Hemalurgy and Vin's earring

[natc]

That hyperlink was butchered so horribly.

You're also misinterpreting the annotation completely. 100% there represents "allomantic potential. Vin was snapped in utero as far as we know, so she was technically born with her abilities, while Elend is lerasium infused. 50% is every other allomancer we know, which need severe trauma to awaken their power, and then there's the ones snapped by the mists, which are even lower.

Elend is still literally stronger in practice than Vin in every conceivable way.

[ccstat he/him]

 

Alright, time to start mathing this. 

[...]

Vin had spent quite a bit of time (maximum bound: 15 years) without the earring in.

[...]

I will use Wax's earring to establish a range of decay rates, and apply them to Vin's case to narrow it down. After getting a final range of decay rates, I will look at the initial decay, and see how much power is lost over the first minute, the first 10, the first hour, and the first day. There should be noticeable power loss over a short period of time, to match up with Marsh's statements in Hero of Ages.

Those are all reasonable assumptions.It's clear from the thread so far that people have varied opinions about the power differences we see, but I think it's important for this discussion to realize where the constraints are and where the numbers are allowed to be fuzzy (and even vary wildly). For example, my opinion is that for Wax's earring 1% charge would still be okay (versus the .01% you propose), but it turns out that parameter hardly matters at all, even over 2 orders of magnitude.

 

Here are the numbers for various half lives, given as a percentage of original charge. I've color coded it for convenience, with >99%, >50%, >1%, and nearly 0. (spoiler for size)

As you can see, any half life that gives appreciable decay in the first 10 minutes (box at upper left) results in basically zero charge after an accumulated year, or even month, outside the body. Going the other direction, anything that allows for more than 1% of the charge to remain after a year (box in lower middle) means at most a 1% loss of power in the first week. After centuries (box on right) practically all of the charge is gone if you assume a half life short enough to notice.

 

The upshot is that a simple logarithmic decay can't possibly meet our requirements under our assumptions. No amount of adjusting relative power levels between Elend, Vin, and Rashek will get us to a scenario where it does. That means we need to rework our assumptions.

 

Aside from changing the math to a new decay model completely, the assumptions from your list that can impact the result are the first few:

 

1. No lower bound on the amount of charge in the spike. As t -> infinity, C -> 0.

2. Allomantic/Feruchemical power available to the user of the spike is proportional to the charge remaining in the spike

3. Giving the spike to someone merely interrupts the decay - after removing a spike, decay resumes from where it left off [...] the decay rate remains a constant.

4. Hemalurgy represents simple addition of strength. You add your allomantic power to the power in the spike.

 

There are some possible modifications to these. Most of them have problems aligning with the text, and for the most part they don't make enough of a difference to solve it completely. Some, such as different decay rates before and after first use, have been discussed quite a bit. (That scenario would basically let you mix and match rows on that chart, using a fast decay row before spiking the first recipient, then switching to a slow decay row if they take it out later.)

 

The other most likely possibility in my view is that strength isn't simply additive, or that a portion of the power goes to making the spike/ability possible at all, with further supplementation being additive.

 

As far as changing the model, my best suggestion is to treat it like real-world radioactive decay. While the decay of element X into element Y follows a simple half life formula, in many cases element Y is also radioactive and decays further into element Z. Element Y can have a dramatically different half life than its precursor did, either faster or slower (examples here). This means that the amount of radiation coming off the sample (now a mix of X, Y, and Z) does not curve-fit to a simple logarithmic decay. If a short-half-life X turns into longer-half-life Y, you get a natural transition between two different decay rates. I don't know if or how you would apply that model to fast-leak and slow-leak investiture, but I can think of several frameworks to justify it. It's early in the morning here, though, so I'm not sure how much I like the approach. Maybe after some sleep and seeing your feedback I will have a firmer opinion.

 

Right now I'm thinking that Brandon was thinking of regular logarithmic decay and just hadn't done the math, so any model we come up with to explain the facts we have will be shoehorned.

[Yata he/him]

 

Right now I'm thinking that Brandon was thinking of regular logarithmic decay and just hadn't done the math, so any model we come up with to explain the facts we have will be shoehorned.

Using a Logarithmic function to express the Hemalurgyc Decay is the only possible way, or at least It's the only that came to my mind.

[Oudeis he/him]

...

 

Thank you. This was intelligent, well-thought-out, and comprehensive. It addresses every concern I've always had with the half-life model with all the math I never bothered to put behind it.

[Seonid he/him]

Thank you, ccstat. That's what I was hoping to end up doing, and your result was what I was afraid of.

Shoehorning may be possible, but it is going to require all manner of unfounded assumptions. I'm not sure where to go from here that is in any way supported by the evidence we've got.

[ccstat he/him]

Okay, this is going to be math and graph heavy, but the short version is that there are two (relatively) simple parameters to change that can get the curve into a shape that looks close to what we want. One uses a very intuitive physical basis to derive the curve, but doesn't get quite as close as we want it to. The other is an adjustment to the equation that  gets a much friendlier shape, but I don't know enough physics to postulate a real-world process to account for the math.

 

(I'll put my guess at it below, but I have already spent a truly inordinate amount of time on this today, and I need to just post this and be done.)
The post turned out super long, so I'll go ahead and spoiler each section as well.
 
Version 1: Chemistry

We've been talking about simple logarithmic decay because that's what we know to be true for a host of natural processes, including radioactive decay, which seems like it should be analagous. These type of reactions have first order kinetics, meaning that how fast it happens depends on how much you have left, and the length of the half-life stays constant. If you've taken chemistry, you know that there are other kinetic orders (most commonly 0, and 2). Second order kinetics is intuitive for chemical reactions, since it basically says that you have to have two somethings bang into each other, so the how fast it happens depends on how much you have of both things. (If it's two of the same thing, you still multiply the terms, making the rate dependent on the amount of stuff you have left squared.) In second order reactions, the half life is not constant, and in fact doubles for each successive step. Higher orders are also possible in other systems, but the likelihood of three things all bumping into each other at the same time is pretty small, so in chemistry there are only a few known third order reactions.
 
Compared to logarithmic decay, higher order reactions drop faster at first, then slower later, exactly like what we want. You can derive the equations from here (and I finally ended up with a simple version, after many many pages of bad messy calculations, so if you are going to do any of this yourself, let me know so I can save you some headache). This is what it looks like generically:

(link)I've scaled these so that their first half lives are the same, meaning that they intersect at 50% of the starting amount.
First, Second, Third, Fourth order.

As you can see, the higher order curves fall a bit faster at the beginning, then take longer to get to zero.

The idea is that we want to choose a half life and order that will scale correctly so that our initial decay is fast, we still have some power after a year outside of the body, but by a century it's all gone. Plugging in the numbers, this is what you get:
(lots of charts--you may want to skip to the summary and come back.)

 
Basically, with 4th and 5th order kinetics, you start to get some almost reasonable possibilities, but even then you have way too much charge left by the time you get to Wax (only 3-4 fold less power than Vin). Still, all of these are a much better fit than the first order kinetics we started with.

Version 2: Manipulating the exponents

The equation for logarithmic decay is essentially y=(1/2)^t, where t is time (in this form, t is measured in # of half lives). That means that the amount of stuff varies as a function of time, and we get there by saying that the rate of disappearance is dependent on how much stuff you have left (same first order reaction as above). If you put modifiers on the t parameter, you can scale the graph to your liking. If you just multiply or divide, all you are doing is changing the half life, and you still have a first order process. But what if you instead start taking roots or logarithms of t? I can't figure out if that has a practical meaning in terms of how the reaction proceeds, but it clearly makes the right sort of changes to our graph.
 
(I'm initially tempted to say that it would be like the inverse square or inverse cube laws in physics, but the parameter being modified here is time, not distance, and it is up in an exponent. I could handwave and say, "Oh, yeah, it's basically gravity, but instead of distance from a planet it's time from a spiking" but that makes no actual sense, and the math probably can't be forced to work that way. Any ideas from the rest of you would be fun to hear.)
 
Note: I'm only going to present the roots since I don't want to bother with the log business, and I suspect I would end up back in higher order kinetics, but with non-rational powers, and I don't know if I can handle that much fun in one day.
 
Once again, we are graph and chart heavy in these spoilers. Feel free to skip to the summary and come back as needed.
Here are the graphs, with explanation.

These are graphs of y=(1/2)^(x^a) where a=1, 0.5, 0.3, or 0.1 corresponding to x and its square, cube* and tenth roots. Once again, they are scaled to intersect at their first half life. This time, 1000 is the starting amount, so 10=1% of original. 
*(obviously not really, but it's close enough for looking at graph shapes, and easier to say. The labeled chart in the next spoiler actually has cube roots, not x^0.3, if it matters to you.) 

 

The reason I'm posting multiple axis scalings is that I realized there are 8 orders of magnitude between 1 minute and 200 years, and in a plot of y vs log x (see 3rd graph below) the curves are symmetric about the half life. That means that if you choose a half life of 1 week (the geometric mean), there will be the same amount left at 200 years as was lost in the first minute. But that also means you drop to 50% a full 2 logs before the range where we want to put Vin.

Alternatively, if you make 1 year the half life, the 200yr residual is the amount lost in several days, constraining how much early decay you can expect.

 
This is a linear plot showing curve shapes. Green is our familiar, first order, vanilla decay curve.

 

This is a log-log plot, showing that it takes ten half lives (one log) after the first for the x^1 curve to reach 0.1%, and 10 times as long (two logs) for the x^0.5 curve to do the same. That geometric relation holds up for higher roots as well.

 

And here is a semi-log plot (y is linear), because that was helpful for me to see how things are shaped symmetrically about the 50% mark... 

 

...and the same semi-log plot zoomed in on the first half-life to see when things start to cross down below a given threshold (e.g. 95%). The extra lines correspond to the fourth root and eighth root curves, which I ended up removing before I took the other screenshots.

And here are the charts.

Summary: Taking the cube root starts to give you some sort-of reasonable decay curves (see the 3 month half life row, for example). Higher roots like 4 and 6 give some possibilities as well, depending on your tolerances for amount of strength left in the spike at different time points, but (especially with 6 or higher) we start running into the same problem as before with Wax's earring having too much power.

Conclusion: there are alternative ways to construct your math to give a curve with a fairly reasonable shape. (Certainly more than the two I've played with here.) I don't advocate any as being the right fit, but I think canvassing some of the space here shows that the right half life with the right decay model could possibly give a curve that fits the text and our assumptions.

I still think the version in the books is a (fuzzy, no-math) logarithmic decay, but if it ever became important I could see Brandon having Peter devise an equation not unlike one of these to govern things going forward.

 

 

EDIT: As Yata points out below, I had forgotten that Pathian earrings are reportedly made by splitting inquisitor spikes into pieces. Wax's earring can be assumed to have a fractional charge already, so if you see a workable scenario where his has too much charge for your liking, go ahead and divide by, oh, let's say 16. 

Yata also mentions that some of the time was unaccounted for--that will make a big difference to the first-order reaction rate, but the others will not be nearly as affected, since we are extending the time by less than half a log.

Edited October 12, 2015 by ccstat

[Yata he/him]

....

Sorry, I have a few doubts.

First of all. AoL take place 341 years after HoA, you stop your table for space problem or quite other 150 years are meaningless to the decay ?

Second, we know that many (k) earrings are made using a single Inquisitor's Spike, therefore Wax's earring must have 1/k of the Hemalurgyc charge left after 341 years of decay.

It's possible that if not splittered the Spike would have a low (but usefull) charge left ?

Edited October 12, 2015 by Yata

[ccstat he/him]

[...are the] other 150 years are meaningless to the decay ?

Second, we know that many (k) earring are made using a single Inquisitor's Spike, therefore Wax's earring must have 1/k of the Hemalurgyc charge.

The point about splitting each spike to make multiple earrings is an important one that I had completely forgotten about. That would absolutely be a factor. 

 

You are also right that the remaining time would make a difference. However, getting Wax's earing down to zero isn't the part that makes the curve not fit well--the charge is almost completely gone by 100 years, even without accounting for the spike being fractured into pieces. The constraint that the model is having a hard time matching is letting the spike decay fast enough that a few minutes make a noticeable difference when you've just made the spike, but still having enough charge left after more than a year for Vin's earring to still be useful.

 

(To address the timeline question specifically, I was also assuming that the spike wasn't just sitting around for the entirety of that time. Other Pathians probably wore the earring off and on, so I figured stopping at 200 years was good enough for these estimates. If we come up with a model we like where that end of the curve matters more, such as some of the ones in my second post, I would be very open to adjusting the Wax end of the timeline.)

[Yata he/him]

Another "unkwon fact" is that we don't know how much a Charge-lost influences the Allomantic/Feruchemist powers and unlucky we don't know how many power is lost also if the Spike is never out-of-body.

 

Example: I like your "version 2 sixth root with 1 year of half life" (also if i suppose we have to go on more than sixth root).

After 1-2 minutes out of body, our Spike lost something like 10% of its power. From the PoV of an Allomancer have the 90% instead of 100% it's a "great lost" ?