What is the "real" second law of thermodynamics?


by moonman239
Tags: real, thermodynamics
A. Neumaier
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Feb28-11, 08:30 AM
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Quote Quote by Andy Resnick View Post
I'm not sure that's a fair criticism- the quantity 'H' (H = Sum(p log p) + const.) isn't the entropy either, S = kH.
H is the entropy in units where the Boltzmann constant is set to 1. That it has a different value than 1 is due to historical accidents in the definition of the temperature scale, similar to accidents that make the speed of light or the Planck constant not exactly 1.
Fra
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Feb28-11, 09:07 AM
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Quote Quote by lalbatros View Post
It just tells us how we play dice.
Life is a game, what else do you ask for besides a rational strategy? ;)

You can see a THEORY as a descriptive model, or, as an INTERACTION TOOL in an inference perspective.

A descriptive model is falsified or corroborated. Corroborated theories lives on. Falsified theories drop dead, leaving no clue as to how to improve.

An interaction tool for inference is different, it either adapts and learns, or it doesn't. Here the falsification corresponds to "failure to learn". The hosting inference system will be outcompeted by more clever competitor. This may be one reason to suspect that the laws of physics, we actually find in nature, does have inferencial status.

It's much more than playing dice.

/Fredrik
Fra
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Feb28-11, 09:15 AM
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To think that we can DESCRIBE the future, is IMHO a very irrational illusion.

All we have, are expectations of the future, based on the present (including present records of the past), and based upon this we have to throw our dice. There is no other way.

In this respect, the second law is one of the few "laws" that are cast in a proper inferencial form as is.

As anyone seriously suggest you say; understand newtons law of gravity, but do not understand the second law? If one of them is mysterious I can't see how it's the second law.

/Fredrik
A. Neumaier
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Feb28-11, 10:04 AM
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Quote Quote by Fra View Post
To think that we can DESCRIBE the future, is IMHO a very irrational illusion.
Both future and past are illusions - real is only the present (if anything at all).

Nevertheless, it is often easier to predict the future than the past. The past of a stirred fluid coming slowly to rest is far more unpredictable than the final rest state.
lalbatros
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Feb28-11, 11:36 AM
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Quote Quote by Andy Resnick View Post
I'm not sure that's a fair criticism- the quantity 'H' (H = Sum(p log p) + const.) isn't the entropy either, S = kH.
I simply don't see what could be obtained from a dimensional argument.
The thermodynamic dimension of entropy is purely conventional.

The factor is there as a connection between a measure of disorder and a measure of energy.
Nevertheless, disorder can be defined without any relation to energy.
The historical path to entropy doesn't imply that entropy requires any concept of thermodynamics.

The widespread use of entropy today has clearly shown that it is not a thermodynamic concept. We know also that entropy finds a wide range of application in thermodynamics.
It should be no surprise that the use of entropy in thermodynamics requires a conversion factor. This factor converts a measure of disorder to the width of a Maxwellian distribution.
Andy Resnick
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Feb28-11, 12:03 PM
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Quote Quote by lalbatros View Post
Nevertheless, disorder can be defined without any relation to energy.
That is true- but then what is the temperature of a sandpile?

http://rmp.aps.org/abstract/RMP/v64/i1/p321_1
http://pra.aps.org/abstract/PRA/v42/i4/p2467_1
http://pra.aps.org/abstract/PRA/v38/i1/p364_1

it's not a simple question to answer.
Andy Resnick
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Feb28-11, 01:42 PM
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Quote Quote by A. Neumaier View Post
H is the entropy in units where the Boltzmann constant is set to 1. That it has a different value than 1 is due to historical accidents in the definition of the temperature scale, similar to accidents that make the speed of light or the Planck constant not exactly 1.
Not exactly- you may set the numerical value of k to 1, but there are still units. Temperature can be made equivalent to energy. One is not primary over the other, just as length and time are equivalent.

Even so, that only holds for equilibrium: thermostatics or thermokinetics. It does not hold in the fully nonequilibrium case. Jou, Casas-Vazquez, and Lebon's "Externded Irreversible Thermodynamics" and Truesdell's "Rational Thermodynamics" both have godd discussions about this.
lalbatros
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Feb28-11, 02:32 PM
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Quote Quote by Andy Resnick View Post
That is true- but then what is the temperature of a sandpile?

http://rmp.aps.org/abstract/RMP/v64/i1/p321_1
http://pra.aps.org/abstract/PRA/v42/i4/p2467_1
http://pra.aps.org/abstract/PRA/v38/i1/p364_1

it's not a simple question to answer.
What is the temperature of a double-peaked velocity distribution?
There need not be an answer.
lalbatros
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Feb28-11, 03:09 PM
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Quote Quote by Andy Resnick View Post
That is true- but then what is the temperature of a sandpile?

http://rmp.aps.org/abstract/RMP/v64/i1/p321_1
http://pra.aps.org/abstract/PRA/v42/i4/p2467_1
http://pra.aps.org/abstract/PRA/v38/i1/p364_1

it's not a simple question to answer.
I have at least two reasons to enjoy this type of sandpile physics:
  1. I work for the cement industry, and there are a many granular materials there:
    limestone, chalk (made of Coccoliths!), sand, clay, slag, fly ashes, ground coal, rubber chips, plastic pellets, ...
  2. I enjoyed reading the book by Pierre-Gilles de Gennes on Sands, Powders, and Grains
It is true that the (excited) avalanche phenomena near the critical repose angle has an analogy with a barrier crossing phenomena that can be associated to an equivalent temperature (fig 85.c in Pierre-Gilles de Gennes). I guess this temperature might indeed represent the probability distribution of the grains energy acquired by the external excitation.

Obviously, this is again an example taken from (statistical) mechanics.
Therefore, the entropy that one might consider here is again related to the distribution of energy.
And therefore this one more energy-related entropy.

If we consider that any information, in the end, needs a physical substrate to be stored, then effectively the whole world is mechanical and , in the end, any entropy could be related to an energy distribution.
As long as there are no degenerate states, of course ...
So the question about entropy and energy could be translated in:

How much information is stored in degenerate states compared to how much is stored on energy levels? (in our universe)

My guess goes for no degeneracy.
Meaning that history of physics was right on the point since Boltzman: it would make sense to give energy dimensions to entropy!
Andy Resnick
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Feb28-11, 08:19 PM
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Quote Quote by lalbatros View Post
I have at least two reasons to enjoy this type of sandpile physics:
Glad to hear it- I find it interesting as well. I was involved with a few soft-matter groups when I worked at NASA.


Quote Quote by lalbatros View Post
Obviously, this is again an example taken from (statistical) mechanics.
Therefore, the entropy that one might consider here is again related to the distribution of energy.
And therefore this one more energy-related entropy.
I think you still misunderstand me- what you say above is of course correct, but I think you missed my main point, which is that temperature and entropy should not be simply equated with energy. Entropy is energy/degree, and so there is an essential role for temperature in the entropy.

How about this example- laser light. Even though laser light has an exceedingly well-defined energy, it has *no* temperature:

http://arxiv.org/abs/cond-mat/0209043

They specifically address the difficulty in assigning a temperature and an entropy to an out-of-equilibrium system:

"Out of equilibrium, the entropy S lacks a clear functional dependence on the total energy
E, and the definition of T becomes ambiguous."

Again, laser light is a highly coherent state, is resistant to thermalization in spite of interacting with the environment, has a well defined energy and momentum, and yet has no clear entropy or temperature.
lalbatros
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Mar1-11, 03:40 AM
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Quote Quote by Andy Resnick View Post
They specifically address the difficulty in assigning a temperature and an entropy to an out-of-equilibrium system:

"Out of equilibrium, the entropy S lacks a clear functional dependence on the total energy
E, and the definition of T becomes ambiguous."
I think there should be no problem to define the entropy, even though the temperature might be totally undefined.

It is clear that entropy is not a function of energy in general.
Just consider the supperposition of two bell-shape distribution.
What is the "temperature" of this distribution?
Even when the two distributions are Maxwellians, you would still be forced to describe the global distribution by three numbers: two temperatures and the % of each distribution in the total.
This is a very common situation.
Very often there are several populations that do not thermalize even when reaching a steady state (open system).
For example the electron and ion temperatures are generally very different in a tokamak.
Even different ion species might have different distributions in a tokamak, specially heavy ions with respect to light ions.
There might even be two populations of electrons, not to mention even runaway electrons in extreme situations.
In quite clear that in all these non equilibrium situations, the entropy is perfectly defined as well as the energy, but the entropy is not a function of energy anymore. Therefore, temperature cannot be defined.

I will read the paper later.
However, the introduction suggests that temperature could be sometimes defined in non-equilibrium situations.
I agree with that with the temporary naive idea that this will be the case when at least approximately S=S(E) .
One can easily built articial examples.
For example, on could constrain a distribution to be Lorentzian instead of Maxwellian, or any suitable one-parameter distribution. Within this constraint S would be a function of E via the one parameter defining this distribution. Temperature should be defined in this situation.
I am curious to see a more physical example in the paper.
I am also curious to think about which "thermodynamic relations" would still hold and which should be removed, if any.

Thanks for the reference,

Michel
A. Neumaier
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Mar1-11, 07:23 AM
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Quote Quote by Andy Resnick View Post
Not exactly- you may set the numerical value of k to 1, but there are still units. Temperature can be made equivalent to energy. One is not primary over the other, just as length and time are equivalent.
The units are arbitrary, since the Kelvin is an independent unit defined only by an experimental procedure. If you set k=1, temperature will have the units of inverse energy, and entropy is unitless.
Quote Quote by Andy Resnick View Post
Even so, that only holds for equilibrium: thermostatics or thermokinetics. It does not hold in the fully nonequilibrium case. Jou, Casas-Vazquez, and Lebon's "Externded Irreversible Thermodynamics" and Truesdell's "Rational Thermodynamics" both have good discussions about this.
The level of nonequilibrium thermodynamics is characterized by local equilibrium (in position space). On this level, dynamics is governed by hydrodynamics, variants of the Navier-Stokes equations. Here temperature exists, being defined as the equilibiruim temperature of an (in the limit infinitesimal) cell. Or, formally, as the inverse of the quantity conjugate to energy.

A more detailed level is the kinetic level, characterized by microlocal equilibrium (in phase space). On this level, dynamics is governed by kinetic equations, variants of the Boltzmann equation. Entropy still exists, being defined even on the more detailed quantum level. Temperature does not exist on this level, but appears as an implicit parameter field in the hydrodynamic limit: The kinetic dynamics is approximated in a local equilibrium setting, by assuming that the local momentum distribution is Maxwellian. The temperature is a parameter in the Gaussian local momentum distribution, and makes no sense outside the Gaussian approximation.
Andy Resnick
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Mar1-11, 09:10 AM
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Quote Quote by lalbatros View Post
It is clear that entropy is not a function of energy in general.

<snip>

In quite clear that in all these non equilibrium situations, the entropy is perfectly defined as well as the energy, but the entropy is not a function of energy anymore. Therefore, temperature cannot be defined.

I will read the paper later.
However, the introduction suggests that temperature could be sometimes defined in non-equilibrium situations.

<snip>

Thanks for the reference,

Michel
My pleasure.

It's possible to recover cleanly defined thermodynamic properties in a nonequilibrium system in certain restricted cases: when the density matrix is block diagonal (if that's the correct term), for example. Conceptually, this is similar to coarse-graining or embedding a dissipative system in a higher-dimensional conservative system.

This only works for linear approximations- the memory of a system is very short (the relaxation time is short), or the Onsager reciprocal relations can be used.

As a more abstract example; we (our bodies) exist in a highly nonequilibrium state: the intracellular concentration of ATP is 10^10 times higher than equilibrium (Nicholls and Ferguson, "Bioenergetics"), which means the system can't be linearized and the above approximation scheme fails. How to assign a temperature? Clearly, there does not have to be a relationship between the "temperature" defined in terms of the distribution function of ATP and 98.6 F.
Andy Resnick
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Mar1-11, 09:13 AM
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Quote Quote by A. Neumaier View Post
The units are arbitrary, since the Kelvin is an independent unit defined only by an experimental procedure. If you set k=1, temperature will have the units of inverse energy, and entropy is unitless.
That's one of the differences between Mathematics and Science. Lots of equations can be nondimensionalized- for example the Navier-Stokes equation- but the scale factors must be retained in order to reproduce experimental results.
A. Neumaier
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Quote Quote by Andy Resnick View Post
That's one of the differences between Mathematics and Science. Lots of equations can be nondimensionalized- for example the Navier-Stokes equation- but the scale factors must be retained in order to reproduce experimental results.
Sure, but this doesn't change anything of interest.

By the way, not mathematicians but scientists called physicists take c=1 and hbar=1 when they discuss quantum field theory. And they actually express temperature (and distance) in terms of inverse energy, not in Meter or Kelvin.

Translating to more traditional units is a triviality that can be done (and is done where needed) at the very end.
Andy Resnick
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Mar1-11, 01:09 PM
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Quote Quote by A. Neumaier View Post
Sure, but this doesn't change anything of interest.

By the way, not mathematicians but scientists called physicists take c=1 and hbar=1 when they discuss quantum field theory. And they actually express temperature (and distance) in terms of inverse energy, not in Meter or Kelvin.

Translating to more traditional units is a triviality that can be done (and is done where needed) at the very end.
This scientist (who is occasionally called a Physicist) is familiar with the system of 'natural units'. No scientist I work with (Physicist or otherwise) would ever confuse a mathematical model with the actual physical system. My students often do, as evidenced by comments like "this data doesn't fit the model, so the data is bad".

Models can be simplified to better suit our limited understanding, at the cost of decreased fidelity to the phenomenon under investigation.
Pythagorean
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Mar1-11, 03:09 PM
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My advisor (a physicist) always suggests for me to use natural units since it's just the general behavior were studying (it's a more theoretical treatment)

I'm not emotionally comfortable with that, but it makes sense for a journal like Chaos. It's an exploratory science, not experimental, more theoretical = more mathematical.

I see a spectrum, not a yes-no situation, but then I'm a dynamicist. My work often involves turning the step function into a hyperbolic tangent.
A. Neumaier
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Mar1-11, 03:17 PM
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Quote Quote by Pythagorean View Post
My advisor (a physicist) always suggests for me to use natural units [...] My work often involves turning the step function into a hyperbolic tangent.
and I guess if you'd use instead an arc tangent you'd use for the resulting angles natural units, too, and not degrees.

Angles in degrees and temperature in degrees are both historical accidents.
An extraterrestrial civilization will not have the same units - but their natural units will be the same.


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