The n-Category Café

Although if you want to be super precise, “kilobyte” in colloquial usage is ambiguous, since in the context of information storage it could mean wither 1000 bytes or 1024 bytes.

$1000^n$ is used by:

• hard drive and storage manufacturers

• measurements of network speeds (1Mbps is 1000000 bits per second)

• online storage providers (Dropbox’s paid plan for 2TB is 2000GB for instance)

• macOS and iOS

• Ubuntu (see https://wiki.ubuntu.com/UnitsPolicy )

• most modern GNOME applications and most GUI applications on desktop Linux

• hard drive manufacturers have been sued over this, and US courts have agreed with hard drive manufacturers that 1 GB = 1000 MB

• the International System of Units (SI) and the International Electrotechnical Commission (IEC) both use the decimal definitions (1 kB = 1000 B)

Microsoft is pretty much the lone holdout.

ISO/IEC 80000 and the JEDEC memory standards introduce new units ki, Mi, Gi, Ti, etc., spelled “kibi”, “mebi”, “gibi”, “tebi”, etc. that are powers of 1024.

Thanks for catching the error—I’ve fixed it.

Maybe I’ll insert a brief remark about how ‘kilobyte’ is ambiguous. I hadn’t known base 10 was winning! I’ve heard people complain that they’re being cheated when their terabyte drive has only 91% of the storage they expected, since they thought they were getting a tebibyte drive.

Those other quantities are important, but they’re not called ‘temperature’. They’re called the ‘thermodynamic conjugates’ of the conserved quantities in question.

I haven’t heard people discuss the thermodynamic conjugate of angular momentum or momentum, but Landau uses these (perhaps implicitly—I forget) to argue that an object in thermodynamic equilibrium with its surroundings will not be moving or spinning. The thermodynamic conjugate of particle number, for conserved particles, is called ‘chemical potential’, and it’s super-important in chemistry.

In the third to last section of my book I discuss the thermodynamic conjugate of volume, and hint at how you can study these ideas more generally. Maybe I should say more. No, that would be the start of another book.

That frequency of $\sim 10$ gigahertz per kelvin seems enormously high: what’s it supposed to mean? Are we saying that if $E = k T$ and a system has temperature $T =$ 1 kelvin, its energy $E$ is such that the phase of its wavefunction is oscillating at a frequency of $E/h \sim 10$ gigahertz? That seems very high.

Hmm, I guess it’s actually not! But now I’m even more confused:

The cosmic microwave background (CMB) radiation is a thermal quasi-uniform black body radiation which peaks at 2.725 K in the microwave regime at 160.2 GHz, corresponding to a 1.9 mm wavelength as in Planck’s law.

How are we getting 160.2 gigahertz? Maybe there’s a factor of $2 \pi$ in here?

I can work this out, and I should. I hadn’t realized 10 gigahertz corresponds to such a low temperature. Just imagine how low the temperature is corresponding to a typical 1 megahertz AM radio frequency!

Could

$Z(X,\beta) = \sum_{i \in X} \exp (-\beta E_i)$

(regarded as a function of $\beta \in \R_+$) perhaps sometimes extend to a (Schwarz) distribution in $\beta$ defined on (some nontrivial part of) the complex plane $\C \supset \R_+$ ?

[Asking for a frind, \cf

https://arxiv.org/abs/math/0611240 ;

there are related questions about heat kernels.

I just posted this back at John’s original questions about partition functions as Euler characteristics but maybe here is where it belongs.]

Thanks! The issue regarding bases of logarithms is quite annoying because there’s no simple way to make everything beautiful except to use base $e$ everywhere… which goes against my desire to allow base 2, to make it easy to talk about ‘bits’. That’s a purely childish desire as far as this book goes: I just figure people often think about ‘bits’ when they talk about information, so sometimes I want to give them bits.

You located the fault line where I give up using $\mathrm{log}$ and start using $\ln$, to make the differentiation simple. Unfortunately not the exact same place where I start putting Boltzmann’s constant $k$ in front of the entropy: I want to wait and introduce $k$ a bit later. So there is Shannon entropy with $\mathrm{log} p$, Gibbs entropy with $k \ln p$, and this half-breed with $\ln p$.

If I weren’t trying to make contact with experimental data, I would have used units where $k = 1$.

Anyway, I have fixed this error in at least a stop-gap manner.

Thanks for catching that mistake! Of course if $E_1 \neq E_2$ then one of them is smaller than the other. But to state the problems it’s convenient to know which one is smaller, so then I assumed $E_1 \lt E_2$, without mentioning it. But that’s ridiculous: I should explicitly assume $E_1 \lt E_2$ from the very start! A corrected version should appear 15 minutes from now.

I’m glad you’re enjoying the book. Yes, entropy is very interesting mathematically!