Reading Response, Week 15 | Thermodynamics and Statistical Mechanics

Reading Response - Week 15

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1: Let's explore some consequences of indistinguishability and probability in the context of a situation we know quite well: rolling two (fair) six-sided dice. For "classical" dice, which are identical but distinguishable, the probability of rolling a "2" is 1/36, while the probability of rolling a "7" is 6/36. Now, consider two (fair) six-sided "quantum" dice, which are indistinguishable. What's the probability of rolling a "2"? What's the probability of rolling a "7"? (Hint: it might be easiest to make an exhaustive list of combinations).

2: In your notes, reproduce an annotated version of the derivations of the grand partition function and occupancy (from the section Distribution Functions on p. 266-268,) related to the (a) Fermi-Dirac (FD) distribution, (b) the Bose-Einstein (BE) distribution, and (c) the Boltzmann (also sometimes called the Maxwell-Boltzmann (MB)) distribution. What questions do you have about the material in this section? Please be specific so I will know how much class time to spend on these derivations.

3. In your notes, copy and annotate the important features of Figure 7.6 and Figure 7.7. On Figure 7.7, the occupancy for the MB distribution when epsilon = mu is shown directly, and the occupancy for the BE distribution is implied. Add in the value for the FD distribution when epsilon = mu. What are those values?

4: Consider the Fermi-Dirac distribution. (a) What's the value of the Fermi-Dirac distribution as T --> 0 (assume epsilon > mu)? (b) What's the value of the Fermi-Dirac distribution as T --> infinity (assume epsilon > mu)? (c) What's the value of the Fermi-Dirac distribution when epsilon = mu?

5. Starting from the Fermi-Dirac distribution given in eq. (7.23), explain (mathematically) how it becomes the step function shown in Figure 7.9, valid for T = 0.

6. How is the Fermi energy related to the chemical potential?

7. In your notes, reproduce an annotated version of the derivations on p. 273 – 275 that lead to eqs. (7.38) and (7.42). What questions do you have about the material in this section? Please be specific so I will know how much class time to spend on these derivations.

8. In eq. (7.38) and eq. (7.40), there is a factor of 2. Where does this factor of two come from? How would this factor change if these particles weren’t electrons (but were still fermions)?

9. What's degeneracy pressure? What is its physical origin?

10. Describe the density of states g(ε) in words. What's the "purpose in life" of the density of states?

11. As T increases, does the chemical potential μ increase or decrease? In your own words, explain why the chemical potential must change as the temperature increases.

What material from this reading (or previous classes) would you like us to go over in more detail?