)) - 1) being an upper limit for any real distribution. The placeholder for 'argument 1' discusses the reasoning behind concluding that µ is between 0 and ϵ0, and it is clarified that this is due to the occupation number being negative for ground states if µ was greater than ϵ0.
Coffee_
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?

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."

## 1. What is BEC?

BEC stands for Bose-Einstein Condensate, which is a state of matter that occurs when a group of bosons (particles with integer spin) are cooled to near absolute zero and occupy the same quantum state.

## 2. How does BEC relate to the number of excited particles?

In BEC, all particles are in the same quantum state, meaning there are no excited particles. As the temperature is lowered, more and more particles will transition to the ground state, resulting in a larger number of particles in the ground state and a smaller number of excited particles.

## 3. What determines the number of excited particles in a BEC?

The number of excited particles in a BEC is determined by the temperature of the system. As the temperature decreases, the number of excited particles decreases, and at absolute zero, there are no excited particles.

## 4. What is the significance of the number of excited particles in a BEC?

The number of excited particles in a BEC is an important factor in understanding the behavior and properties of the condensate. It is also a key factor in determining the critical temperature at which a BEC forms.

## 5. Can the number of excited particles in a BEC be controlled?

Yes, the number of excited particles in a BEC can be controlled by adjusting the temperature of the system. By cooling the system to lower temperatures, the number of excited particles can be decreased, and by heating the system, the number of excited particles can be increased.

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