In this lecture we revisit the statistical mechanics part, but now that you are a bit used to the Boltzmann distribution we will derive it properly without any assumptions about any special system.

In fact, this turns out to be a pretty good and not too difficult exercise about working with equations and models too. The hardest part is that there aren't a ton of things we are told to use, so instead we have to set up a general system and start defining things ourselves - which can lead to some pretty powerful results.

The equations might look daunting if you are not used to it, but they are NOT difficult - just don't assume you can understand a slide full of equations in 30 seconds; you might need to give it 5 minutes of thinking!

This time we make a better formal definition of "microstate" vs. "macrostate", and discuss energy fluctuations in a general system.

The Boltzmann distribution is closely related to the "partition function", which is simply just the sum of all possible states in the system. If we know the partition function, we can calculate any property of the system we want.

Once we are ready with the derivation, we also start to look at some plots of S(E), i.e. entropy as a function of energy - as you hopefully recall the thermodynamic defintion of temperature was 1/T=dS/dE, which is the slope of this curve. These curves give us more insight about processes that can happen and those that can't, and what happens when we e.g. raise the energy so much that we force the system to undergo a phase transition.

Up to this point, we have only talked about the Boltzmann distribution to determine what states are most stable at equilibrium, but we can also use the energy landscapes to gain insight about kinetics, i.e. the speed with which processes occur.

If we have time we will also apply this to alpha helices, but that will likely have to wait until lecture 6!

Reading instructions

Book chapter 8: statistical mechanics

Book chapter 9: helices

Study questions/tasks (longer ones)

Repeat all the free energy stuff: Understand E, H, S, F, G - what do they represent?
Why do we have k_B (or the gas constant R)?
How does a systems’ properties change with E (or T)?
What is a phase transition?
What are transition rates? How are the related to barriers?

Slides

Handout-format slides for lecture 5

Full-format slides for lecture 5