Can pressure be negative?

[Ziggurat]

Just to be certain, I assume that last expression should be dS₂/dE₁, the omitted d being an oversight.

Yes, that was a typo, it should indeed be dS₂/dE₁.

Thanks to your help and Tim Thompson's, I think I'm getting somewhere. Question: How should I interpret the above? Why do we "want" to maximize entropy? Is it because over time it will "seek a maximum?

Yes.

The fundamental postulate of statistical mechanics is that for an isolated system at equilibrium, every accessible microstate is equally likely. That means that for a system composed of two subsystems, every energy distribution between the two subsystems is in principle possible. But some energy distributions have relatively few corresponding microstates, and some have very many corresponding microstates. For a macroscopic system, the number of accessible microstates as a function of energy distribution will be VERY sharply peaked. The numbers involved are just ginormous, but the relative width of the peak is really, really, REALLY small (roughly speaking, the relative width will scale as N^-1/2, where N is the number of particles. For (say) 10₂₀ particles, then, the relative width of this peak is around 10^-10. So even though it's possible to end up far from the peak of your distribution, in practice it just doesn't happen because the probabilities are too small.

Now, in the above discussion I talked about the number of accessible states, not the entropy (which is, aside from a factor of k, equal to the natural log of the number of accessible states). The reason that we use entropy and not the number of accessible states directly is for simplicity. For a system made up of two subsystems, the total number of accessible states is equal to the product of the number of accessible states for each subsystem, but the total entropy is the sum of the entropies of the two subsystems. That makes entropy easier to work with. And if one does the sort of calculus approach to finding the maximum of the number of accessible states as a function of energy distribution, the condition you get can be easily expressed in terms of a derivative of a logarithm of that quantity. So we work with entropy because it's simpler, but we're still just operating off the fundamental postulate, along with the knowledge that our accessible states (and thus our macrostate probabilities) are VERY sharply peaked.

Over time, we can expect to drift around randomly. If we're at or near equilibrium, the probability of staying near equilibrium is really high, because there's so many more microstates there. If we're off equilibrium by any significant amount (and "significant" is, per above, quite small), then there are so many more microstates available if we move towards equilibrium energy distribution than if we move away from it, that even with individual microstate transitions being completely random, the probability that we'll drift towards equilibrium over time is huge. REALLY huge.

 

[Ziggurat]

Actually, I'll redo the calculus starting from the number of accessible states, to illustrate why we end up with a logarith. Let W be the number of accessible microstates of the entire system. We have
W = W₁*W₂
Take the derivative, apply the product rule, and set it equal to zero:
dW/dE₁= W₂*dW₁/dE₁ + W_1*dW₂/dE₁ = 0
W₂*dW₁/dE₁ = -W₁*dW₂/dE₁
Switch variables on the derivative (as before):
W₂*dW₁/dE₁ = W₁*dW₂/dE₂
Divide both sides by W₁*W₂:
(dW₁/dE₁)/W₁ = (dW₂/dE₂)/W₂
Now we note that using product rule, d ln(f(x)) = (df/dx)*(1/f)
Which means that
d (ln W₁)/dE₁ = d (ln W₂)/dE₂
so we find that using the logarithm of the number of accessible states is quite natural. For historical reasons we don't use ln(W) directly, but rather S = k_B*ln(W), but that just changes the units. And thus we recover the same thing that we got starting from entropy directly.

 

[Perpetual Student]

This is all really good and the fog is lifting, thanks to Ziggurat and Tim Thompson.
I was (and still am) very much enamored with mathematics (BA and MS in math -- almost 50 years ago); but I must admit this forum has led me to wish I studied more physics. This is such fascinating stuff!

 

[Perpetual Student]

Is k_B some particular constant which is used universally?

 

[Ziggurat]

Is k_B some particular constant which is used universally?

k_B is Boltzmann's constant. Essentially, k_B just sets your temperature scale. It mostly appears alongside temperature - for example, the Boltzmann factor e^-E/k_BT. One can also write this as e^-Beta*E, so if one works with Beta directly (rather than temperature), one can often avoid using k_B at all.

 

[Tim Thompson]

Fundamental Constants

Is k_B some particular constant which is used universally?

Yes. See the entry for Boltzmann's Constant from the Fundamental Constants page, courtesy of the National Institute of Standards and Technology (NIST). The values of the various physical constants are determined through extensive studies via the Committee on Data for Science and Technology (CODATA), which studies are published in detail every few years. The latest studies were published in 2006, simultaneously in Reviews of Modern Physics and the Journal of Physical Chemistry (both papers linked from the NIST papers page). Figuring out what the correct value should be for the various physical constants is itself quite an involved industry and probably involves the most precise family of laboratory experimentation.

 

[Perpetual Student]

Well, that was a stupid question! Of course: Boltzmann's constant. LINK

 

[Calrid]

Your knowledge and expertise are exceeded only by your condescending attitude and superiority complex.
Carry on.

I quite agree the guys far too used to being sucked up to which has caused him to adopt heirs and graces he doesn't warrant or deserve. He's also very wrong about temperature having actual physically measurable infinities. The spins are not infinite, the kinetic energy is not infinite, nothing in an actual experimental setting is actually infinite, and a quick Google will tell you as much. I supply an excerpt from a work below.

He's just wrong and the sheer number of people who have told me so now is very revealing. Sometimes you have to just say well there is a difference between a maths model and actual experiment. Well you do if you have any integrity. This guys clearly been labouring under a mistaken belief for ages and now is so invested in the mistake that he can't admit his error. It doesn't help he has a cohort of lackies to defend his erroneous claims. I think he imgaines himself superior, and I get the impression this arrogance is totally misplaced.

Quantum physics formally assumes infinitely positive or negative temperatures in descriptions of spin system undergoing population inversion from the ground state to a higher energy state by excitation with electromagnetic radiation. The temperature function in these systems exhibits a singularity, meaning the temperature tends to positive infinity, before discontinuously switching to negative infinity.[3] However, this applies only to specific degrees of freedom in the system, while others would have normal temperature dependency. If equipartitioning were possible, such formalisms ignore the fact that the spin system would be destroyed by the decomposition of ordinary matter before infinite temperature could be reached uniformly in the sample.

C. Kittel, H. Kroemer (1980). Thermal Physics (2 ed.). W. H. Freeman Company. ISBN 0-7167-1088-9.

Basically this means that like there is an absolute cold, there is also an absolute hot; a physical limit on how hot something can be and thus a limit on physical temperature, usually taken to be the planck temperature or the temperature at the big bang, where forces had yet to become discreet.

Anyway I'm not hanging around to hear more of this guys nonsense, despite other experts telling them they are wrong (and yes the guy who wrote that email is educated to Ph.D level and has worked in the field for some 30 years), I presume he'll just continue plowing on with scientific inaccuracies. Not worth my time.

I presume you will now tell me that the authors of this text book don't have the requisite merits to tell you you are just wrong.

I tell you what go to an actual physics forum such as www.physcisforums.com and try and make the claims you have made, then show me the results. Maybe you will learn something, the difference between experimental models and maths perhaps? The moderators there are professionals in the field so you need not worry about their credentials at least.

 

[Calrid]

This is all really good and the fog is lifting, thanks to Ziggurat and Tim Thompson.
I was (and still am) very much enamored with mathematics (BA and MS in math -- almost 50 years ago);...[]

Just reading the backposts. The irony here is great.

Yeah for the most part what they have said is true however the fog descends when they resort to describing maths limits or infinities as experimental reality. The limits of integrals aren't called limits for a laugh they are the limit to which numbers approach, in the same way infinities in physics are as said evidence of a mistake and usually denote that the equation is not 100% accurate but is accurate within its limits (hence renormalisation) although tbh we have no idea if its a pictoral representation anyway and the equations are inductive not deductive. Anyway I would get your information from a reputable physics forum. Some anonymous nobody no matter how highly educated they are (or claim to be). is not going to be all that reliable. This is probably particularly important if you are likely to be examined on this stuff. Because it could cost people passes in degrees. Which is why I find this whole debacle particularly worrying. Sometimes people think calculus = reality, sometimes it is as close as it gets to perfect, and sometimes it has to have real bounds and conditions placed on it usually when referencing it directly to experiment. Like a foot note saying of course the matter would disasociate before such infinite limits were reached and thus in practice the equation is only reliable up to the planck limit of temperature aka absolute hot or temperature.

An important and valuable lesson for me is not to trust anyone's credentials, especially when they spend much of their time attacking other peoples and don't seem to have formed any sensible conclusions on particular issues. That is after all what distinguishes good science from crackpots. Integrity and openness and a willingness to learn. I will learn but I will not blindly accept nonsense from people. Hell I think half the drivel Stephen Hawking comes out with is highly speculative horse **** even though I respect his work on gravitation and black holes, some of what he says is just arm waving. (well not literally but you get what I mean.) He should definitely stick to physics not philosophy though, I'll say that much. He's no Dennett.

Trust no 1.

Btw the link I gave above is a forum of a high standard, if you really want to learn this stuff properly. This forum is not specialised in physics and peoples opinions should probably be taken with a pinch of salt. Including mine. Hell go to a Physics Professor/expert or a reputable physics forum and ask this question yourself just to be sure. I would. Actually I already did on both counts.

Although ironically the book link is a result of just typing infinite temperature into google. Google is of course God, and Wiki is his prophet PBUH.

"It's good that we have met with a paradox, now we may begin to make progress."

"Your theory is crazy, but the question we have now to answer is, is it crazy enough to be true?"

Niels Bohr.

...[] but I must admit this forum has led me to wish I studied more physics. This is such fascinating stuff!

what we don't know is far more interesting than what we do know, there has never been a better time to be alive in physics. Which is ironic as the subject is being studied less and less and physics departments are shutting all of the place in my country. Our university lost ours and the professors were either layed off or merged with the maths department. Which is tragic really. Means I can't study this locally and have to go through the OU, or travel all the way to Southampton to get on a course, which is just too expensive to do anyway £9000; I guess I'll have to go sell my cow if I want to climb that beanstalk? *********** Tories. Bunch of silver spooned, only out for themselves school tie mavericks. I predict a riot or two more in the future, when their dumb assed policies are revealed as the usual, help the rich screw the poor crap they always are. The coalition is pointless.

Equal opportunities my great big fat hairy ass, meritocracy, don't make me laugh. The next Einstein will probably end up working not in a patent office but as a bus driver or bin man or something, because of these tits.

 

[Ziggurat]

The spins are not infinite

I never said they were.

the kinetic energy is not infinite

I never said it was.

nothing in an actual experimental setting is actually infinite

Except the temperature. But then, you still don't know what temperature is.

He's just wrong and the sheer number of people who have told me so now is very revealing.

Argument ad populum is a fallacy.

Sometimes you have to just say well there is a difference between a maths model and actual experiment.

And sometimes, you just need to learn the definition of a term before you engage in a debate about it.

C. Kittel, H. Kroemer (1980). Thermal Physics (2 ed.). W. H. Freeman Company. ISBN 0-7167-1088-9.

So according to them, a system of spins in a magnetic field can exhibit temperatures which diverge to infinity. Which is... exactly what I said. Not terribly surprising, since (along with Reif) it's one of the textbooks I actually learned thermodynamics from.

I presume you will now tell me that the authors of this text book don't have the requisite merits to tell you you are just wrong.

The problem is not the authors. The problem is your understanding of them.

I suggest you consult that textbook and spend some time learning what temperature is, and why they state that "the temperature tends to positive infinity, before discontinuously switching to negative infinity," in agreement with what I said.

The moderators there are professionals in the field so you need not worry about their credentials at least.

I never worry about people's credentials, only their actual ability. But I also know that there are several professional physicists on this board too, including some who have participated in this thread. And yet, for some reason, none of them have come rushing to your rescue.

Now, would you care to (finally) tell us what temperature actually is?

 

[Calrid]

Edited by Professor Yaffle: 

Edited for rule 0 and rule 12

Do what I said go to an actual physics forum, make these claims. You seriously need to learn this subject properly before you ruin other peoples educations with total ******** about infinite temperatures actually being measured in experiment.

I'd love to see you debunk that text book anyway your drivel is laughable now and your just digging an ever deeper hole. It clearly says that in normal matter the material would become disassociate as it approached high temperature and so physically in a medium these results could never happen.

Edited by Professor Yaffle: 

Edited for rule 0 and rule 12

So absolute hot is the highest achievable temperature AKA the Planck temperature, AKA the temperature at the big bang, but you can achieve infinite temperature in an actual experiment and measure it, could you be any more wrong?

Edited by Professor Yaffle: 

Edited for rule 0 and rule 12

.[3] However, this applies only to specific degrees of freedom in the system, while others would have normal temperature dependency. If equipartitioning were possible, such formalisms ignore the fact that the spin system would be destroyed by the decomposition of ordinary matter before infinite temperature could be reached uniformly in the sample.

Edited by Professor Yaffle: 

Edited for rule 0 and rule 12

 

[Hercules Rockefeller]

...

 

[Skwinty]

Try reading sometime Calrid.

http://www.sciencenews.org/view/generic/id/66435/title/Negative_temperature,_infinitely_hot

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/neg_temperature.html

 

[W.D.Clinger]

Just reading the backposts. The irony here is great.

Yes, the irony here is great.

Yeah for the most part what they have said is true however the fog descends when they resort to describing maths limits or infinities as experimental reality. The limits of integrals aren't called limits for a laugh they are the limit to which numbers approach, in the same way infinities in physics are as said evidence of a mistake and usually denote that the equation is not 100% accurate but is accurate within its limits (hence renormalisation) although tbh we have no idea if its a pictoral representation anyway and the equations are inductive not deductive.

A definite integral's limits of integration are just the endpoints of an interval. The fundamental theorem of calculus says

[latex]\[ \int_a^b dF = F(b) - F(a) \][/latex]

You've gotten that wrong before, and were corrected. You ignored that correction. Getting freshman calculus wrong and continuing to get it wrong is not the best way to establish your credibility.

Some anonymous nobody no matter how highly educated they are (or claim to be). is not going to be all that reliable.

An important and valuable lesson for me is not to trust anyone's credentials, especially when they spend much of their time attacking other peoples and don't seem to have formed any sensible conclusions on particular issues.

That is after all what distinguishes good science from crackpots. Integrity and openness and a willingness to learn. I will learn but I will not blindly accept nonsense from people.

Yes, the irony here is great.

Which is ironic as the subject is being studied less and less and physics departments are shutting all of the place in my country. Our university lost ours and the professors were either layed off or merged with the maths department. Which is tragic really. Means I can't study this locally and have to go through the OU, or travel all the way to Southampton to get on a course, which is just too expensive to do anyway £9000; I guess I'll have to go sell my cow if I want to climb that beanstalk? *********** Tories.

I'm sorry to hear of your country's misfortune and its tragic consequences for your university. I feel special sorrow for your cow.

Physics is still being taught at my current university, at the universities I attended as a student, and throughout my country. Calculus also.

 

[sol invictus]

I'd love to see you debunk that text book anyway your drivel is laughable now and your just digging an ever deeper hole. It clearly says that in normal matter the material would become disassociate as it approached high temperature and so physically in a medium these results could never happen.

No Calrid, that's not what it says. That quote - and everything else in that book - is completely consistent with what Zig has been trying to explain to you.

If you would calm down and pay attention, you might actually come to understand something. K&K say "if equipartitioning were possible" infinite temperature would be impossible. But the experiments that achieve infinite and negative spin temperature do so by isolating the spin degrees of freedom, so that (at least during the time the experiment is conducted) they do not equilibrate with any of the colder degrees of freedom around them.

 

[Tim Thompson]

Infinite Temperature

This is all really good and the fog is lifting, thanks to Ziggurat and Tim Thompson.
I was (and still am) very much enamored with mathematics (BA and MS in math -- almost 50 years ago); ...[]

Just reading the backposts. The irony here is great.
Yeah for the most part what they have said is true however the fog descends when they resort to describing maths limits or infinities as experimental reality.

Now we all know that if you stick a mercury thermometer into what ever it is, the mercury column will not streak to infinity, and if we stick a digital thermometer into what ever it is, we won't see a sideways 8. That's what some people are thinking when they insist that the temperature cannot be infinite, and to that extent they are correct. But of course, those instruments don't measure temperature anyway, they measure something else and that is in turn interpreted as a temperature, over some limited and defined set of conditions. Temperature, like anything else in real physics, is whatever the equations define it to be. As you can see from the example above, the temperature becomes infinite, but there is nothing even close to "physically infinite" in the system.

I don't see anyone, and certainly not myself, talking about an infinity as an experimental reality. Indeed, in the passage above I have made it quite explicit that such is not the case. When we talk about temperature being infinite, we are talking about the temperature as it is mathematically defined, not as it is physically measured. Nobody in this thread is suggesting an infinite measurement, that is just something you made up because you don't know what the word "temperature" means. You could clear this all up for yourself by simply looking in that book you cited (Kittel & Kromer, 1980), which I assume you must have, and seeing how they define temperature.

 

[Emet]

You could clear this all up for yourself by simply looking in that book you cited (Kittel & Kromer, 1980), which I assume you must have, and seeing how they define temperature.

Perhaps that will happen in the next month.

 

[Perpetual Student]

Just reading the backposts. The irony here is great.

Yeah for the most part what they have said is true however the fog descends when they resort to describing maths limits or infinities as experimental reality.
...

Calrid:

How carefully did you read the "backposts"? If you followed the discussion carefully, you would have learned why a more intuitive notion of temperature (a measure of average kinetic energy) is not the basis of the definition which has been discussed above. You would have also learned that (quoting Tim Thompson) "if you stick a mercury thermometer into what ever it is, the mercury column will not streak to infinity." The modern definition has a specific utility because "the mathematical description of thermal equilibrium forms a basis for the definition of temperature." It is a universally accepted definition! You cannot argue against a definition! I suppose you could choose to use some different definition for some particular purpose, but that is not what we are discussing here. T as dE/dS is the basis of this discussion. The fact that it becomes infinite in some particular system is merely a consequence of that definition. AGAIN: "If you stick a mercury thermometer into what ever it is, the mercury column will not streak to infinity"
The definitions of modern physics have utility in describing and analyzing physical processes and systems, that's it! They are not necessarily designed to suit our intuition. You and people like Mozina (and others) seem to get stumped by how modern physics defines certain quantities, you refuse to abandon your intuitive notions and consequently you refuse to learn.
I suggest you read this thread again with an open mind and confirm what I have said.
I participate in this forum instead of some "physics forum" (as you have suggested) because I am a layman and I have discovered over time that the knowledgeable participants here are reliable (I have cross referenced ideas presented here many times) and they will take the time to discuss some esoteric (for me) concepts with laymen (like me).

 

[Reality Check]

C. Kittel, H. Kroemer (1980). Thermal Physics (2 ed.). W. H. Freeman Company. ISBN 0-7167-1088-9.

Basically this means that like there is an absolute cold, there is also an absolute hot; a physical limit on how hot something can be and thus a limit on physical temperature, usually taken to be the planck temperature or the temperature at the big bang, where forces had yet to become discreet.

Basically you are wrong: This quote from a 1980 textbook (31 years old) states that the experimental status at the time the book was written was that decoherence would prevent infinite tempertures from being reached unifiormly in an equipartitioned sample.

You have forgotten that science progresses. I am fairly sure that the techniques have progressed in the last 30 years so that systems can be equipartitioned and separated from ordinary matter so that decoherence is not a problem.

Personally I do not think of the experiments as having actually measured infinite temperatures. The actual measured temperatures are the high positive and negative temperatures on either side of the transition that includes infinite temperatures.

 

[Ziggurat]

Edited by Professor Yaffle: 

I'd love to see you debunk that text book anyway

Edited by Professor Yaffle: 

Why should I debunk it when it agrees with what I said?

So absolute hot is the highest achievable temperature AKA the Planck temperature,

That's not what the Planck temperature is. The Planck temperature is (roughly speaking) the temperature at which the blackbody radiation power spectrum peaks at the Planck wavelength. If blackbody radiation were to reach that temperature, the energy densities would be so high that it would test the limits of our current theories. But that doesn't mean that it's impossible to go higher. Furthermore, that's specific to blackbody radiation. But blackbody radiation doesn't support negative temperatures, and it has a heat capacity which increases with temperature. The limit is not temperature, but energy density.

In contrast, systems which support negative temperatures have heat capacities which vanish with increasing temperatures, and finite energy densities even at infinite temperature. So none of the problems associated with blackbody radiation at diverging temperatures exist for such systems.

Of course, you'd know this if you knew what temperature is, and if you knew that temperature and energy are very different quantities. But you don't. You don't know what temperature is. Have you actually read Kittel and Kroemer? Or did you just find that quote from a Google search?