Title: Physics Toolbox 2: Taking the inverse Date: 14/09/2020 Category: Physics

One of the underappreciated challenges of science is finding useful metrics - things that we measure by. It is, after all, why we have the Imperial system - yes, pints and coppers and barrels and fluid ounces don’t make any sense together. But they did all represent a way to solve a real problem, at one point.

However, we can go further than that. With simple things, like a lump of stuff, the choices tend to be (relatively) simple - usually a matter of choosing whether to measure by weight or by volume. Usually, both will be relevant at different times. With derived units, like “speed” or “fuel economy”, however, the choice isn’t quite so obvious.

One of the biggest examples of this is fuel economy in cars: I rarely want to know how far I can go on a tank (I’m not even sure how big my tank is). I want to know: how much will it cost to do this journey? And is it worth buying a greener car? More politically, is it worth punishing “gas guzzlers” or eking out some marginal gains across the entire fleet? This is where the inverse comes in:

Don’t like something? Invert it!

A change from 10 to 15 mpg doesn’t sound like much - it’s still poor - whereas a change from 50 to 70 sounds like a lot. But, it isn’t really. Because we’re measuring the inverse of what we’re actually interested in. Convert it to gallons per mile (gpm) and the differences are:

Old New Difference 0.1 0.067 0.033 0.02 0.014 0.006

Notice the difference - the actual fuel not used per mile - in going from 10 to 15 mpg is 0.033 gallons not used per mile. Or slightly more than the total fuel use of the economic car. In other words, even making the eco-car totally fuel-free would not have as much impact!

Of course, this isn’t the only factor. There are not many bentleys in the world, and many fiestas, and the bentleys that do exist tend to do low mileage. Indeed, if humans are rational economic actors, you’d EXPECT eco-cars to do high mileage. But the mathematics is interesting nonetheless.

A slightly heavier use case is temperature. Temperature is interesting because we have an intuitive grasp of it: Ow things hot, not ow things less hot, sometimes ow things very cold. Things not want be very cold or very hot. Also, things only get so cold. So… wait, what’s all this about negative temperature?

Well, slightly less intuitively, we might get the concept of temperature as “how concentrated heat is”. As an equation, that might look like:

T(K) = Heat (J) / Volume (m<sup>3</sup>)

Since heat and volume are both scalars, negative values still don’t make sense. Unfortunately, this doesn’t quite work - materials can have different “heat capacities”, or resistance to change in temperature. Water, for instance, has a very high heat capacity, and is fairly conductive, making it an excellent tool for heat transfer. Also, why rain in temperate climates can straight-up kill you.

scientifically, temperature is really defined as - wait for it:

T = dU/dS (keeping volume and number of moles constant)

Where U refers to “work” - energy you’re putting in - and S refers to “entropy”. (if you don’t know what moles means, look here, but it roughly means “amount of stuff” in a more rigorous way than weight or volume). That’s not, I’ll grant you, very intuitive. Entropy is, again very roughly, the number of ways you can organise a body of things whilst keeping their overall properties the same.

. A negative value is now possible, albeit not helpful, and it’s not obvious why T = 0 isn’t.

What if we flip it?

β = dS/dU (again, keep volume and number of moles constant)

Congratulations, you’ve found thermodynamic beta.

So: a high beta means each change in entropy requires a lot of work. A low beta means each change in entropy requires little. A negative beta , and an infinite beta means an infinite change .

Success!

Wasn’t that easy?