Problem :

Take a system with a multiplicity function such that g(N, U) = 100. What is the entropy of the system, and in what units?

The entropy is σ(N, U) = log g(N, U) = log 100 = 4.61. There are no units, since entropy is a dimensionless quantity. Remember that log means ln!

Problem :

What is the conventional entropy S of the system in the above problem?

Recall that S = kBσ, so we calculate S = 6.360×10-23J/K.

Problem :

Say that we had a system in which adding a small bit of energy actually decreased the entropy. What can you say about the temperature of the system?

Since =   , a small increase in energy causing a decrease in entropy means that   is negative. Therefore the temperature is negative! But doesn't that violate our understanding of absolute zero? It turns out that this solution really does exist, but only in systems of nuclear spins and similar examples where one cannot actually "feel" the temperature of the system.

Problem :

Say that in a large system, adding one Joule of energy increases the entropy by 1020. What is the approximate temperature of the system?

We know that =   . We approximate the partial derivative by , and can therefore determine that τ = 10-20J.

Problem :

What is the temperature in kelvin of the system in the previous problem?

We use the relation τ = kBT and find that T = 1000K.