Not quite, but close. It should be a + instead of a—in the denominator. Nice work, though.
You have the right formula for the entropy. Notice that it is nearly identical to the Bernoulli distribution entropy. That should make sense: there is only one state with energy 0 or Nε, so the entropy should go to 0 at those limits. It’s maximum is at Nε/2. Past that point, adding energy to the system actually decreases entropy. This leads to a negative temperature!
But we can’t actually reach that by raising its temperature. As we raise temperature to infinity, energy caps at Nε/2 (specific heat goes to 0). To put more energy in, we have to actually find some particles that are switched off and switch them on. We can’t just put it in equilibrium with a hotter thing.
Not quite, but close. It should be a + instead of a—in the denominator. Nice work, though.
You have the right formula for the entropy. Notice that it is nearly identical to the Bernoulli distribution entropy. That should make sense: there is only one state with energy 0 or Nε, so the entropy should go to 0 at those limits. It’s maximum is at Nε/2. Past that point, adding energy to the system actually decreases entropy. This leads to a negative temperature!
But we can’t actually reach that by raising its temperature. As we raise temperature to infinity, energy caps at Nε/2 (specific heat goes to 0). To put more energy in, we have to actually find some particles that are switched off and switch them on. We can’t just put it in equilibrium with a hotter thing.