# Temperature, Colder than Absolute Zero



## PeterB (Jan 10, 2013)

This is interesting.
Link


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## tups (Jan 10, 2013)

The theory underpinning this has been known for decades, but to achieve it in practice is forbidden by the laws of thermodynamics - and given the amount of effort/energy needed to create a minute amount of gas with negative temperature int he first place, that is not bound to change anytime soon


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## Herbiev (Jan 10, 2013)

Another wacky theory. It is a simple fact that you can not get an engine to run at greater than 100%. Imagine hooking up hundreds of these engines in series then one could power Las Vegas with a 9 volt battery. Anyone got the plans Rof}Rof}


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## OrangeAlpine (Jan 13, 2013)

Yes, interesting.  I do not understand it, but that does not mean it is not real.

Bill


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## Entropy455 (Jan 13, 2013)

To cool something down to zero degrees kelvin, you need a heat-sink thats colder than zero degrees kelvin  I.e. to flow the heat out and down to zero degrees. Heres problem one:  at zero degrees kelvin, everything stops moving. That means that electron clouds collapse, and atoms shrink into extremely dense clumps of strange-matter (or so it's theorized). Because of the lack of electrons, zero-degree material would be incapable of interacting with other real matter.

Heres problem two: at zero degrees kelvin all motion stops (again, matter collapses down into its highest possible level of order). So how exactly do you cool it further? Answer, you cant. Not even theoretically. This article is hog-wash. Just like cold fusion a few years back. . . . .

In order to reduce the entropy of a system, you must disproportionately increase the entropy within an adjacent system  as it is written within the laws of thermodynamics.


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## OrangeAlpine (Jan 13, 2013)

Supposedly published in Science, the preeminent peer reviewed scientific publication.  Does not mean it is correct, but I certainly would not reject it simply because it flys in the face of science as I understand it.  Be skeptic?  Sure.  It is like the Theory of Relativity and Quantum Mechanics when they were first published.  And my favorite (because I watched the controversy) Continental Drift.

For the present time, it will be filed in my mind under the category of  "Interesting things that may be true and if it is, totally upsets my understanding of science".

Bill


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## Tonnetto (Jan 13, 2013)

What's your theory Mr. Spock. It's life Jim, but not as we know it.


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## Entropy455 (Jan 13, 2013)

Every perpetual motion machine design relies on one critical design aspect &#8211; which is that the machine must decrease entropy.

Or said another way - the reason perpetual motion machines don&#8217;t exist, is because you cannot extract power from an energy reservoir without increasing entropy.

Articles like this pop up pretty regularly. Normally they fizzle pretty quickly too. However sometimes the article gains momentum resulting in upset investors, and some scientists with a lot of egg on their faces. (I&#8217;m still not sure how they intended to &#8220;boil&#8221; water with a &#8220;cold&#8221; fusion reactor. . .)

Combustion engines, solar cells, atomic weapons, nuclear reactors, particle accelerators, etc. &#8211; aka real machines - do not violate the laws of thermodynamics. But perhaps if we tweak some gas atoms with a laser, under a high vacuum. . . . . Yeah, right. My recommendation &#8211; invest your money elsewhere.

I&#8217;m not sure how things run within a physics lab, but in the engineering community, the worst possible thing that can happen to an engineer is to have his name associated with perpetual motion &#8211; bar none. An engineer that proposes or supports a perpetual motion idea will be discredited, tossed out of the engineering community, and humiliated for life by his peers.

Anyone who claims to be able to destroy entropy by creating sub-zero-kelvin matter, is proposing perpetual motion - and has clearly watched one too many episodes of Star Trek. . . .


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## OrangeAlpine (Jan 14, 2013)

As I read the article, I do not see a promise of perpetual motion.  Mostly, I see a new paradigm, where the rules are changed, making such things as perpetual motion at least theoretically possible.  

No need to go into the impossibility of PM, I get it.  I also understand that as science progresses, we sometimes have to alter our understanding of "truth" or "reality".  This may (most likely not) be one of those occasions.  At this time, I do not know enough to make that judgment and I also doubt that anyone here does.   My initial reaction is that the phenomenon may exist, but the control and energy requirements are such that it will have no practical application.  At least not in my (now somewhat limited) lifetime.

Bill


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## OrangeAlpine (Jan 14, 2013)

A second musing.  Lets say that it is possible to create PM at sub zero temperatures, but the energy requirements to achieve those conditions are greater than any energy "created", does that invalidate the claim?

Bill


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## Entropy455 (Jan 14, 2013)

Perhaps 14 billion years ago, an alien civilization was experimenting with sub-zero matter. Long and behold, they actually created a small batch of matter that was 57 degrees below zero Kelvin. And when they inserted a thermometer to verify their results, the infinite release of energy resulted in a big bang. . . . Possible? Sure &#8211; but likely more on the religious side of theoretical studies, and not that of science. . . .

From the article: &#8220;When temperatures go below zero or above infinity. . .&#8221; This statement pretty much sums up the validity of the article. Infinity is without bound. By definition, you cannot exceed something that is without bound &#8211; it&#8217;s a &#8220;perpetual&#8221; limit.

Temperature is the measure of thermodynamic disorder within a system &#8211; basically the rate in which things are moving around. The engineering equation for temperature (derived from the laws of thermodynamics) is as follows: temperature is equal to the partial derivative of Energy, divided by the partial derivative of Entropy. Within engineering text, Entropy is often described as the quality of work extraction from a thermodynamic reservoir - which is more valid than stating it's the thermodynamic disorder of a system.

What does the temperature equation really mean? It means that when you hit zero degrees Kelvin, things STOP moving, because dS/dE = 0 degrees Kelvin. Question - how can you move something at less than zero movement? Answer &#8211; you can&#8217;t - for the same reason you cannot heat something to infinity plus one degrees Kelvin. The article is hog-wash. . . .


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## tups (Jan 14, 2013)

Orangealpine's last post is exactly correct.

And Entropy, if you wish to speak with authority on a subject, then make sure you master it before you do, especially when calling it "hogwash".

Absolute thermodynamic temperature is defined as follows through the Boltzmann distribution:

N/N° = e^(-deltaE/kT)

This signifies that the number of molecules at a higher energy level +deltaE , N, with respect to those at a lower energy level, N°, uniquely defines a property "T", temperature, with the aid of a constant k, Boltzmann's constant, of which the value is well-known. Through statistical thermodynamics also entropy follows from this proportion.

At low temperature, the proportion of high-energy particles will be very low. As temperatures get higher and higher, the ratio will go up, until, at infinite temperature, the proportion will be exactly 1, and there will be equal populations at all energy levels. Entropy, at the same time, will rise.

Now imagine what happens with the mathematics if you succeed in creating a system where the higher energy level has a higher population than the lower energy level (that's your "infinity plus one degree").

Exactly - the expression at the right hand side will go positive, implying that the temperature has now gone formally negative. Entropy, in this "virtual" temperature range, behaves exactly the same as it does in the real temperature range - if the temperature rises to approach zero (which is not the same zero as in the positive temperature range, as here the population between low and high energy levels would be completely inverted) the entropy lowers.

You can think through the consequences yourself - how will this system behave when left to its own devices ? The high energy particles will lose energy, and the system will formally "cool down" quickly and its entropy will rise ... from negative temperatures, to very negative temperatures, and finally it will reach an asymptote at negative infinity, at which point it will switch to positive infinity and cool down further from there, with the entropy now lowering.

So from the mathematics, negative temperatures have nothing at all to do with "zero movement" and "collapsed electrons" - they are states with an internal energy higher than infinity.

Impossible, you say ? Well, apparently these dudes have succeeded in creating some. Which is why they are published in Science in the first place.
The impossible is only defined through the limits of one's imagination.


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## Propforward (Jan 14, 2013)

If we freeze ice down to absolute zero, will it make my cocktail taste better?


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## Omnimill (Jan 14, 2013)

From another site:_ 

"What's that you say - perpetual motion is impossible?   My, you're a difficult one to please.   The electrons in the molecules of rock formations have been spinning steadily for millions of years without stopping - at what point will you agree that they are in perpetual motion?"_

I don't know much. But what I do know is that just because you don't know how something works doesn't mean it doesn't exist!


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## Entropy455 (Jan 14, 2013)

I stand by my words. Its hogwash.

http://en.wikipedia.org/wiki/Boltzmann_distribution

Negative temperature (Inverted Boltzmann Distribution) is mentioned on the above link. Heres the quote I like best: In January 2013, German scientists reported having achieved an "inverted Boltzmann distribution" with the ultracooling of atomic gas, creating negative absolute temperature. The experiments may shed light on the nature of dark energy, and indicate that a 100 percent energy-efficient internal combustion engine, which had previously been considered impossible, might in fact be achievable.

Yes, you heard it here first  perpetual motion combustion engines might in fact be achievable! Now if only I could get my hands on some of that antimatter gasoline  the fuel that produces negative heat when burned. . . . . .Rof}


http://en.wikipedia.org/wiki/Negative_absolute_temperature

The above link is a reality check:

Key quote: The possibility of decreasing in entropy with increasing energy requires the system to "saturate" in entropy, with the number of high energy states being small. These kinds of systems, bounded by a maximum amount of energy, are generally forbidden classically. Thus, negative temperature is a strictly quantum phenomenon. Perhaps the Germans were looking to construct a quantum internal combustion engine?


The half-life of a proton is 6.6×10^33 years via antimuon decay and 8.2×10^33 years via positron decay. No proton means no electron orbit. Electrons themselves will decay at 4.6×10^26 years. I.E. even atomic particles are not perpetual.

Perpetual devices do not exist. . . .


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## Entropy455 (Jan 14, 2013)

Above is another valid temperature equation:

Where: E is kinetic energy in Joules
kB is the Boltzmann constant (1.3806504(24)×10&#8722;23 J/K)
T is the thermodynamic temperature in Kelvin.

The only way to get negative temperatures, is with negative energy.


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## lensman57 (Jan 14, 2013)

Hi All You Egg Heads,

Unless I have missed something in the last 35 years  since I went to UnI to do MECH ENG, at absolute zero all the molecules and atoms are at a stand still and there is no migration between the orbital paths, zero energy state, the only way this negative energy thing could theoretically work is to prove that at a stand still, zero energy state of the atoms , some hidden energy could be released from the atoms, some how. Is this really possible, plausible or not? Energy from nothing!!!


A.G


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## OrangeAlpine (Jan 15, 2013)

My scientific training is on the biology (entomology) side, but it appears our understanding of absolute zero is (horrors) not quite right.  It appears be more more of an average of energy states of the atoms.  Deviate from average one way, you have positive temperature.  Deviate the other way, negative temperature.  

But I could be entirely wrong.  The subject is so far over my head it sounds like techno-babble.

Bill


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## MuellerNick (Jan 15, 2013)

Just look at the right part of the Boltzman distribution:
 e^(-deltaE/kT)

What happens if T gets close to zero? And what happens if it is zero?
You won't be able to get over that point. Well, just in mathematics.


Nick


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## OrangeAlpine (Jan 15, 2013)

MuellerNick said:


> Just look at the right part of the Boltzman distribution:
> e^(-deltaE/kT)
> 
> What happens if T gets close to zero? And what happens if it is zero?
> ...


Nick, you don't see a way around that problem and I don't.   This guy claims there is a way and he has found it.  The science in me says that I have to entertain the possibility he is correct.  When others have tried and failed, then it becomes doubtful.  When he cannot demonstrate mastery of his technique to others, it becomes false.

Bill


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## MuellerNick (Jan 15, 2013)

> Nick, you don't see a way around that problem and I don't.



It might sound contradicting:
The researcher found *distinct* atoms with a negative temperature. Considering that atoms interact with each other with all the quantum stuff (I don't understand at all), it might be possible for some atoms to have negative temperature. As soon as you have a bunch of them, in sum, they are at or above absolute zero.
This is just my explanation, I'm not a physics professor at all, nor do I claim to be one or try to pretend to be one.

The wiki-link repeatedly *excluded* quantum-effects (that's what I understood).

Edit:


> more of an average of energy states of the atoms.


That's what you wrote, and I think the same. And that's what the Boltzman distribution says.
And that's what would explain the observation (within my *very* limited knowledge on that matter).

Nick


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## tups (Jan 15, 2013)

OrangeAlpine and MuellerNick are on the right path -

Classically, there is no way to get past the "infinite energy asymptote" - but you can, for a small group of atoms (temperature is in any case a collective phenomenon) by employing quantum effects.

Whereas Entropy455's implication that it is "all hogwash" is correct where it concerns perpetual motion and negative energy, the mathematics does not forbid negative temperatures - they are just not what you would intuitively expect.
The equation Entropy455 quotes, 3/2kT = E is true for an ideal gas, or any mass of non-interacting volume-less particles, but it cannot be used to generally define temperature. The Boltzmann equation is a much more general - whereever there is an energy difference it can be used to define a temperature. The E = 3/2kT follows from the Boltzmann distribution by applying the postulates of the kinetic gas theory to it. Reaching a negative temperature is quite impossible under the conditions where the postulates retain their validity, so in that sense, you are right, it is nonsense.

But calling negative temperatures hogwash is throwing out the baby with the bathwater. Of course the scientists will speculate about what it mathematically means. And of course the most sensational quotes will get used and distorted in an article that is pointed towards the general public. Do the authors of the research genuinely believe that they will be able to decrease entropy in the universe by creating negative temperatures, or be able to create a perpetual motion engine ? I would be genuinely surprised. The bottom line is you cannot use mathematics where it suits you, and then disregard its conclusions if and when you do not like them. Either you discard the entire model and build a better one, or you accept its quirks where they apply to states of the universe that do not generally occur. And very occasionally the model will predict an anomaly that leads to actual new and surprising knowledge. Read up about the kinetic gas theory and the viscosity of gases - it led to the realisation that atoms actually do exist.

The "negative temperature and what we could do with it" discussion is somewhat comparable with that about "Boltzmann's demon" - if you could construct a very small machine that would be a one-way valve for high energy gas atoms, you could create "free energy". But the energy needed to make the machine work makes the whole of the system obey the second law of thermodynamics anyway. There is no free lunch, and it is exactly the same here - you need a prodigious amount of energy to set up the boundary conditons for making a very small sample of matter with a negative temperature, and you can never win. And the scientists who did it know that very well too.


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## Omnimill (Jan 15, 2013)

PeterB said:


> This is interesting.
> Link



Now look what you've done Peter! You've got some of these guys fit to bust a blood vessel! You won't do that again will you! Rof}


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## Entropy455 (Jan 15, 2013)

tups said:


> If you could construct a very small machine that would be a one-way valve for high energy gas atoms, you could create "free energy".


 
It&#8217;s time for a sanity check. If this check-valve device were possible, you could install it on the intake manifold of a piston engine, then proceed to extract massive amounts of mechanical power. The machine would ingest atmospheric air, output usable mechanical work, and exhaust nothing but cold air.

In the engineering profession, we call this idea a Clausius Violator of the Second Kind &#8211; in that it&#8217;s impossible to construct a device that extracts work from a single thermal reservoir (see the Second Law of thermodynamics). Not only would the device demand that heat flow from cold-to-hot, it would also require system entropy to decrease - both of which are not possible.

I took gas dynamics in college. It was one of my favorite classes. I really liked the topic of supersonic nozzle design, and also determining shock-angles from various object geometries within super-sonic flight. FWIW, if an engineering student proposed a free-energy check-valve device within a gas-dynamics class - the Dean of Engineering would probably kick the student out of the engineering program for life. Does this seem extreme? Ask yourself the following question: would you want the engineer who designed the wing of the airplane you&#8217;re riding in, to (A) believe in imaginary machinery and impossible technology? Or (B), have a firm understanding of the laws of physics, and capable of differentiating between achievable and impossible machine designs?

Heat flows from hot to cold. Entropy always increases. Absolute zero is &#8220;absolute&#8221;. There is no such thing as free energy - as perpetual motion is not possible.


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## Entropy455 (Jan 15, 2013)

tups said:


> There is no free lunch, and it is exactly the same here - you need a prodigious amount of energy to set up the boundary conditons for making a very small sample of matter with a negative temperature, and you can never win. And the scientists who did it know that very well too.


Apparently not. The experiments may shed light on the nature of dark energy, and indicate that a 100 percent energy-efficient internal combustion engine, which had previously been considered impossible, might in fact be achievable. Source: Braun, S.; Ronzheimer, J. P.; Schreiber, M.; Hodgman, S. S.; Rom, T.; Bloch, I.; Schneider, U. (2013). "Negative Absolute Temperature for Motional Degrees of Freedom". Science 339 (6115): 5255.:fan:


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## tups (Jan 15, 2013)

Entropy455 said:


> Its time for a sanity check. If this check-valve device were possible, you could install it on the intake manifold of a piston engine, then proceed to extract massive amounts of mechanical power. The machine would ingest atmospheric air, output usable mechanical work, and exhaust nothing but cold air.
> 
> In the engineering profession, we call this idea a Clausius Violator of the Second Kind  in that its impossible to construct a device that extracts work from a single thermal reservoir (see the Second Law of thermodynamics). Not only would the device demand that heat flow from cold-to-hot, it would also require system entropy to decrease - both of which are not possible.
> 
> ...



I will make one more post on this topic, Entropy455, and then I will simply stop.
1st point.
The entire post above summarizes succinctly our knowledge about thermodynamics acquired in the last two centuries. My argument about "Boltzmann's demon" is a *thought experiment* (are you familiar with those ?) used by Peter Atkins in his famous book "The Second Law" to lead to the conclusion that all you write above is exactly true. We are disagreeing on exactly nothing in this discussion, except on your calling negative temperatures "hogwash". You know, thought experiment in the sense of "if you could construct a one-way valve - but you can't, because if you could, x, y and z would follow, and we know that that is impossible". This book by Peter Atkins (which I heartily recommend) also derives the existence of negative temperatures if I remember correctly, and this was in 1984. It follows from the mathematics employed, you can do the calculations yourself on the back of an envelope, but instead of opening your eyes and seeing, you rather choose to parrot what you have heard in a thermodynamics class I don't know how many years ago. Use your imagination, for heaven's sake!

2nd point.
Please understand there is a difference between the interpretation of a scientific article by journalists and what the authors actually wrote. The sentence you quoted in your other post right after the one above is nowhere to be found in that article. I'll PM you a copy if you like to see for yourself. If it would have been in there, perhaps because all the authors would have been delusional nutcases, the peer review process to which all scientific articles are subjected prior to publication would have very quickly weeded it out. And incidentally, the authors do a pretty good job at explaining the basics behind it too, so perhaps you should at least read the introduction of the article for yourself.
Whatever interpretation journalists give to it is mainly meant to be thought-provoking - or perhaps provocative, especially to people like yourself I suppose. They do not mean at any time that this could be a practical way to a free energy machine. Because those don't and can't exist. Yes, we know you know that. And I know it too. And so do these authors. The journalist who wrote that sentence - well perhaps. I do hope so.

3rd point.
I especially take offense at the mention of me being kicked out of an engineering curriculum for life by the Dean on mentioning "Boltzmann's demon" - which, once more, is a very instructive thought experiment to demonstrate the validity of the second law of thermodynamics - it says you cannot win, even if you try to cheat. You really should read that Atkins book, and get rid of this "It cannot be done because it's the (second) Law" attitude, and instead try to breed a little understanding of the reasoning, process and mathematics behind it. 

And that is where my contribution to this shouting match will end. I think I'll go do some model engineering to "cool down" (hee hee).


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## Entropy455 (Jan 15, 2013)

tups said:


> The negative temperatures in the article are a way of cheating - they will never provide a way to break the second law of thermodynamics (except locally - to be compensated in the surroundings, as in so many other processes that do exist), and the authors never stated otherwise. Because they are serious scientists, not driveling journalists.
> 
> So, if you were following my thermodynamics class, which I teach in an engineering curriculum with the express purpose of making my students realize which machines can and cannot be built, and which chemical reactions are or are not spontaneous, based on a firm understanding of the laws of thermodynamics and their correct application, I think I'd personally kick you out, for (a) sustained and stubborn demonstration of lack of ability or willingness to understand and (b) not being able to process new information efficiently and place it in its proper context.
> 
> http://arxiv.org/pdf/1211.0545v1.pdf


 
There&#8217;s no need to send me private messages. This is an important part of the discussion. I hope you don&#8217;t mind me posting a portion of the private message you sent me.

Why didn&#8217;t you say you were a thermodynamics professor?

Before you kick me out of class, perhaps you can clearly answer one very specific question for me - in the link you provided to the actual article (thank you for finding that by the way) &#8211; the author makes reference to the Carnot cycle being able to exceed an efficiency of one. He does so not once, but twice.

How can this be anything but an acknowledgement by the author(s), that the Second Law of Thermodynamics can be violated &#8211; clearly implying that perpetual motion heat engines are feasible?


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## terrywerm (Jan 16, 2013)

Entropy455 said:


> The only way to get negative temperatures, is with negative energy.


 

Doggone!   My wife says I'm FULL of negative energy,  and it's worst first thing in the morning before I've had my coffee!  stickpoke


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## MuellerNick (Jan 16, 2013)

I think, this is a discussion between an engineer and a physics.
The engineer says, that you can't go below absolute zero, and he is right.
The physics says, that you can go below absolute zero, and he is right.
The contradition is in the scale both look at the problem. The engineer looks at it as a whole, practical thing. The physics looks at it at a atomic level.

How can this work out?
It is statistics! If you take a sample big enough (engineer), you can't go below absolute zero. If you take a sample small enough and pick out only what you want (physics), you can observe below zero.

This explanation might well be wrong (I'd like to be corrected) and a bit "popular science".


Nick


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## lensman57 (Jan 16, 2013)

Entropy455 said:


> Its time for a sanity check. If this check-valve device were possible, you could install it on the intake manifold of a piston engine, then proceed to extract massive amounts of mechanical power. The machine would ingest atmospheric air, output usable mechanical work, and exhaust nothing but cold air.
> 
> In the engineering profession, we call this idea a Clausius Violator of the Second Kind  in that its impossible to construct a device that extracts work from a single thermal reservoir (see the Second Law of thermodynamics). Not only would the device demand that heat flow from cold-to-hot, it would also require system entropy to decrease - both of which are not possible.
> 
> ...


 
Hi,

I am with you there, all my training as an ex student of engineering tellls me that this negative energy is not feasible to either muster or utilise. Theoretical physics will have you believe that impossible things could happen and the next decade someone else comes along and says that no it can not happen. I guess the example of an elephant being able to fly under its own power in theory would be too tacky to mention. For decades they have us believe that the universe started with a big bang, now they are saying that it couldn't have happened that way and there must have been  another mechanism at work. All this might be theoretically correct or achievable but we do not trust our machines to do a job based on unproven abstract physics. We do not create energy, we are not capable of doing so, all we can do is convert one type to another with questionable degrees of efficency much the same as that we can not create wealth, we move it around from one place to another and from one type of people to another yet day in day out people go around talking about creating wealth.

I for one would leave the theories to the people that research these and untill there is soild proof that it exists and can work, would leave the engineers alone to deal with reality.

Regards,

A.G


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## Entropy455 (Jan 16, 2013)

This is not an issue of cup-half-full, verses cup-half-empty. The issue here is the clearly stated implications made by the scientists - specifically with being able to exceed a Carnot efficiency of 1. 

Please read this link: http://en.wikipedia.org/wiki/Carnot_heat_engine

Key equation: 
	

	
	
		
		

		
			





Where:
n is efficiency
W is the energy exiting the system as work
Qh is the heat energy entering the system
Tc is the absolute temperature of the cold reservoir
Th is the absolute temperature of the hot reservoir

Here's why negative temperatures are hogwash: per the right-side Carnot equation, if the cold reservoir (Tc) is negative, then the Carnot Efficiency is capable of exceeding 1 (which is 100% engine efficiency). This also means (using the left side equation) that its possible to get more work out of the engine (W), than heat energy going into it (Qh). That's like getting 5 crankshaft horsepower out of an engine, with only a 3 horsepower thermal input. . .  It's an indisputable perpetual motion claim, as exceeding a Carnot Efficiency of one IS the textbook definition of perpetual motion.

Read this link: http://en.wikipedia.org/wiki/Perpetual_motion

Key quote: No heat engine (an engine which produces work while moving heat from a high temperature to a low temperature) can be more efficient than a Carnot heat engine.

They also claimed to be able to destroy entropy, which is just as absurd as their claim of exceeding a Carnot efficiency of one.


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## Herbiev (Jan 16, 2013)

This is a model engine building forum not a site for plucking bits of info from Wikipedia so let's just wait for those in the know to supply a nice little PDF on building a machine that converts 1Kw input to 2Kw output. Then the horrendous power bill funds can be used to buy more toys ;D


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## Philjoe5 (Jan 16, 2013)

OK, I think we have covered this topic pretty well.  Considering all the negative energy expended in this discussion consider it locked.  

However, anyone who has built a model engine with 100% efficiency is encouraged to start a new thread.  A video would be mandatory, of course.

Cheers,
Phil


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