The pleasures of enough time

One of the joys of retirement is the ability to take the time to fully understand the math behind statistical mechanics and thermodynamics (on which large parts of chemistry are based — cellular biophysics as well). I’m going through some biophysics this year reading “Physical Biology of the Cell” 2nd Edition and “Molecular Driving Forces” 2nd Edition. Back in the day, what with other courses, research, career plans and hormones to contend with, there just wasn’t enough time.

To really understand the derivation of the Boltzmann equation, you must understand Lagrange multipliers, which requires an understanding of the gradient and where it comes from. To understand the partition function you must understand change of variables in an integral, and to understand that you must understand why the determinant of the Jacobian matrix of a set of independent vectors is the volume multiplier you need.

These were all math tools whose use was fairly simple and which didn’t require any understanding of where they came from. What a great preparation for a career in medicine, where we understood very little of why we did the things we did, not because of lack of time but because the deep understanding of the systems we were mucking about with simply didn’t (and doesn’t) exist. It was intellectually unsatisfying, but you couldn’t argue with the importance of what we were doing. Things are better now with the accretion of knowledge, but if we really understood things perfectly we’d have effective treatments for cancer and Alzheimer’s. We don’t.

But in the pure world of math, whether a human creation or existing outside of us all, this need not be accepted.

I’m not going to put page after page of derivation of the topics mentioned in the second paragraph, but mention a few things to know which might help you when you’re trying learn about them, and point you to books (with page numbers) that I’ve found helpful.

Let’s start with the gradient. If you remember it at all, you know that it’s a way of taking a continuous real valued function of several variables and making a vector of it. The vector has the miraculous property of pointing in the direction of greatest change in the function. How did this happen?

The most helpful derivation I’ve found was in Thomas’ textbook of calculus (9th Edition pp. 957–> ). Yes Thomas — the same book I used as a freshman 6o years ago ! Like most living things that have aged, it’s become fat. Thomas is now up to the 13th edition.

The simplest example of a continuous real valued function is a topographic map. Thomas starts with the directional derivative — which is how the function height(north, east) changes in the direction of a vector whose absolute value is 1. That’s the definition — to get something you can actually calculate, you need to know the chain rule, and how to put a path on the topo map. The derivative of the real valued function in the direction of a unit vector turns out to be the dot product of the gradient vector and any vector at that point whose absolute value is 1. The unit vector can point any direction but the value of the derivative (the dot product) will be greatest when the unit vector points in the direction of the gradient vector. That’s where the magic comes from. If you’re slightly shaky on linear algebra, vectors and dot products — here’s a (hopefully explanatory) link to some basic linear algebra — https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/. This is the first in a series — just follow the links.

The discussion of Lagrange multipliers (which is essentially the relation between two gradients — one of a function, the other of a constraint in Dill pp.68 -> 72 is only fair, and I did a lot more work to understand it (which can’t be reproduced here).

For an excellent discussion of wedge product and why the volume multiplier in an integral must be the determinant of the Jacobian — see Callahan Advanced Calculus p. 41 and exercise 2.15 p. 61, the latter being the most important. It explains why things work this way in 2 dimensions. The exercise takes you through the derivation step by step asking you to fill in some fairly easy dots. Even better is  exercise 2.34 on p. 67 which proves the same thing for any collection of n independent vectors in R^n.

The Jacobian is just the determinant of a square matrix, something familiar from linear algebra. The numbers are just the coefficients of the vectors at a given point. But in integrals we’re changing dx and dy to something else — dr and dTheta when you go to polar coordinates. Why a matrix here? Because if differential calculus is about anything it is about linearization of nonLinear functions, which is why you can use a matrix of derivatives (the Jacobian matrix)  for dx and dy.

Why is this important for statistical mechanics. Because one of the integrals you must evaluate is of exp(-ax^2) from -infinity to + infinity, and the switch to polar coordinates is the way to do it. You also must evaluate integrals of this type to understand the kinetic theory of ideal gases.

Not necessary in this context, but one of the best discussions of the derivative in its geometric context I’ve ever seen is on pp. 105 –> 106 of Callahan’s bok

So these are some pointers and hints, not a full discussion — I hope it makes the road easier for you, should you choose to take it.

 

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