Roger’s sources at the end here are great, and they are ones I hadn’t read.
In principle, Legendre duality (the exchange between function arguments and
derivatives with respect them) is a main theme in all the thermodynamics
textbooks. I am so far out of date that I have no idea what I seeing used to
teach these days, but when I was an undergrad, it was old war horses like
Kitten and Kroemer, which are quite serviceable, though slow as one becomes
older and more impatient:
https://www.amazon.co.jp/Thermal-Physics-Charles-Kittel/dp/0716710889
<https://www.amazon.co.jp/Thermal-Physics-Charles-Kittel/dp/0716710889>
I say “in pcinriple” because there is one thing in the standard teaching that I think is a less-than-ideal choice, and it is repeated nearly universally. That choice is to introduce thermodynamic potentials as outgrowths of the Newtonian potential energy, and then work with everything in energy units. So usually the gradients are not explicitly shown to be of the entropy with respect to its arguments, but rather “of the internal energy”, with respect to other variables that aren’t actually its arguments, since (from the perspective that emphasizes states and ensembles) it isn’t a function but rather just a boundary condition. That approach introduces entropy as aa kind of secret sauce that gets added to conservation relations, for incomprehensible reasons, makes the rules for when to set the outcome of an irreversible transformation to the same values as the outcome of a reversible transformation obscure and confusing, and in general gives the whole field the aura of some
kind of black art.
So — and sorry to do this — as the only remedy I knew at the time, I tried to
write things in a less tangled order in the book with Harold:
https://www.amazon.co.jp/Origin-Nature-Life-Earth-Emergence/dp/1107121884
<https://www.amazon.co.jp/Origin-Nature-Life-Earth-Emergence/dp/1107121884>
The only version available to me right now is an old electronic proof copy, but
I am sure the sections are right and the pagination may even be correct.
The general section where I try to start at some kind of a sensible beginning
and work in a linear order is Sec.7.3, maybe on p.436. It’s all
large-deviations framing and some examples for several pages, but interacting
systems and the origin of intensive variables is introduced in Sec.7.3.2,
p.439. The general relation, introduced in its own name, without digressions
and examples, is Eq.(7.6) on p.441.
Again, speaking of too long for the old and impatient…
But, to try to get to the point of introducing all the standard content, in a
way that (for me, at least) is more sensible and scannable, I try to put all
this into a self-contained pane in Box 7.1 starting on p.443, and only running
a couple of pages. That box is written to be free-standing, and includes
everything I made reference to in my earlier post here. The prepending text
would only be needed if one wanted to know what various letter of the alphabet
(S, U, V, N) actually _stand for_ in operational terms. If one takes the
letters as given and familiar, and just wants to see the relations, the box is
sufficient.
To use the energy framework to make contact with the measurable phenomenology
of thermodynamics, starting from a premise that mechanics (or, more
specifically, thermometry and calorimetry, hearkening back to brewer Joule)
will be more widely known, isn’t necessarily bad, and one could do it very
compactly as Fermi does:
https://www.amazon.co.jp/Thermodynamics-Dover-Books-Physics-Enrico/dp/048660361X
<https://www.amazon.co.jp/Thermodynamics-Dover-Books-Physics-Enrico/dp/048660361X>
I view Kittel and Kroemer as fulfilling a similar role, and it deserved to
become a gold-standard text, for its very good design in that role.
After one has realized, however, that the old conversation about energy was not
the semantics for these questions, and that the conversation has shifted to one
about stability, you would think it must then seem natural to stop clinging to
antiquity, and just reformulate the presentation in its natural language. Why
physics teaching is so slow to do this, I do not understand. They say they do,
when they shift away from “thermodynamics as a phenomenological discipline” to
“statistical mechanics as the basis for thermodynamics”, as in the usual shift
from undergrad to early grad texts like Huang:
https://www.amazon.co.jp/Statistical-Mechanics-2E-Kerson-Huang/dp/0471815187
<https://www.amazon.co.jp/Statistical-Mechanics-2E-Kerson-Huang/dp/0471815187>
Yet they retain the energy framework and the energy-denominated intensive state
variables, so the obscurity remains. As if it were believed that students
would never actually become capable of shifting frames of mind and simply
seeing phenomena in new terms. I fear this is a case of professors projecting
their own inflexibility onto their students, who could do much better if given
appropriate supports.
Eric
On Mar 31, 2022, at 3:24 AM, Roger Critchlow <[email protected]
<mailto:[email protected]>> wrote:
I think the trap of essentialism is that there exist contexts where it works,
and works magnificently, but in most contexts it's nothing but pet
rattlesnakes, often being waved around with no caution at all. And I suppose
the back side of the trap is that we have an innate essentialist heuristic
which we use for organizing essentially everything we encounter in the world.
So in certain contexts -- mechanics, chemistry, thermodynamics, electronics,
computation -- we have refined our naive essentialism into categories and
operations which essentially solve or are in the process of solving the
context. And in other contexts, we have lots of enthusiastic application of
naive essentialist theories, lots of ritualistic imitations of the procedures
employed in the contexts which are succeeding, and lots of proposals of ways
that the unresolved contexts might be reduced to instances of the solved.
EricS's dimensional analysis in a nutshell, which is an essential description
of a successful essential analysis of a context, leaves a lot of problems for
the reader to work out if taken as a recipe for action. How do you identify
the units of aggregation? What are the rules for forming larger aggregates
from smaller and vice versa? What is entropy, anyway, and what is the correct
entropy (*dynamic potential) in this context?
Thermodynamic state functions as derivatives with respect to entropy are all
over JW Gibb's On the Equilibrium of Heterogeneous Substances. It is the
point. PW Bridgman's Dimensional Analysis essentially summarizes all of
physics up to 1922 as a problem of combining and factoring units of
measurement, one of my favorite library discoveries as an undergraduate. Both
available in the internet archive.
-- rec --
On Wed, Mar 30, 2022 at 12:12 PM Marcus Daniels <[email protected]
<mailto:[email protected]>> wrote:
Here is a situation I frequently experience with software development where
I try to adopt some code, even my own. I stare at the code and..
1) It becomes clear how to assemble it into to what I want
2) I become confused or frustrated. As a ritual, I remove it from my
sight and open a blank editor window to start over. Sometimes I must walk away
from the screen to think, until I want to type.
I think the reason I dwell in #2 space is because I believe in #1. That
is, when I have just the right combinator library things just snap into place.
I seem to spend a lot of time trying to convince myself of why it can't work,
and whether it is a bad fit or something that needs to be fixed in the
platform. What is important, in this value system, is that platforms are good,
not that this or that problem gets solved. I think it is basically the
Computer Science value system in contrast to the Computational Science value
system.
To [re]abstract and [re]concretize can be expensive and those who don't do
it have a productivity advantage, as well as the benefit of having particulars
to work from. I don’t think it is a case of confusing the sign for the
object. It is a question of what kind of problem one wants to solve.
In contrast, I have met several very good computational people that hate
abstraction and indirection. They want code to be greppable even if it that
means it is baroque and good for nothing else.
-----Original Message-----
From: Friam <[email protected] <mailto:[email protected]>>
On Behalf Of glen
Sent: Wednesday, March 30, 2022 8:40 AM
To: [email protected] <mailto:[email protected]>
Subject: Re: [FRIAM] To repeat is rational, but to wander is transcendent
Of all the words being bandied about (quality, property, composition, domain, continuity, intensity, general, special,
iteration, etc.) EricC's "contextless" stands out and reflects EricS' initial target of dimension analysis. The
conversation seems to be about essentialism. Maybe that's a nice reflection that we're sticking to the OG topic "analytic
idealism". But maybe it's Yet-Another example of our pareidolia to see patterns in noise and then to *reify* those patterns.
[Re]Abstracting and [re]concretizing heuristics across contexts may well be what separates us from other life forms. But
attributions of the "unreasonable effectiveness" of any body of heuristics is the most dangerous form of reification.
The superhero ability to [re]abstract and [re]concretize your pet heuristics convinces you they are "properties" or
"qualities" of the world, rather than of your anatomy and physiology. Arguing with myself, perhaps Dave's accusation is
right. Maybe this is an example
of swapping the sign for the object, or reworded prioritizing for the
description over the referent, confusing the structure of the observer with the
structure of the observed.
Those of us with less ability tend to attribute (whatever haphazard
heuristics they've landed on) to the world *early*. Those of us with more
ability continue the hunt for Truth, delaying attribution to the world until we
get too old to play that infinite game any more.
I think Possible Worlds helps, here, too:
https://plato.stanford.edu/entries/possible-worlds/
<https://plato.stanford.edu/entries/possible-worlds/> Patterns are simply
(non-degenerate) quantifiers over possible worlds.
Regardless, I'd like to ask whether the formulation of intensive properties
as derivatives of entropy w.r.t. extensive properties is formalized somewhere?
If so, I'd be grateful for pointers. I'm used to the idea that the intensives
divide out the extensives. But I haven't seen them formulated as higher order
derivations from entropy.
Thanks.
-glen
On 3/29/22 14:37, David Eric Smith wrote:
> [snip]
> 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.
>
> [snip]
>
>> On Mar 30, 2022, at 5:04 AM, Eric Charles <[email protected]
<mailto:[email protected]> <mailto:[email protected]
<mailto:[email protected]>>> 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.
>>
