Energy and particle conservation in Maxwell-Boltzmann statistics

Question

I was working with a derivation of Maxwell-Boltzmann statistics occupation probability today and I got this problem with the constraints:

We know that the particle number is conserved when we are increasing the value of $N_i$ by a little amount, so we write that, $$\sum_{i}^{N}\delta N_i=0$$ where $N$ is the total number of particles.

Also, since if the number of particles of a particular energy $E_i$ increases, then by energy conservation, that same energy change has to be accompanied by some other particles, in different quantum states. So, for that we write: $$\sum_{i}^{N}E_i \delta N_i=0$$ However, I am not being able to understand this last equation. For example- Suppose we have $1000$ particles with energy $2 eV$ and $2000$ particles with energy $1 eV$, the total energy of the system is $4000\ eV$. Now, assuming that $1$ and $2$ $eV$ are the only energies that can be possessed by the particles, let us decrease the number of particles in the state $1 eV$ by $100$. So now we have $900$ particles in the $1 eV$ state, and since the energy of the system has to be conserved, the number of particles increased in the state $2 eV$ is 50.

This is where I start facing problems. The last argument shows that the number of particles having energy $2 eV$ is now $2050$, which, if added to the other ones, gives $2950$ particles, whereas the total number, in the beginning, was $3000$.

What am I missing here?

Answer 1

Looking at your constraints, it seems like you're dealing with a micro-canonical ensemble description of statistical physics. The constraints you give have physical meaning. The first one $$\sum_i^N \delta N_i = 0$$ tells you that the number of particles in your system is fixed. The second one $$\sum_i^N \delta N_i = 0$$ tells you that the total energy of your system is conserved.

Now be careful. N IS NOT the number of particles in your system, but the number of states that a particle can occupy. In the case you're describing, N = 2, because you have two possible energies (I' m not considering eventual degeneracies).

In you' re example you cannot satisfy the constraints, because particle conservation tells that $$\delta N_1 + \delta N_2 = 0,$$ while energy conservation implies that $$\delta N_1 + 2 \delta N_2 = 0.$$ They can' t be satisfied at the same time.