Gold Standard

My brother-in-law Alex is a fan of anime, and recently I saw him watching One Piece, a show I wrote about long ago. Since making that post, I learned about a better model for the situation, and I thought I'd revisit it, and see whether the writers know their statistical mechanics!

As a reminder, the main character in the show, Monkey D. Luffy, is described as a Rubberman. Rubber is a type of polymer, which means it consists of a chain of repeating units, called monomers. As we physicists often do, we can take the absolute simplest form of this: Each link of length a either goes up or down, with equal probability.

We can find the length of the chain and the total number of segments in terms of the number that go up, and the number that go down:

What's interesting about this model is that these equations do not specify specific links in the chain, only the total number that go up or down. That means we have a system with indistinct microstates, and entropy becomes relevant. For a system like this, the entropy is given by

What does all of this have to do with stretching though? For that, we turn to the first law of thermodynamics:

where U is the internal energy of the system, dQ is the heat added to the system, and dW is the work done on the system. For our system, we want to keep the energy constant, so we can set dU = 0, and the second law of thermodynamics gives dQ = T dS. The work done on the system is a force applied by the chain multiplied by a displacement, or dW = -f dL. The work is negative because the chain pulls in the opposite direction it's stretched. Putting all this together, we get

To find this derivative, we can solve the first 3 equations together to get S in terms of N and L. Skipping all the algebra involved, we end up with

where L0 is the length with no force applied. To find a and N, we can look back at the diagram from last time:

The sphere's mass is 29 million kilograms, and we can multiply by g to get the force. The temperature is around 300 K. We can get L from the diagram, and according to DaVinci, Luffy's unburdened arm span L0 should be the same as his height.

Wikipedia

Rearranging the equation above,

Plugging in the values, B = 26.8 m, but A = exp(1.3729469e+29 m^-1), which is an absolutely enormous number. That suggests that for a and N to be close to the same order of magnitude, we would need around a quadrillion links, each on the order of femtometers. I thought maybe a more detailed analysis would make this situation a little more understandable, but that ball is just too damn heavy!