Link Between the Density Operator and the Partition Function and Boltzmann Distribution in Quantum Statistical Mechanics

I have a very limited knowledge of statistical mechanics, but I seem to running into some related concepts for my background readings for the research project this summer. For example, see the expressions for $\rho_{B}$ and $Z_{B}$ between equations $(C.19)$ and $(C.20)$ in the attached image:

I have no idea how one has made the link between the density operator and the partition function and the Boltzman Distribution. I found the following notes (link: http://bohr.physics.berkeley.edu/classes/221/1011/notes/density.pdf) that seem to explain this topic in section 16, but I failed to understand them at all.

I won't be comfortable with statistical mechanics unless I take a full course starting next semester. In the mean time, if anyone could either link me to better explained notes or explain these concepts over here, that'd be great.

• What exactly do you want explained? How much about Stat Mech do you know? For example do you now what the partition function is and where the factor $\frac{\exp\left(-\beta H\right)}{Z}$ comes from? Or are you just asking about the density matrix, $\rho$, is? Jun 17, 2016 at 12:41
• I know of the basic formulas; so yes, I know about the partition function. I can't seem to recall the above formula, though I think it's a derived quantity from the partition function, like the average value of the energy perhaps. I know about the density matrix; I don't know about the link between it and the aforementioned stat mech concepts Jun 17, 2016 at 12:46
• There is a neat list already here on Stack Exchange: Resources for introductory quantum statistical mechanics Jun 17, 2016 at 22:36