So, this is the last year of my undergrad degree, and in the spirit of finishing my homework (to write something informal and informative about each week’s topic for my Statistical Mechanics class), I will be blogging my classes. Perhaps I may even make a few posts that I won’t get credit for! Perhaps I will turn these posts into the lyrics of an epic post-rock concept album! We shall see. So, on that note, let me begin cramming knowledge-photons down your eye holes (or phonons and ear holes for any blind readers out there).
Oh boy, Ben! What are Degrees of Freedom?
Well, dear reader, it is good you asked, because that is what I need to write about this week. Such serendipity. So, what exactly is this concept: “Degrees of Freedom”? Well, the way I would explain it is by first introducing the concept of “configuration space”. Most people are familiar with a simple Cartesian space, like a plane or cube of volume. Each dimension within this space is orthogonal to the others, in other words, moving up-down doesn’t affect your left-right motion. If we extend this concept of spaces containing orthogonal dimensions to include things other than just position, like the rotation of an object along a certain axis, or any other way of describing the state of an object, each of these dimensions is a degree of freedom. So, a particle confined to a plane, and allowed to move around in straight lines, has two degrees of freedom (up-down motion and left-right motion). Adding a third dimension adds another degree of freedom, and allowing arbitrary rotations would add three more (as any rotation can be described by rotation about three axes).
So who cares?
Well, beyond being an interesting way of visualizing the behavior of an object, considering the degrees of freedom is a way of understanding the thermal motion of collections of objects. One of the most essential concepts is the equipartition theorem. This states that the thermal energy of the system consists of equal contributions from each degree of freedom. This allows us to quantify the nature of thermal energy, and a little thought reveals that a world without equipartition would be a bit silly (you could blow into a balloon and only have it inflate in one dimension!)
So, I hope this post has been clear, and feel free to post any questions in the comments. Expect a new post next week!