[FRIAM] To repeat is rational, but to wander is transcendent

Tue Mar 29 17:37:57 EDT 2022

I wrote a big long tedious thing at the branch point in this thread from Frank’s post, but then had the good sense to delete it without sending.

The great thing about technical terms is that one can do lots of work to get an idea clear, assign the terms _at the end_ as tags for the properly-formulated idea, and then send those terms out into the world where they can be joyously used as metaphors, in contexts where none of what was needed for the precise meaning need be checked to have any pertinence to the topic under discussion.

To the wide world, intensive and extensive are available metaphorical tokens, understood in terms of the general phenomenology that we use to kinda-sorta make them intuitive when we tach them.  For me that is a pit of endless hazard, so they are technical terms, usable in a few places but not obviously indefinitely more widely, unless one can do all the legwork to justify them.

In the technical usage, I would say:

1. One first has to have a notion of a macrostate; all these terms only come into existence with respect to it. (They are predicates of what are called “state variables” — the intensive ones and the extensive ones — and that is what the “state” refers to.)

2. One needs some criterion for what is likely, or stable, which in general terms is an entropy (extending considerably beyond the Gibbs equilibrium entropy, but still to be constructed from specific principles), and on the macrostates _only_, the entropy function (which may be defined on many other states besides macroststates as well) becomes a _state function_.

3. Then (actually, all along since the beginning of the construction) one needs to talk about what kind of aggregation operator we can apply to systems, and quantities that do accumulate under aggregation become the arguments of the state-function entropy, and the extensive state variables.  (I say “accumulate” in favor of the more restrictive word “add”, because what we really require is that they are what are termed “scale factors” in large-deviation language, and we can admit a somewhat wider class of kinds of accumulation than just addition, though addition is the extremely common one.)

4. Once one has that, the derivatives of the entropy with respect to the extensive variables are the intensive state variables.  It is precisely the duality — that one is the derivative of a function with respect to the other, which is the argument of that function — that makes it not bizarre that both exist and that they are different.  But as EricC rightly says, if one just uses phenomenological descriptions, why any of this should exist, and why it should arrange itself into such dual systems, much less dual systems with always the same pair-wise relations, seems incomprehensible.  For some of the analogistic applications, there may not be any notions of state, or of a function doing what the entropy does, or of aggregation, or an associated accumulation operation, or gradients, or any of it.  Some of the phenomenology may seems to kinda-sorta go through, but whether one wants to pin oneself down to narrow terms, is less clear.

But that’s just me.  Other people should not be pressured to do things in ways that I find preferable.

Even internally, I don’t feel a need to be a nazi about this.  I like metaphors as much as the next guy, and I am willing to press technical terms into metaphorical service.  It is just that there they become (as a friend once said about a Texan boss of us-both) like pet rattlesnakes, to be handled accordingly.

Eric

> On Mar 30, 2022, at 5:04 AM, Eric Charles <eric.phillip.charles at gmail.com> wrote:
> 
> That is a bizarre distinction, that can only be maintained within some sort of odd, contextless discussion. If you tell me the number of atoms of a particular substance that you have smushed within a given space, we can, with reasonable accuracy, tell you the density, and hence the "state of matter". When we change the quantity of matter within that space, we can also calculate the expected change in temperature. 
> 
> For example, when there are 25 moles of propane in the tank under my grill, the state of matter is liquid. 
> 
>  <mailto:echarles at american.edu>
> 
> On Fri, Mar 25, 2022 at 8:35 PM Frank Wimberly <wimberly3 at gmail.com <mailto:wimberly3 at gmail.com>> wrote:
> From the web:
> 
> Intensive properties do not depend on the quantity of matter. Examples include density, state of matter, and temperature. Extensive properties do depend on sample size. Examples include volume, mass, and size.Dec 4, 2019
> 
> How is any of those variables emergent.
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz, 
> Santa Fe, NM 87505
> 
> 505 670-9918
> Santa Fe, NM
> 
> On Fri, Mar 25, 2022, 5:26 PM Nicholas Thompson <thompnickson2 at gmail.com <mailto:thompnickson2 at gmail.com>> wrote:
> Hi Jon, I am sitting outside at a Ohori‘s. Chris tells me that there’s some sort of giant fork cook up that’s going to happen around six or maybe seven and then we all should come. Not the kind of thing I could talk Kenny into I don’t think but it might be fun to bring the kid to. They all smell is much more bearable when there’s an actual smell of food mixed up in the afternoon is very mild and pleasant. I wanted to take a few moments to try and get my mind around the intensive extensive distinction I now have a way to remember which is which which is the intensive properties are independent of the size of the system. So at least I have that little bit of grip.  Song the number of altruists in a group is an intensive sorry sorry extensive property of the group it changes with the size of the system according to Siri, the functional organization of the group the degree of functional organization if you will varies nonlinear early with the number of altruist  but it does vary with the number vouchers and so would be also an extensive property? No according to Winsett poop so far as I’m concerned is the only one who makes any sense on the subject an emergent property contrast with an aggregate property so the number of altruists in the group is an aggregate property because it is insensitive to the arrangement of the parts when you get a property which is sensitive to the arrangement of the parts then you get an emergent property. So I’m trying to think of those two distinctions are orthogonal related to One another. I will send this message along now that’s where I am at the moment. I will send a message along now so you get the info about the pork. Since I have no computer I will probably keep dictating these messages to you which you are entirely entitled to ignore but give me a medium in which to Think. Nick
> 
> Sent from my Dumb Phone
> 
> On Mar 24, 2022, at 4:16 PM, Jon Zingale <jonzingale at gmail.com <mailto:jonzingale at gmail.com>> wrote:
> 
> 
> https://en.wikipedia.org/wiki/No-wandering-domain_theorem <https://en.wikipedia.org/wiki/No-wandering-domain_theorem>
> 
> https://www.youtube.com/watch?v=DQkZVPU2txg&ab_channel=TheAbelPrize <https://www.youtube.com/watch?v=DQkZVPU2txg&ab_channel=TheAbelPrize>
> 
> Been thinking about fractals and analytic continuations for recursive algorithms, lately. I would love to read some thoughts.
> 
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