Rings a Bell

A couple weeks ago, it was announced that this year's Nobel Prize in Physics was going to a group of scientists who experimentally tested Bell's Theorem. As with many concepts in quantum mechanics, this can be a bit tricky to understand, so I wanted to build it up piece by piece (as much for myself as for you).

When Quantum Mechanics was first developed, many people (including Einstein) were disturbed by the implication that, not only did interactions have random results, but that entangled particles could communicate those results instantaneously, breaking the speed of light. One explanation that was proposed to avoid this problem was the "Hidden Variable Hypothesis", which claimed the final states of the entangled particles were actually determined by some unknown property present at their joint creation, and therefore no information needed to be exchanged. John Bell came up with a way to test for the presence of a hidden variable, which I'll outline below.

First, suppose we have some collection of measurements. Each measurement has three properties: A, B, and C, which can either be +1 or -1. If each property is assigned randomly, we can think about the probability of two properties being the same, e.g. P(A = B). Now we can write

If you're not sure about this, you can try a couple sets of values, but the key is that we only have 2 choices, +1 and -1, for 3 properties. Now these properties are exactly the type of hidden variables that we're suggesting may exist. If we can come up with an experiment that can measure yes/no for three different properties, then we can simply count the outcomes and check if this inequality holds.

Looking again at that old post I linked to, we can imagine the following experiment: We produce entangled particles with opposite spin, and send them in opposite directions. Each goes into a Stern-Gerlach box and gets measured as spin-up or spin-down on the box's axis. However, we vary the angle of each box between 3 possibilities: 0°, 120°, and 240°. Our three properties are "spin-up along n-degree axis", and negation represents spin-down.

With this setup, we can think of the hidden variables as a set of rules for how the particles respond to each of the three angles. When a pair is produced, each is assigned one of these rule sets, e.g.

Since the particles have opposite spin, comparing the two detectors means the equalities in the equation above become not-equal. Now we can look at all the possible combinations of A, B, C between the two detectors, as well as each set of rules the pairs could be assigned, and find the probability that the two measurements are opposite:

In the inequality above, each term contributes 1/3, which gives us a total of 1 and satisfies the relation.

Quantum mechanics, though, predicts a different result. When one of the entangled particles is measured to be spin-up along a particular axis, we immediately know the other one is spin-down along that same axis. Knowing the second particle's orientation, we can find the probabilities of measuring opposite spin along each of the possible axes:

Adding up the terms again, according to Quantum Mechanics we only get 3/4, violating Bell's Inequality! That means we have an experimental method to test for hidden variables. Unfortunately, dealing with entangled particles is a delicate process, and the experiment needs to be repeated several times to accurately measure the statistical distributions, which is why it has taken more than 50 years to confirm this result. A well-deserved Nobel Prize for these scientists!

A Churning Ring of Water

[Title with apologies to Johnny Cash.]

The sink in our new home has an interesting setting that I was curious about:

It sprays a thin film outward, but the water curves back to meet in the middle again. When the water leaves the sprayer, there are only two forces acting on it: gravity pulling it down, and surface tension pulling the droplets together. I mentioned surface tension long ago, but I've never dug into the mechanics of it.

Surface tension is a force that acts to decrease the surface area of a fluid. For a given volume, a sphere has the smallest surface area, which is why water forms drops, and why shot towers can make round bullets. The magnitude of the force is given as

where γ is a constant that depends on the two materials being considered (air and water in this case) and L is the length of the edge that F will act to reduce. The sink is spraying out a ring of water, so if we take a cross-section, L is the inner plus the outer circumference of the ring. We can rewrite the force as

where m is the total mass of water, a is the acceleration, ρ is the density, A is the area of the ring, and Δh is the small vertical slice we're considering. Now this a refers to the radial acceleration of each water molecule, but we want to relate it to back to L. To do that, we can write two equations expressing L and A in terms of the inner and outer radii of the ring:

Since the ring is thin, we can take r1 approximately equal to r2, and after a bunch of algebra write

Since the flow of water is constant, A must be constant, so we can use the above equation to get a timeseries for L, then find r1 and r2.

In order to integrate this, we need initial values for L and Ldot. We can approximate the opening on the faucet to get r1 and r2, and find the initial L and the constant A. Then we can use A with the typical flow rate of 2.2 gallons/minute to find Ldot. Something didn't quite work out with my estimations, since the scale is way off in the following plots, but the shape matches great. Here's a side view of the spray:

and an animation descending through cross-sections:

So far I haven't found much use for this setting when cleaning dishes, but it did give me something interesting to think about!

A Charge of Battery

This week, I have a question from my mother Sally about their Chevy Bolt: I was thinking about the most efficient way to drive the Bolt on a long trip: Do you drive more slowly, so that your miles/kWh are more and you don’t need to charge as much? When you do charge do you stop frequently and only charge at the highest rate (the rate declines as the battery % is higher, e.g. 42kWh, declining to 26kWh), or do you stop as little as possible and stay until you’re at 80%?

As Sally outlines, there are two states the car will be in during a trip: charging and driving. The amount of time it takes to add a certain amount of charge to the battery depends on how much charge is in it – As it fills up, it gets harder to add more. During charging then, we can write

that is, the rate of charging is proportional to the space remaining. While driving, we have to push against a wind resistance that depends on our speed:

Drag typically depends on the square of velocity, meaning that increasing speed can quickly become counter-productive.

Now, both these equations I've written are just proportionalities, which need specific scales and shapes. Normally I'd have to make up something based on wild assumptions of a clueless physicist, but lucky for me, Bolt drivers are my kind of crazy, and have measured the behavior of their cars in detail!

For the first equation, I found this chart from InsideEVs:

and for the second, I found one from ChevyBolt.org:

Since these are just graphs, I had to do a bit of fitting, but I'm pretty happy with the results. To answer Sally's question, I chose a bunch of different driving speeds and max/min charge levels. We drive the car at the given speed until the battery hits the minimum charge level, then we stop and charge until it gets to the maximum charge. We continue for 1000 miles, then check the average speed during the trip, and the number of stops.

First, I plotted a heatmap of the average speeds for the different max/min charge values, planning to mark the point where we get the best speed and the fewest stops. Unfortunately, it turned out both of those always occurred with a min charge of zero, so it's easier to see them as lines:

I found this a bit difficult to take in, so I tried plotting only the zero min charge case for each speed:

This shows some really interesting results: Because of the difficulty in getting up to the battery's capacity, high max charge can result in spending most of your time charging, even at high speeds. However, if your max charge is too low, you have to stop over and over, even if it's only for short time.

Thanks for a great question, Sally!