Syllabus

• We are going to focus on the equilibrium properties that are described best through

thermodynamics. We start with thermodynamics. Basically, it's a phenomenological

approach, so you essentially look at something as a kind of black box, without knowing what

the ingredients are, and try to give some kind of description of how its function and properties

change. These can be captured, for the case of thermal properties of matter, through the laws of

thermodynamics, which we will set out in this first section, which will roughly take us the first

four lectures of the course.

• Then we learn that statistical mechanics is a probabilistic approach – to deal with a large

number of degrees of freedom.

• But then we said large number of degrees of freedom. So what are these degrees of freedom?

In thermodynamics, you look at the system as a black box and try to develop laws based on

observations. We know that this box contains atoms and molecules that follow very specific

laws, either from Newtonian mechanics or quantum mechanics. And so if we know everything

about how atoms and molecules behave, then we should be able to derive how large collections

of them behave. And get the laws of thermodynamics as a consequence of these microscopic

degrees of freedom and their dynamics. And so that's what we will discuss in the third part of

the course that is devoted to kinetic theory.

• It is beneficial to, rather than follow individual particles in a system, adopt a probabilistic

approach and think about densities and how those densities evolve according to Liouville's

Theorem.

• What we will also try to establish is a very distinct difference that exists between

thermodynamics, where things are irreversible and going in one direction, and Newtonian, or

quantum mechanics, where things are reversible in time. And we'll see that really it's a matter

of adapting the right perspective in order to see that these two ways of looking at the same

system are not in contradiction.

• So having established these elements, we will discuss statistical mechanics in terms of some

postulates about how probabilities behave for systems that are in equilibrium. And how based

on those postulates, we can then derive all the laws of thermodynamics and all the properties

of thermodynamics systems.

• Classical systems-- description of particles following classical laws of motion. And, again, as a

first simplification, we will typically deal with non-interacting systems, such as ideal gas. And

make sure that we understand the properties of this important fundamental system from all

possible perspectives.

• Realistic classical systems where there are interactions among these particles. And there are

two ways to then deal with interactions.

• You can either go by the way of perturbation theory. We can start with the ideal system and add

a little bit of interaction and see how that changes things. Or, you can take another perspective,

and say that because of the presence of interactions, the system really adopts a totally different

type of behavior.

• And there's a perspective known as mean-field theory that allows you to do that. Then see how

the same system can be present in different phases of matter, such as liquids and gas, and how

this mean field type of prescription allows you to discuss the transitions between the different

types of behavior.

• Quantum description of the microscopic degrees of freedom. We will see how the differences

and similarities between quantum statistical mechanics and classical statistical mechanics

emerge.

• The place where quantum statistical mechanics shows its power is in dealing with identical

particles, which classically, really are kind of not a very well-defined concept, but quantum-

mechanically, they are very precisely defined, what identical particles mean. And there are two

classes-- fermions and bosons--and how even if there's no interaction between them, quantum

statistics leads to unusual behavior for quantum systems of identical particles, very distinct for

fermions and for bosons.

Recommended Texts

• Pathria, Beale - Statistical Mechanics. (totally followed)

Other Texts

• Mehran Kardar - Statistical Mechanics of particles.

• Reif - Statistical Physics.

Similar Course

• Statistical Mechanics of particles - Mehran Kardar, MIT: https://ocw.mit.edu/courses/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2013/.

All Files