In the B-E formula for the average number of identical bosons that can occupy a single particle state ψj , out of a set of such states
‹n›j = 1/{exp[β(ϵj – μj)] – 1}
there appears the chemical potential μj . Let me remind the definition of the chemical potential:
The chemical potential of a species in a mixture is the derivative of the free energy of the system with respect to the number of moles of that species, under constant temperature, and given that the concentrations of all the other species in the mixture remain constant.
But, in a gas of identical bosons, there is one single species. Also, since the particles are supposed not to interact, there are no reactions in the gas.
Bottom line, what has to do the number of particles occupying a certain orbital ψj , with chemical potential?
Could it be that the particles occupying different orbitals are considered as different species? But if so it is, how can this number change under a constant temperature? Constant temperature is a state of equilibrium, the average number of particle per orbital doesn't change.
(My question may be naïve, but it's a very long time since I learnt statistical mechanics.)
Your definition is from chemistry. Chemists like moles. The energy for them is also in J/mole. Physicist use particles, so the chemical potential is per particle.
Chemical potential is important parameters for the equilibrium. Two systems that can exchange particles are in equilibrium if their chemical potentials are equal.
Therefore, \mu cannot have an index. It is the same for the whole system. Otherwise the particles would flow from one state to another.
What we do next? If the system has N particles, then \sum n_i = N. This is the equation to determine \mu. Solving that equation, we are set.
\mu should be smaller than any of the energies. Otherwise some levels would be populated with negative number of particles.
If, solving the equation for \mu we obtain a value bigger than the energy of some levels, we are in a serious trouble. Then this distribution is wrong and the system degenerates to the Bose-Einstein condensate.
Your definition is from chemistry. Chemists like moles. The energy for them is also in J/mole. Physicist use particles, so the chemical potential is per particle.
Chemical potential is important parameters for the equilibrium. Two systems that can exchange particles are in equilibrium if their chemical potentials are equal.
Therefore, \mu cannot have an index. It is the same for the whole system. Otherwise the particles would flow from one state to another.
What we do next? If the system has N particles, then \sum n_i = N. This is the equation to determine \mu. Solving that equation, we are set.
\mu should be smaller than any of the energies. Otherwise some levels would be populated with negative number of particles.
If, solving the equation for \mu we obtain a value bigger than the energy of some levels, we are in a serious trouble. Then this distribution is wrong and the system degenerates to the Bose-Einstein condensate.
Thank you Igor, very much!
But I have a question: you see, the Bose-Einstein statistics is obtained in the assumption that the gas is a grand canonical ensemble. Therefore, the total number of particles, N, is not fixed. What we can say, I suppose, is that at equilibrium the average of the total number of particles is fixed. Am I right?
Assuming, tentatively, that I am right, I try to reformulate the definition of chemical potential, as you say, for physicists:
The chemical potential of an energy level in system (a gas in my case) is the derivative of the free energy of the system with respect to the number of particles of that energy, under constant temperature, and given that the average number of particles on all the other energy levels in the system remain constant.
Can this be correct?
I repeat, my doubts come from the fact that in the grand canonical ensemble the number N is not fixed, only in the canonical ensemble it is fixed. But, the trouble is that the B-E formula is not valid for the canonical ensemble, except for the asymptotic case of huge N. What I saw in the literature is that people don't assume that N is enough huge even for numbers of particles of the order 106.
So, can you tell me how to modify properly the definition?
With thanks in advance,
Sofia
As usual, statistics can be applied to a closed system assuming that it is big enough, so that one part of it can serve as a reservoir to another. The neat equation of this kind is obtained for the big number of particles. If people say that 10^6 is not big enough, it could be. One may claim this only after proper estimates. The derivation I remember from university is based on the Stirling approximation to factorial. But then we did not bother with the system of 10^6 particles. It was like 10^23 or more.
The contribution of the surface energy to the bulk Gibbs free energy per atom of a cluster of n atoms in a given phase is strongly affected by the number of atoms itself.
Source - Alessandro Latini, et all (2014), High Temperature Stability of Onion-Like Carbon vs Highly Oriented Pyrolytic Graphite, PLOS, Published: August 25, 2014, DOI: 10.1371/journal.pone.0105788
This article should help you.
http://www.physics.udel.edu/~bnikolic/teaching/phys624/PDF/chemical_potential.pdf
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