What is the "real" second law of thermodynamics?

lalbatros

Dear All,

I feel much more confortable to consider that the second law is an engineering heuristics.
And I even don't feel totally confident with this view!
Sorry for that!

To go back again to the Clausius formulation, I believe the full construct of thermodynamics can be derived from this statement. It can lead to the existence of an entropy state function and a thermodynamic temperature scale. But this is all a direct consequence from defining the notion of "hot" and "cold", a consequence of defining hot and cold as the direction for heat. Heat was already known before the second principle.
Where is there real physics in the Clausius statement?
Where is there real information on our physical world?
This Clausius statement is more like instructions for engineers to contruct thermodynamic tables from experiment in an organized way, and first of all it instructed engineers to built a consistent thermometer!
What could you do with the Clausius principle if you had no thermodynamic tables?
You reap the result of the Clausius statement only when a huge amount of experiments have been tabulated (recorded) in a rational way.
That's a huge achievement, but I don't really see any physical law in the Clausius statement.

For me the real physics comes with the statistical thermodynamics, with Boltzmann and Gibbs.
Thermodynamic tables can be calculated ab-initio thanks to their work.
And their work acknowledges clearly the existence of fluctuations.

Michel

Andy Resnick

Spoken like a true mathematician! :)

Andrew Mason

Perhaps you could explain why, in the Boltzmann equation for entropy, the Boltzmann constant has units of Joules/Kelvins. Or why not just rewrite second law of thermodynamics without reference to temperature?

AM

atyy

I think he's referring to the case of ground state degeneracy.

lalbatros

The Boltzmann constant doen't appear in the H-function or in the H-theorem, and it even doesn't appear in the Boltzmann equation.

The Boltzmann equation can, within its range of validity, predict the evolution of any distibution. The distribution doesn't need to be a Maxwellian and therefore, the temperature simply doesn't play any role in the Boltzmann equation, the H-function and the H-theorem. The same applies for other or all master equations in statistical mechanics, except if they are specialized to near-equilibrium solutions. The Maxwellian distribution comes into play as a special stationary solution, and for this special solution the temperature can be taken into consideration. The Boltzmann constant is introduced in the Boltzmann S-function for mere convenience. By doing so the results for (near) equilibrium distributions can be compared with thermodynamics.

This illustrates my point that the second law is more like an "engineering heuristics".
In addition, it also shows that the "historical formulations of the second law" are only a very special case of a much broader "second law". The H-theorem applies to the thermalisation of particle distributions, which is not in the scope of the "second law" of thermodynamics as formulated by Clausius or Kelvin or Carathéodory.

Concerning the second law (of thermodynamics), it is the basis to construct the thermodynamic scale of temperature.
Having defined this scale, the recipe to built entropy tables is the famous law dS=dQ/T, where the temperature factor testifies of an assumption of equilibrium. Note that two of the three classical formulations of the second law make no explicit reference to temperature.

In summary, there are many overlapping definitions of entropy!

Andy Resnick

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.

A. Neumaier

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

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

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

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

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

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

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

What is the temperature of a double-peaked velocity distribution?
There need not be an answer.

lalbatros

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

Glad to hear it- I find it interesting as well. I was involved with a few soft-matter groups when I worked at NASA.


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

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