It’s natural to ask what it means to take these limits, given that you can’t actually change the constants of nature. The answer lies in the fact that these aren’t dimensionless constants. It doesn’t make sense to ask what would happen as since is dimensionless. But has units of action, has units of velocity, and has units of force times distance squared per mass squared.

So, if we’re studying a physics problem where all the velocities involved are small compared to the speed of light— is very small for all velocities in your problem—you typically can get a good approximate answer by taking the limit where And by changing units in a clever way, this can be reinterpreted as taking a limit where is held constant but The latter approach is often more convenient, because you just need to let one thing go to infinity (namely ) instead of letting lots of things go to infinity (all the velocities in your problem).

Similarly, in a physics problem where all the actions involved are large compared to classical mechanics is typically a good approximation to quantum mechanics. And it’s often convenient to study this by holding the actions that apprear in your problem constant but letting

In short, it involves a clever use of dimensional analysis.

]]>Naive question: isn’t ‘h bar’ (reduced Planck constant) a constant by itself? What does limit of h bar tending to zero mean? What is the motivation?

]]>I’ve ended on this site pondering a similar analogy and problem. My starting point was the problem of “correlations” in the many-body wavefunction, which are a bane for DFT calculations. The gist of the problem is: you have two interacting electrons, say, in a potential. Their effective wavefunction will have a certain degree of correlation built in; namely, if I know the position of electron 1, I now have SOME more knowledge about electron 2, though not certainties, of course (it’s not full on entanglement after all). This lowers the Von Neumann entropy of the wavefunction, of course. What makes me think is how there really is a “tradeoff” at work here. If there is only ONE potential well, for example, the energy would be minimized by both electrons sitting into it. But of course there is repulsion among them, so in that situation the energy would be higher. So in order to lower it they sacrifice some of the entropy of the wavefunction. Of course no one says that if I were to express Schroedinger’s equation as a minimization principle analogous to the free energy one Von Neumann’s would be THE entropy that appears in it – it could just be a function which happens to be somewhat monotonous with the ‘true’ entropy. But looking at it, it should be possible to recast the time independent Schroedinger equation with your formalism if one performs a Wick rotation and calculates only the closed loops in the path integral.

Now of course I say all that but the thought of the math scares the hell out of me.

]]>his comment23 by stating that he would not have written some of his

papers had he been aware of Gibbs’ comprehensive treatise14 .”

So, historically, there has been confusion between S_G and S_B … yet they are essentially identical and equivalent for large, macroscopic, classical systems. (see below; its an off-by-one counting problem, for an N-particle system)

It turns out that if you plug these two into the standard textbook equations, you get two different definitions of a temperature. (call them T_G and T_B). You can measure T_B by using an ideal classical gas. If you couple the ideal classical gas to a quantum simple harmonic oscillator, you find that the T_B of the oscillator can be negative, when your thermometer is small enough… Whoops. By contrast, T_G stays positive.

Similarly, heat capacity stays positive if calculated from the S_G definition, and so on. The rest of the paper articulates and explores all the consequences of this.

So, anyway, my universe stands unshaken: I always use S_G. Its possible that S_B was in my textbooks too, I don’t recall.

BTW, later on, they state/show that T_B is the temperature of an N-particle system when you accidentally counted only up to (N-1). So it’s an off-by-one counting problem.

]]>If I look at the wikipedia entry for the microcanonical distribution then first “it looks” as if only the what is called volume (surface) entropy satisfies some kind of condition which relates the differentials of energy and entropy. That is there is no remark that the Boltzmann entropy would satisfy a similar relation (which would be different from what you say). So I wonder about that.

Now “it looks” on Wikipedia, as if only more the volume entropy satisfies this condition, or may be I had overread something. Anyways thanks for pointing out the article it seems one might find more information on that issue in the article. It seems on afirst glance however that that what’s called Boltzmann and Gibbs entropies there is again different from what is mentioned in the Wikipedia article.

]]>• E. T. Jaynes, Gibbs vs Boltzmann Entropies, *Amer. J. Phys.* **33** (1965), 391–398.