A paradox for bosons with non-degenerate states?

[Philip Koeck]

The BE-distribution for the case of only one state per energy level (g_i = 1) is
n_i = 1 / (exp^{(u_i - μ)β} - 1)
This is a reasonable and well defined distribution as far as I can see.

On the other hand the number of possibilities to realize a given distribution of bosons among k energy levels with g_i states in level i is given by
W = ∏_{i = 1 to k} (n_i + g_i -1)! / (n_i! (g_i-1)!)
If g_i = 1 for all i then W = 1 no matter what the n_i are. In other words all distributions are equally likely.
How can the BE-distribution be well defined then for this case?

 

[Demystifier]

The BE-distribution for the case of only one state per energy level (g_i = 1) is
n_i = 1 / (exp^{(u_i - μ)β} - 1)
This is a reasonable and well defined distribution as far as I can see.

On the other hand the number of possibilities to realize a given distribution of bosons among k energy levels with g_i states in level i is given by
W = ∏_{i = 1 to k} (n_i + g_i -1)! / (n_i! (g_i-1)!)
If g_i = 1 for all i then W = 1 no matter what the n_i are. In other words all distributions are equally likely.
How can the BE-distribution be well defined then for this case?

Your number of possibilities says nothing the total energy in the system. You must add a constraint saying something about the total energy, and this constraint changes the probabilities. It is the temperature (i.e. the parameter $\beta$) that encodes the knowledge of this constraint. Further constraint comes from a knowledge about the total number of particles, which is encoded in the parameter $\mu$.

 

[Philip Koeck]

Your number of possibilities says nothing the total energy in the system. You must add a constraint saying something about the total energy, and this constraint changes the probabilities. It is the temperature (i.e. the parameter $\beta$) that encodes the knowledge of this constraint. Further constraint comes from a knowledge about the total number of particles, which is encoded in the parameter $\mu$.

Does that really solve the paradox? If I require that E_tot = ∑ n_i u_i and N = ∑ n_i I can still say that all sets of n_i within these constraints have W = 1. Obviously this will only be possible for certain spacings of energy levels.

 

[Demystifier]

Does that really solve the paradox? If I require that E_tot = ∑ n_i u_i and N = ∑ n_i I can still say that all sets of n_i within these constraints have W = 1. Obviously this will only be possible for certain spacings of energy levels.

You are computing a wrong W. Your W counts in how many ways a given microscopic state can be formed, so it's not surprising that you get W=1. Instead, you must count in how many ways a given macroscopic state can be formed. Typically, a given macro state can be realized by many different micro states, even if all g_i=1.

 

[Philip Koeck]

You are computing a wrong W. Your W counts in how many ways a given microscopic state can be formed, so it's not surprising that you get W=1. Instead, you must count in how many ways a given macroscopic state can be formed. Typically, a given macro state can be realized by many different micro states, even if all g_i=1.

If the total energy and particle number is fixed how would you define a macro state? I always thought in that case a macro state should just be a distribution, i.e. how many particles are on which energy level. The micro states are simply the different ways to realize a given distribution. Obviously if each level only has one state there is only one micro state per macro state. Notice that the states belonging to an energy level are something entirely different to micro and macro states.
This paradox is mentioned in lecture notes by Feder that have been uploaded in another thread I recently started (so not my idea), but I don't understand how Feder resolves it, I'm afraid.

 

[Demystifier]

If the total energy and particle number is fixed how would you define a macro state?

For instance, suppose that the whole system has $10^{23}$ particles and energy 10 J. Consider a macro subsystem containing $10^{22}$ particles. It's expected energy will be 1 J, but it can also happen that it has 0.9 J or 1.1 J. Nevertheless 1 J is more probable than 0.9 J or 1.1 J, because there are more micro realizations of 1 J than that of 0.9 J or 1.1 J.

 

[Philip Koeck]

So you fix the energy of the larger system, but not the energy of the smaller subsystem. The latter finds it's most likely energy by itself so to say.
I'm struggling to see what's wrong with my description. Take a container with an ideal gas in it. N is constant and T can also be constant which means that the total inner energy is constant. And I don't want to consider any smaller subsystem. To my way of thinking there can still be many macro states simply defined by the distribution of atoms among "energy levels" (kinetic energies). The micro states are then the different ways each macro state can be realized, for example due to atoms flying in different directions with the same velocity.

 

[Demystifier]

On the other hand the number of possibilities to realize a given distribution of bosons among k energy levels with g_i states in level i is given by
W = ∏_{i = 1 to k} (n_i + g_i -1)! / (n_i! (g_i-1)!)
If g_i = 1 for all i then W = 1 no matter what the n_i are. In other words all distributions are equally likely.
How can the BE-distribution be well defined then for this case?

I've just checked out K. Huang, Statistical Mechanics, page 182. He writes down exactly the same equation for W, but he uses a different interpretation of g_i. For him, g_i is not the number of states with the same energy. Instead, g_i is the number of states in a cell, where cell represents the set of all states that have energy in a certain finite range. In other words, g_i is the number of states with approximately equal energy. At the macroscopic level one cannot measure energy with perfect precision, so it makes sense to consider all states within the same cell as different micro realizations of the same macro state. Hence it does not make sense to consider the case g_i=1.

Now you may be worried that then physics depends on the choice of the size of cell. But actually it doesn't, provided that the size is not too small. For instance, it can be shown that a modification of the size of cell modifies entropy by an additive constant. This constant does not play any physical role because a change of entropy $\Delta S$ in a thermo-dynamical process does not depend on that constant.

 

[Demystifier]

So you fix the energy of the larger system, but not the energy of the smaller subsystem. The latter finds it's most likely energy by itself so to say.

Yes.

I'm struggling to see what's wrong with my description. Take a container with an ideal gas in it. N is constant and T can also be constant which means that the total inner energy is constant. And I don't want to consider any smaller subsystem. To my way of thinking there can still be many macro states simply defined by the distribution of atoms among "energy levels" (kinetic energies). The micro states are then the different ways each macro state can be realized, for example due to atoms flying in different directions with the same velocity.

I see nothing wrong with that, but see my post #8 above.

 

[Philip Koeck]

I've just checked out K. Huang, Statistical Mechanics, page 182. He writes down exactly the same equation for W, but he uses a different interpretation of g_i. For him, g_i is not the number of states with the same energy. Instead, g_i is the number of states in a cell, where cell represents the set of all states that have energy in a certain finite range. In other words, g_i is the number of states with approximately equal energy. At the macroscopic level one cannot measure energy with perfect precision, so it makes sense to consider all states within the same cell as different micro realizations of the same macro state. Hence it does not make sense to consider the case g_i=1.

Now you may be worried that then physics depends on the choice of the size of cell. But actually it doesn't, provided that the size is not too small. For instance, it can be shown that a modification of the size of cell modifies entropy by an additive constant. This constant does not play any physical role because a change of entropy $\Delta S$ in a thermo-dynamical process does not depend on that constant.

What about if we restrict ourselves to systems with truly discrete energies. Not sure how realistic that is, but one could imagine binding sites with different binding energies for example. One would also have to require that many particles can bind to the same site for "bosons", otherwise the system would be "fermionic". Notice that I don't mention the quantum mechanical definition of bosons and fermions. I just say that many or just one particle can occupy each site.
I think we could have a situation then that there is a finite number of sites with a given energy level, so the g_i could be all 1 in principle.
I actually simulated a "bosonic" system like this last autumn and I wondered about its behaviour then. With large values for the g_i I get very nice BE-distributions, but for small g_i they become very noisy. For g_i = 1 the distribution is completely random. The way I see it the maximum of W is sharp for large g_i, but becomes very round for smaller g_i. Finally, for g_i = 1 all possible distributions give the same W.

 

[Demystifier]

What about if we restrict ourselves to systems with truly discrete energies.

It doesn't prevent you from organizing energy levels into cells, each of which has a relatively large g_i.

For instance, if the discrete energy levels are 1,2,...,100, you can define the first cell as (1,2,...,10), second cell as (11,12,...,20), etc.

 

[Philip Koeck]

I've just had another look at the derivation of the BE distribution and realized that the expression for W I give in post 1 of this thread is valid for all integer g_i and n_i, whereas the BE-distribution is only valid if both the n_i and the g_i are all much larger than 1. So there's no paradox. One just has to take care of the BE distribution's range of validity.

 

[Demystifier]

I've just had another look at the derivation of the BE distribution and realized that the expression for W I give in post 1 of this thread is valid for all integer g_i and n_i, whereas the BE-distribution is only valid if both the n_i and the g_i are all much larger than 1. So there's no paradox. One just has to take care of the BE distribution's range of validity.

Yes, because at some point factorials are approximated by exponentials through the Stirling's formula.