Differences between Boltzmann and Fermi-Dirac distributions

[Urmi Roy]

Hi All,

In relation to the Boltzmann distribution vs the FD/BE distributions in different applications, I have 2 basic questions:

1. The Boltzmann distribution comes most easily from the Canonical Ensemble (constant N, V,T) while the FD/BE come from the Grand Canonical ensemble (constant .mu, V, T).

Intuitively, I picture the Boltzmann distribution being used for systems such as a container of an ideal gas. This makes sense to me, but in my h/w I got a question where I had to apply the Boltzmann distribution to a DNA molecule which increases energy by 'e' every time a bond breaks. I don't understand however how the Boltzmann distribution is applicable here, as this problem has nothing to do with V or T.

2. For an electron, I don't see why the grand canonical ensemble is applicable because I don't see how a 'chemical potential reservoir' is applicable to an electron nor how the temperature of an electron is fixed.

Please help me understand these concepts!U

 

[DrClaude]

1. The Boltzmann distribution comes most easily from the Canonical Ensemble (constant N, V,T) while the FD/BE come from the Grand Canonical ensemble (constant .mu, V, T).

Intuitively, I picture the Boltzmann distribution being used for systems such as a container of an ideal gas. This makes sense to me, but in my h/w I got a question where I had to apply the Boltzmann distribution to a DNA molecule which increases energy by 'e' every time a bond breaks. I don't understand however how the Boltzmann distribution is applicable here, as this problem has nothing to do with V or T.

In the canonical ensemble, the system is connected to a reservoir at fixed T, with which it can exchange energy. Therefore, the system is taken at being at a fixed temperature, instead of a fixed energy. I guess that in the problem, you had to calculate probabilities of a DNA molecule being broken at a certain T.

2. For an electron, I don't see why the grand canonical ensemble is applicable because I don't see how a 'chemical potential reservoir' is applicable to an electron nor how the temperature of an electron is fixed.

Going to the grand canonical ensemble can be seen as a "trick" that allows to take into account the peculiarities of the quantum statistics. You can still work with a fixed number of particles by setting the chemical potential such that the total average number of particles is equal to the (fixed) desired number, with fluctuations too small to matter. As for temperature, it is the same thing: the system can exchange energy with a reservoir at a given T, and hence at equilibrium will also have a temperature T.

 

[Urmi Roy]

In the canonical ensemble, the system is connected to a reservoir at fixed T, with which it can exchange energy. Therefore, the system is taken at being at a fixed temperature, instead of a fixed energy. I guess that in the problem, you had to calculate probabilities of a DNA molecule being broken at a certain T.

Yup, you're absolutely right and I realize that now. I also find from my notes that the Grand canonical system also considers constant T but as you mentioned, it is a trick to restrict number of particles (by fixing the chemical potential= reservoir chem. potential). So is it true that for the DNA case, I could choose either the canonical or the grand canonical ensemble, since I'm not defining anything other than the T of the system...Volume doesn't make any sense here and number of particles N doesn't make sense either...I could write out the sum of probabilities, make it =1, and find the partition factor. It would result in a different distribution that looks a lot like the Boltzmann distribution, just with the mu in it.

Going to the grand canonical ensemble can be seen as a "trick" that allows to take into account the peculiarities of the quantum statistics. You can still work with a fixed number of particles by setting the chemical potential such that the total average number of particles is equal to the (fixed) desired number, with fluctuations too small to matter. As for temperature, it is the same thing: the system can exchange energy with a reservoir at a given T, and hence at equilibrium will also have a temperature T.

Ok so regarding an electron, I still understand what exactly the chemical potential of an electron is...isn't chemical potential defined for molecules and ions?

Thanks for your help, I think I'm getting there!

 

[DrClaude]

Ok so regarding an electron, I still understand what exactly the chemical potential of an electron is...isn't chemical potential defined for molecules and ions?

The chemical potential is defined for any particle that is part of the system, especially if its number can fluctuate. For example, in a mixture of neutral atoms and the corresponding ions and electrons, all three particle numbers will fluctuate as atoms get ionized and ions and electrons recombine into neutral atoms.
All three species will have a chemical potential, and the fraction of neutral atoms at equilibrium will depend on all three μ's.