There's no such thing as "mathematically impossible" in the physical world. If it's impossibility is evident in a mathematical model, that's still physical impossibility.

> There's no such thing as "mathematically impossible" in the physical world.

I think that this is not quite true: there is such a thing as mathematical impossibility in the real world, but it applies only to mathematical models; that is, models cannot exhibit the behaviour. This doesn't prevent the real world from exhibiting that behaviour, but it does mean that, if the real world exhibits that behaviour, then the model is inaccurate.

These caveats might make the notion seem useless (as you seem implicitly to be arguing), but I'd argue that they are quite useful. The whole pursuit of quantum gravity comes from the realisation that relativistic and quantum mechanics are mathematically incompatible. That fact alone doesn't tell us which one, if either, is correct, but it certainly tells us that they can't both be correct in all regimes, and that's indisputable information about the real world that we wouldn't have without idealising it through mathematical models.

By the way, did you edit your post? I thought the post to which I responded only had the sentence above, but on preview I saw:

> If it's impossibility is evident in a mathematical model, that's still physical impossibility.

This sentence I don't understand at all. I would argue instead that there is no such thing as physical impossibility; the real world will do what it likes, and we may not forbid it, only observe that certain things haven't happened yet. Any impossibility we observe in a mathematical model is, I would say almost by definition, a mathematical impossibility, even if we deduce physical consequences from it.

Hmm okay you changed my mind, so I guess there's no such thing as physically impossible then. Either way, maybe we should probably leave off the qualifier, since there's only one kind of "impossible" in this context.

Not sure why you changed your mind away from a perfectly valid claim. (Please see my response to OC).

> I think that this is not quite true: ...

No it is. It is quite true that mathematical impossibility is mathematical and has nothing to do with the real world.

> ... but it applies only to mathematical models; that is, models cannot exhibit the behaviour. This doesn't prevent the real world from exhibiting that behaviour, but it does mean that, if the real world exhibits that behaviour, then the model is inaccurate.

You are talking about the validity of a mathematical model (with respect to a real world phenomena). That is a completely different notion.

Mathematically impossible means that, using the rigors of a mathematical proof, a contradiction is reached starting from the original mathematical assumptions. You started with math and ended with math. No real world involved.

> [quantum mechanics and gravity]

First of all, you're not making your life easier by defending your point using the argument of incompatibility of quantum mechanics and gravity.

And second of all, again you're mixing validity of a mathematical model as applied to observed phenomena, with the notion of mathematical impossibility. If the theory of quantum mechanics and the theory of gravity are incompatible and each on its own is unable to explain all of the observed physical world, that means both are likely inaccurate (rather incomplete) models of reality and we need a new model (maybe string theory) to explain the physical world.

I agree completely. How about mathematically proven to be physically impossible?

It should be "mathematically proven to physically impossible under this model/these assumptions".

Every math-based prediction about the real world will be based on two parts:

1) The world behaves like this model [up to our measurement accuracy].

2) The model implies this prediction.

Their claim would only be speaking to 2) -- that, under the model, you cannot reach absolute zero. The model can still be wrong.

> How about mathematically proven to be physically impossible?

I would argue (as I did in a sibling thread https://news.ycombinator.com/item?id=13878107) that it is this that can't happen. Mathematics has no power over the real world, only over mathematical models (optionally of the real world). A proof of mathematical impossibility does not constrain the real world; it only allows us to deduce that, if the real world does the mathematically impossible, then our models are wrong (or at least incomplete).

Still incorrect. You can say 'mathematically shown to be physically impossible'.

Mathematical proof is mathematical. It has nothing to do with the physical world.

P.S.: Note that 'mathematically shown to be physically impossible' still allows the possibility that the mathematical model was invalid. E.g., what if, after you have mathematically shown it to be physically impossible, you conduct an experiment and the "physically impossible" happens? That means you just provided an evidence that the mathematical model is invalid.

P.S.: There are tons of resources on this topic [1]. Read and make up your own mind.

[1] https://www.google.com/search?q=mathematical+proof+of+physic...

I am not a mathematician and I did not read the actual paper, so forgive me if I beclown myself with this question.

Isn't this a nothingburger? How would you hit absolute zero anyway when the minimum joint uncertainty of any given particle's position and momentum must be greater than Planck's constant / 2?

Temperature in this sense no longer has anything to do with kinetic energy. A quantum system at absolute zero would simply be in its ground state, which is perfectly possible. In open space you only run into problems that there's not a particularly clear minimum.

You do run into some problems with quantum fluctuations which may or may not randomly add energy to the system, I guess that's what this new result is about.

It seems to me that uncertainty is more uncertain a theorum to base your beliefs about the universe on than entropy.

True perhaps, but we're relatively certain about uncertainty.

Basing it on entropy makes certain assumptions on topological properties of time and space though.

Also, all the other particles in the universe will exert a force on the particle (through e.g. gravitation), which will probably result in (small) vibration of the the particle.

> Hitting Absolute Zero Has Been Declared Mathematically Impossible

Wait ... wait ... this isn't about mathematics, it's about a mathematical model of physics, of reality. The title is both confusing and confused. Also, if something is mathematically impossible, declaring it so means nothing (because mathematical truths don't rely on human authority).

There's an interesting scientific finding behind the title, but the title is a disaster.

In any case, there's a simple heat-energy-transfer equation that can be used to show that achieving absolute zero isn't possible for the simplest of reasons:

    Q(t) = Δq * e^(-t * f)

    Q(t) = temperature at time t
    Δq = initial temperature difference
    f = a rate-of-heat-transfer factor
    t = time, seconds
    e = base of natural logarithms, equal to lim n -> ∞, (1+1/n)^n (2.171828....)

If you plot the temperature given by Q(t) over time (the temperature of one body being heated or cooled by another), you see that the temperature never gets to zero, because the rate of heat transfer depends on the remaining temperature difference, and as that difference approaches zero, so does the rate -- and neither becomes zero.

It's simple mathematics, but its meaning is located in reality, in physics. If that were not true, another mathematical relationship could be substituted. So it's not mathematically impossible, it's physically impossible. This kind of mathematics describes reality, it doesn't substitute for it.

Where does this expression come from? Proving this is the point.

The equation I posted is the solution to a differential equation that expresses the idea that a rate of change depends on the difference between two quantities -- temperature, voltage and so forth. It is a very common solution in physics, describing the rate at which water moves between connected reservoirs, gas moves between connected tanks, current moves between connected circuits in electronics, and many other physical systems.

More here: http://arachnoid.com/sage/differential1.html

I'm kind of confused. For example, if you have a glass sphere and then you take all of the atoms out of the sphere so that it's a complete vacuum, isn't that some sort of absolute zero? Or does that not count since you need at least one particle to 'have' the temperature (i.e. a measurement of thermal energy)?

A perfect vacuum still displays activity in the form of "vacuum energy" - a counterintuitive property of the universe where particles apparently blink into existence for a short duration in otherwise empty space.

https://en.wikipedia.org/wiki/Vacuum_energy

Right -- it's probably more accurate to say that a perfect vacuum has no temperature than to say it's at absolute zero.

Wow!

In technical terms temperature is conventionally understood to relate to the average kinetic energy of particles in a region.

If a region contains no particles it would be more accurate to say that it doesn't have a temperature than to say that its temperature is zero.

There would still be the Higgs field permeating the universe and any radiation emitted nearby permeating that transferring energy to your sphere's surface, and normalizing the temperature inside.

In its most general sense temperature describes how the entropy of a system increases/decrease as a reaction to temperature, if there is only 1 possible state then entropy is 0 and can neither increase or decrease so the notion of temperature doesn't really make sense.

Some of your replies are pointing out that empty space actually still has some states, but even ignoring that, it's just not really sensible to talk about the temperature of nothing.

I'm a total layman to this stuff, but from my understanding isn't thinking about temperature in terms of linear scales (Fahrenheit, Celsius, Kelvin) fairly misleading for understanding how hard it is to "hit" absolute zero. Wouldnt describing temperature as a logarithmic scale make it much more obvious, from a layman's perspective, that absolute zero is impossible to "hit". Ie, negative infinity degrees of "temperature"?

I've heard arguments to this effect, that absolute zero is more obviously impossible when you think in terms of Thermodynamic beta - which is standard physics notation for 1/(k_B * T) and appears in many places including in statistical mechanics.[1]

That also makes it more clear that while reaching 0 (i.e. infinite Beta) is impossible, negative temperatures are possible (this occurs in processes that undergo a state inversion like in a laser / maser). [2]

[1] https://en.wikipedia.org/wiki/Thermodynamic_beta

[2] https://en.wikipedia.org/wiki/Negative_temperature

Interesting! Thanks, great answers (both this post and your response to its sibling post)

Even if that's a vaild approach (I have no idea if it is) I suspect that such a system would be hilariously impractical for any real-world day-to-day usage. Therefore there would be no reason for a layman to even learn about it in the first place (much like the Planck unit system).

Actually, I'm not so sure. If you used inverse temperature (i.e. "coldness") and scaled it properly, it wouldn't actually be all that different, in practice, from a positive temperature scale.

Here is a plot of Temperature vs. Coldness from 0-100 C, scaled such that water freezes at 100 coldness units and boils at 0 coldness units[1]: https://i.imgur.com/fluOSY2.png

It traces out a pretty gentle curve, which seems like it's not appreciably less linear than our ability to perceive differences in temperatures (from eyeballing this: https://biology.stackexchange.com/a/28522), so probably for day-to-day use inverse temperature would be a perfectly acceptable substitute.

Of course, this is just to say that if we were starting de novo today in determining how to discuss temperature, coldness could be a reasonable choice. I don't think it actually offers substantial advantages over the current system and it would likely be very costly to switch everyone over just for the sake of making it a bit easier to think about why temperature has an absolute zero.

[1] Python source code: https://bpaste.net/show/76bf4f7c36cb

Cool (pun intended) but somebody maybe remembers from three years ago that it's possible to go below absolute zero. And it's hotter than positive temperature there. An explanation at http://physicscentral.com/explore/action/negative-temperatur...

tl;dr: The negative values described in that article are possible because a broader definition is used:

> In statistical mechanics and thermodynamics, temperature is defined as follows:

> 1/temperature = change in entropy/change in energy

> This definition of temperature can go from positive infinity to negative infinity.

For context, the more common definition for absolute temperature is the average kinetic energy of the particles in the system, which of course can't be negative.

Can anyone explain why it took 100 years to 'prove' this?

Like a really basic proof is:

- It takes an infinite amount of time and resources to remove energy to reach absolute zero

- You can't reach infinity

- Therefore reaching absolute zero is impossible.

What deeper knowledge am I missing?

> It takes an infinite amount of time and resources to remove energy to reach absolute zero

Since particles not at absolute zero have finite energy, this point is non-trivial and proving this point was actually the focus. (Tangentially, I think this has been a common and strong intuition, and suggested by extrapolation from empirical evidence, that you could asymptotically approach but not reach absolute zero because it would be progressively harder as you got closer. But there's a difference between that and a mathematical proof grounded in assumptions that are well-accepted.)

Yes, if you have this as a known truth, you're basically done.

Ah! Thanks for the clarification.

That first premise is the one that hasn't always been "proven." Including it your "proof" is begging the question.

>- You can't reach infinity

That's the hard bit. The infinity could simply be an artefact of your description. You could for instance describe direction on a map as 'miles east / miles north' and conclude that it's impossible to move due east or west but that's clearly not the case.

A more sophisticated example would be the event horizon of a black hole, where several quantities seem to become infinite, but it turned out this wasn't the case with the right coordinate system, meaning that it's likely all too possible to cross the event horizon.

You learn why reaching absolute zero is impossible in an introductory thermodynamics class, but apparently this is a more rigorous proof.

So if entropy must increase, and temperatures must fall but can't reach zero... Does that imply the universe must keep expanding?

There's no rule that temperatures must fall. In fact there are rules that temperature can't simply fall without moving energy somewhere else.

But my mom (and other assorted guardians and authority figures) told me anything's possible.

So "if", and this is a big if, we did cool a sample to absolute zero by some yet to be discovered process, it's total energy should shrink to nothing and as so the sample should shrink to nothing and disappear?

No. Quantum issues aside, a system at absolute zero would still have energy in the form of its mass. You can think of mass as 'frozen' energy.

Note that in some systems it's possible to have negative temperatures, which are "hotter" than infinite temperatures. See https://en.wikipedia.org/wiki/Negative_temperature

Thermodynamics is cool!

> "It's the speed of cooling."

Maybe I missed it but did the article actually define this "speed of cooling"?

The "speed of cooling" is a layman's way of describing the rate of temperature change. For two connected masses at different temperatures, that rate is described like this:

    Q(t) = Δq * e^(-t * f)

    Q(t) = temperature a time t
    Δq = initial temperature difference
    t = time, seconds
    f = heat transfer efficiency factor
    e = base of natural logarithms (2.71828...)

If you plot this equation (http://i.imgur.com/DiZsY0G.png), you realize that the rate of cooling depends on the remaining temperature difference, and as the difference declines, so does the rate. This means both the temperature difference, and the rate, can never become zero.

The linked article addresses some more deep reasons why one can never achieve absolute zero, but this well-known and conventional (i.e. non-quantum) equation is a sufficient reason.

Thank you!

The fact that it is technically possible to reach temperatures lower than absolute zero because of quirks in the math really makes this "declaration" less than meaningless to me.

Math is a description of the physical universe not a definition. As another commented, what is mathematically possible and impossible is entirely our own invention.

> what is mathematically possible and impossible is entirely our own invention

That's actually quite a controversial statement that many professional mathematicians might beg to differ with.

Viewing mathematical results and axioms as being totally invented, opens the possibility of being able to just will contradictory proofs into existence, and that is simply not the case in practice. Mathematics might be more for description than definition, but what it's describing isn't usually the physical world, it's the logical limitations of certain principles. And if these principles are found to be reflected in some aspect of the "real world", then the corresponding mathematics becomes applicable. But you can't just pick any arbitrary axioms/principles and have them automatically be mathematically coherent either, so again we're limited in the maths that we can "invent".

link to original work: http://dx.doi.org/10.1038/ncomms14538

Or maybe possible but unobservable ?