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Stream: deprecated: id my structure

Topic: empirical formulas

Kalan Kucera (Mar 07 2023 at 22:44):

In engineering here, and have been trying to think my way through identifying mathematical structures embedded in some of the models we use for different microscale phenomena. Most of the examples I read are much more abstract, so perhaps I'm needlessly confusing myself over something and it's actually simple, but how do you go about id'ing structures in empirical formulas?

For instance, take the basic Arrhenius formula $k=Aexp^{-E_a/RT}$, what is it's mathematical structure? First, the activation energy $E_a$ is a measured quantity, so it's just a Real number assigned units of J or eV for each substance. The temperature $T$ is the only independent variable, and everything else is a measured constant, so there seems as though there would be a set of temps in the domain, a function (can you call it a morphism?) mapping each value of $T$ to a rate $k$ in the codomain, but I can't see that this function would have any special properties, or structure other than perhaps the exponential.

Am I approaching this incorrectly?

John Baez (Mar 07 2023 at 22:46):

This is a fun question. The first step to a nice answer - in my opinion, anyway - is to replace $1/R T$ with $1/k T$ (getting the moles out of the problem) and then write $1/k T = \beta$. The reason is that $\beta$, 'coolness', is more fundamental than temperature $T$.

John Baez (Mar 07 2023 at 22:48):

Then an expression like $\exp(- c \beta E)$ is a homomorphism from the additive group reals to the multiplicative group positive numbers. You can think of this either as a function of $\beta$ with $E$ fixed (as you want to) or as a function of $E$ with $\beta$ fixed (which is also useful). Either way it's a group homomorphism of the sort mentioned.

John Baez (Mar 07 2023 at 23:12):

Moreover functions like $\exp(-c \beta E)$ are the only homomorphisms like that, at least if we rule out crazy ones that require the axiom of choice to 'construct'.

John Baez (Mar 07 2023 at 23:13):

So the next step is to understand why you want a homomorphism from the additive reals to the multiplicative positive numbers.

John Baez (Mar 07 2023 at 23:17):

(This is all just in my opinion. There are probably lots of ways you could go with this, but I've thought about this general sort of stuff and developed some opinions.)

Kalan Kucera (Mar 07 2023 at 23:18):

My initial reaction would be that it represents an evolution of physical behaviour from temperature to the more complex activation energy-based interpretation of a driving force of a reaction. Energy has more degrees of freedom than temperature and... this may be a stretch... needs a more versatile set to represent it?

Kalan Kucera (Mar 07 2023 at 23:18):

Thank you, btw, I appreciate your opinion immensely.

John Baez (Mar 07 2023 at 23:20):

Thanks! I could keep on going in the direction I outlined, but I'm not sure it'll help much... I feel I should have written something about this already, focusing on the meaning of the "Boltzmann factor" $\exp(-\beta E)$.

Kalan Kucera (Mar 07 2023 at 23:24):

Perhaps in "Getting to the Bottom of Noether’s Theorem?" Going to give it a read, and appreciate the help!

John Baez (Mar 07 2023 at 23:27):

No, definitely not that paper.

John Baez (Mar 07 2023 at 23:28):

Don't read that!

John Baez (Mar 07 2023 at 23:30):

A bit better are the week 12 and especially week 13 lectures here, where I explained a temperature-dependent number system.

John Baez (Mar 07 2023 at 23:31):

But I was doing that just as part of some other story, so I zipped through it pretty fast.

Kalan Kucera (Mar 07 2023 at 23:32):

Ok, will look at those!

John Baez (Mar 07 2023 at 23:32):

I'm afraid I've thought about this a lot more than I've written about it.

Kalan Kucera (Mar 07 2023 at 23:34):

Seems like something that will necessarily change as I look through the empirical formulas I work more with from day to day. I chose that one though because it is all over the place, and is the same basic form as equations for grain growth in polycrystalline metals, or diffusion. So, maybe I should start with looking at the number scales of measured quantities and going from there?

Kalan Kucera (Mar 07 2023 at 23:34):

At least, in trying to discern their mathematical structure?

John Baez (Mar 07 2023 at 23:42):

It all depends on where you want to go.

The most interesting thing about the Arrhenius formula, to me, is that it's an approximate formula that seems to work pretty well in lots of cases, but not an iron-clad law of nature, so when you try to understand it you wind up needing to figure out under what situations it approximately holds, and why. And I've never seen an account that's completely satisfying - probably since I'd like to see a careful derivation of it from axioms (which might hold approximately in some situations but not others), and most the people who use it aren't the kind of people who care much about that.

John Baez (Mar 07 2023 at 23:43):

This is quite different than the appearance of the Boltzmann factor $\exp(-\beta E)$ in statistical mechanics, which has a very clear and detailed explanation.

John Baez (Mar 07 2023 at 23:45):

So if I decided to work on the Arrhenius formula (and I've been tempted), I'd try to derive it from some assumptions that are "chemically realistic" at least to some approximation.

John Baez (Mar 07 2023 at 23:45):

I've seen people try this, but I'm not satisfied.

Kalan Kucera (Mar 07 2023 at 23:47):

I'm looking at more complicated derived equations (if you're familiar with the phenomena of Creep, I've been looking at diffusional flow creep) and trying to do an analysis of the creep model on the basis of the mathematical structure of the underlying phenomena, one of which is naturally diffusion. But it's been more complex than I imagined, and perhaps the approximate nature of the empirical models is part of it.

Kalan Kucera (Mar 07 2023 at 23:48):

In your opinion, can this kind of analysis be accomplished on the foundation of things like Arrhenius-type models that are mathematically "incomplete" in some form?

John Baez (Mar 07 2023 at 23:51):

You're asking if more complicated models can be clarified even though they're based on quasi-empirical, incompletely understood things like the Arrhenius formula?

John Baez (Mar 07 2023 at 23:52):

If so I'd say yes: it seems chemists and physicists make progress doing this sort of thing quite often.

John Baez (Mar 07 2023 at 23:52):

It's just because I'm a mathematician, or mathematical physicist, that I'm drawn to clarifying the simplest things that aren't already clear.

John Baez (Mar 07 2023 at 23:53):

I just find that tremendously satisfying compared to working with half-understood ideas.

Kalan Kucera (Mar 07 2023 at 23:54):

Well, certainly no lack of them in materials!

John Baez (Mar 08 2023 at 00:01):

Right! People gotta get stuff done.

Kalan Kucera (Mar 08 2023 at 00:10):

I thought it was pretty funny when one of my first forays into trying to discern the origin of all the variables in my creep model, one of them was '42.' I spent a week tracking the original source and ended up it was a fudge factor to account for the assumed sinusoidal shape of grain boundaries sliding across one another. Art imitates life.

John Baez (Mar 08 2023 at 00:16):

There you go! I've given a talk about the importance of the number 42, but it has nothing to do with creep.

James Deikun (Apr 08 2023 at 14:42):

My take on Arrhenius is that it breaks down into two assumptions: "log likelihood crossing the rate-determining energy barrier when approached is bilinear in activation energy and coolness" and "log likelihood of approaching the rate-determining energy barrier is constant in coolness". The first is basically ironclad, the second not so much. Justifying each of the assumptions is interesting separately and putting the justifications together you get an interesting statement about the mean-field energy landscape of the "typical" reaction, which exact interesting statement you get varying with how you justified the second assumption.

John Baez (Apr 09 2023 at 00:05):

Nice! Sometime I'd like to learn more about that, since I can imagine using chemical reaction networks where the rate constants are given by the Arrhenius formula.