Minesweeper thermodynamics

The article discusses Boltzmann's formula exp(-E/kT). I was recently looking at the same formula in the context of semiconductors and I realized that Boltzmann's constant k is only needed because temperature uses bad units. If we measured temperature in energy instead of degrees, then Boltzmann's constant drops out. For instance, you could express room temperature as 25 meV (milli electron volts) or 2444 joules/mole and the constant disappears. Likewise, the constant in the ideal gas law disappears if you measure temperature as energy rather than degrees Kelvin. In other words, degrees Kelvin is a made-up unit that should be abandoned. (I'm not sure I believe this, but I don't see a flaw.)

Yes! Units are up to you (with some constraints)! Physicists do this in practice a lot: https://en.wikipedia.org/wiki/Natural_units

This is true of any units. A lot of physicists say things like "set c=1", does that mean that meters/feet are bad units and we should instead be measuring our height in fractions of c? That sounds inconvenient to me.

Physicists also sometimes deal with inverse temperature aka "coldness".

I don't think temperature would be measured as energy, but rather as energy per information; e.g. joules per bit. Boltzmann's constant defines the degree as one joule per a very large number of bits, to get numbers convenient at macroscopic scales.

The argument of the exponential E/kT needs to be dimensionless. If E and T are not homogeneous you will still need a k to “cancel” the units.

See: Temperature as Joules per Bit https://arxiv.org/abs/2401.12119

Another example that shows up in mid school is how we measure angle.

Like the Greeks and Babylonians we usually measure it in degrees. Later around 18th century radians started getting used, especially in power series expansions.

In India, historically, angle was measured in the units of length (for a standardized circle). That made functions like sin be a function from length to length.

I just calculated that I’m about 6.24 nanolightseconds. A nanolightsecond is just over a foot, so at least Americans should easily get use to the unit.

This is to an extent a party trick to befuddle lay people. Physicists know perfectly well that temperature is not a well-defined concept out of equilibrium. And when in a population inversion experiment when "temperature" is determined to be negative (or "beyond infinite" if you will) it arises because for a short while you have a non-Boltzmannian distribution.

Ah, but most natural unit systems don't measure time in seconds either. See https://en.wikipedia.org/wiki/Natural_units

I don't think they were talking about negative temperature, they just mean that sometimes the convenient quantity to work with is β = 1/T.

Oh, right, yes. But that's just because 1/T occurs in many formulas and thus it's often convenient to work with it instead. No exciting new physics hiding there. ;)

Right, using beta is more convenient and also it's better behaved since it doesn't have the weird "goes to infinity then comes back from negative infinity as you keep increasing it" behaviour.

> angle was measured in the units of length

i.e. a variant of radians? A radian is the circular arc whose length is equal to the radius. If we standardise the unit circle then a radian is a length of 1.

Landauer's cost is one of those things that frighten me because it makes clear how little I understand about how the world works.

Yes. Similar idea. Sometimes they would also use minutes. Whether this was as a result of contact with the Greeks I do not know. Indian trigonometry however has a different flavour from the trigonometry of Hipparchus.

What Indian mathematicians typically used was a circle with radius 3438 units. Where units would be one of the standard units of length.

Why 3438 you may wonder.

They also wanted to divide the circle into 360 x 60 minutes. For the standard circle they wanted each of those minute arcs to be of 1 unit length. The radius that would accomplish this is (360 x 60)/ 2pi ~= 3438 units.

An angle of 1 minute would then be described as arc length 1 unit on that standardized circle of radius 3438 units.

Indian version of sine and cosine were not expressed as ratios but the corresponding (half) chord for a hypotenuse of 3438 units.

Very neat. Alternative-length radians seem quite common. Radians are by definition one radius, but NATO mils are 0.00098 radii (and apparently the Finnish piiru are 0.00105 radii). What you describe are effectively a unit that is 0.0018 radii. Makes sense.

I see. Today I learned about piiru from you.

I do dislike the fact the libc sin takes argument in radians. For two reasons. One, the angles in the application are rarely in radians, so they need to be converted before the function call. Two, I would like the standard angles, such as the multiples of 15 degrees to have an accurate representation in 32 bits (or 64 bits).

Anyhow this is way off topic.