# Bosons in a box

1. Mar 11, 2013

### Boltzmann2012

Hi guys,
Let us consider this thought experiment of having say Avogadro number of bosons in a box. According to statistical mechanics, it is equally probable to find every distribution of bosons in the box.
But, say we wait really long enough to find that at one point of time, we find all the bosons on one side of the box. Now, bosons as they are, have the feature that the probability of finding n of them in a state, is proportional to n+1. Hence even if we add one more boson to the box, it would most probably join the other bosons and still maintain the previous configuration.
Now, we can see that by adding more and more bosons, instead of getting more distributed, the particles are maintaining the configuration. Is there something wrong with my logic or is this what happens with bosons?

2. Mar 11, 2013

### Staff: Mentor

It is not.

n+1 is just a relative probability, and you have many other states - compared to a single one with your bosons inside. Most bosons will get a different state, unless you cool them significantly to reduce the number of available states.

3. Mar 12, 2013

### Boltzmann2012

Could you explain why the states are not equiprobable?

Also, is there any interaction between the bosons which we are missing out here?

4. Mar 13, 2013

### Staff: Mentor

It is a result of a calculation, I don't know if there is an intuitive way to explain it.
I would expect that this interaction was neglected in your setup.

5. Mar 13, 2013

### HomogenousCow

"Bosons in a box", there's a jingle out there waiting to be written with that title.

I have not studied that much QM, but the "boson in a box" system sounds very unspecific.
Energy will be quantized in the box, but what superposition of these states?
Also why do you claim that all distributions are equally probable?

6. Mar 13, 2013

### Boltzmann2012

Thank you all very much for replying. Let me make my question more specific
There are N bosons in a box which are at a sufficiently low temperature so that the entire box exists at a specific constant energy. According to statistical mechanics, for such a distribution of bosons, it will happen(very unlikely, but IS PROBABLE)that all of the bosons occupy one half of the box and the other half is empty at some time.
Now, if at the very next instant we start adding one boson at a time, then according to the probability condition of bosonic behaviour, it is more probable that the newly incoming boson would get into the fuller side of the box. My question is that is this configuration possible and if yes wouldn't this imply that our box now would be in this static situation of one side full and the other side empty for ever?

7. Mar 13, 2013

### Bill_K

What you're describing is Bose-Einstein condensation, and you're arguing that it will always happen. Well it will always happen - at absolute zero!

What you're neglecting is the effect of temperature. You need to remember that a single macrostate such as "the left side of the room" consists of many microstates, and the most probable configuration will occur when the particles are spread uniformly over the microstates. Even for bosons. At a finite temperature, this overcomes the tendency for bosons to want to occupy the same state.

8. Mar 13, 2013

### DrDu

In principle that is what is happening in a superfluid or superconductor. Once you have created a state with a nonvanishing current, it won't decay although states being energetically vavourable (i.e. those with vanishing current) are available.

9. Mar 13, 2013

### Boltzmann2012

OMG, so all this time, was I simply describing the BEC? Anyway, one more doubt, does the probability rule hold only at 0 K?

10. Mar 13, 2013

### DrDu

The point is that you are talking about very long lived non-equilibrium states. They can decay into something more probable, but their livetime is very long as this requires a whole fraction of all bosons to change state simultaneously.