Statistical thermodynamics

[mycroft]

I think I get the concept of fermi-dirac and bose-einstein statistics can follow the derivation of their distribution functions as per Stastical physics by Guénault but I'm having severe trouble trying to apply them.For what I imagine is a simple question, two identical particles are placed into four single particle states with energies 0, 0, E0 and 2E0. I'm asked to calculate the partition function, possible energy levels and their degeneracies for each distribution. I haven't been able to find a simple example on the internet or in the books I've searched so a simple clear explanation or even a good reference would be greatly appreciated.

 

[Jhero]

Dear poster,

I understand your struggle in trying to apply fermi-dirac and bose-einstein statistics to a specific problem. These concepts can be quite abstract and difficult to grasp at first. However, with some guidance and practice, I am confident that you will be able to successfully apply them to various scenarios.

To start, let's briefly review the basic concepts of fermi-dirac and bose-einstein statistics. Both of these are statistical distributions that describe the behavior of particles at the quantum level. Fermi-dirac statistics applies to particles with half-integer spin, such as electrons, while bose-einstein statistics applies to particles with integer spin, such as photons. These distributions take into account the fact that particles at the quantum level can occupy different energy states, and the probability of a particle occupying a particular energy state is given by the distribution function.

Now, let's apply these concepts to the specific scenario you mentioned. You have two identical particles placed into four single particle states with energies 0, 0, E0, and 2E0. To calculate the partition function, we first need to determine the possible energy levels and their degeneracies for each distribution.

For fermi-dirac statistics, the possible energy levels are given by En = nE0, where n is the number of particles in the state. In this case, n can only be 0 or 1, since there are only two particles. So, the possible energy levels are E0 and 2E0. The degeneracies for these levels are 2 and 1, respectively, since there are two ways to arrange the two particles in the E0 state and only one way to arrange them in the 2E0 state.

For bose-einstein statistics, the possible energy levels are also given by En = nE0, but n can now be any positive integer. So, the possible energy levels are 0, E0, 2E0, and 3E0. The degeneracies for these levels are 2, 2, 1, and 1, respectively, since there are two ways to arrange the two particles in the 0 and E0 states, and only one way to arrange them in the 2E0 and 3E0 states.

Now, to calculate the partition function, we use the formula Z = Σe^(-βEn), where β = 1/(kT