Hi. I'm the author of the post. I realize that it's a somewhat challenging read, at least compared to my usual style, but my goal was to stay as true to the physics as possible. I was a little disappointed with some of the popular science coverage, because some sources made it seem like this is the first time that physicists reached negative temperatures (not true), or that negative temperatures aren't physical (they are, but they're weird!). I hope you find it interesting.
That is a great exposition, it gets a bit bogged down in the money/happiness analogy at times, but overall I felt like I understood what you were saying. To test that, let me echo back what I think you said.
Physicists decided to define the temperature of a system not simply by the amount of energy in the system, rather to define it as the ratio of the energy in the system to the total entropy of that system. In so doing, they created a situation where systems that go to a lower entropy state when energy is added, are described as having a negative temperature.
Small correction, not the ration of the energy to the entropy but the derivate of the energy by the entropy - that is, the rate at which the energy increases as the entropy increases.
(Of course, it's slightly more confusing since it's actually the inverse of the temperature that's defined as dS/dE, which is the rate at which the entropy increases as the energy increases, a much more understandable definition. And as the physics professor who taught this to a room of students from various departments said of the transition from
1/T = dS/dE
to
T = dE/dS
"The mathematicians in the crowd will be in an uproar now. But we can do stuff like that in physics")
Keep in mind that the relations T = ∂E/∂S and 1/T = ∂S/∂E are normally associated with different levels of abstraction:
T = ∂E/∂S is implied by the fundamental relation of (phenomenological) thermodynamics, whereas 1/T = ∂S/∂E gets derived from a microscopic theory via statistical mechanics.
The problem is in the assumption that the inverse derivative exists, which is true for most functions in the real world but not for all (non-smooth) mathematical functions.
I thought you did a great job of using metaphor to explain a dense topic. In fact, my main complaint is that your allusions to human behavior actually confused, or at least got in the way of the explanation...I think the metaphorical part should be less elaborate, because your primary audience is going to be (I'm being solipsistic here) are going to be people like me with a casual interest in physics who still remember the implications of absolute zero at the atomic level...no need to extend the metaphor in a way to appeal to people who think absolute zero is 0 celsius.
Thank you. What I liked about the metaphor is that it fairly accurate. But you may be right about the overuse - it's hard for me to work out the right balance when explaining physics, and I perhaps err on over-explaining. Then again, no one really complains if you make something too easy to understand! :)
I would improve it by defining the relationships between money/happiness and energy/entropy before giving the example. I would have found it a lot easier to read.
If you understand entropy (which can be thought of as a measure of disorder or randomness), then I think the Wikipedia definition is pretty clear.
Systems with positive temperature increase in entropy as
one adds energy to the system. Systems with negative
temperature decrease in entropy as one adds energy to the
system.
Not, and moreover, you've mixed apples and oranges.
A system's entropy in statisical mechanics is k log W, where k is Boltzmann's constant and W is the number of microstates. The microstate is a configuration of, say, electrons in energy levels.
Information theory has a quantity that behaves much like entropy in stat mech, but is not actually entropy in stat mech.
BTW: The statement in stat mech would be - Adding energy to a system with negative temperature reduces the number of microstates.
>>> Let's try more specific question. Laundauer's principle requires kT ln 2 of heat for every 1 bit of randomness erased from the system. What about systems with negative T? I can't erase bits?
There is no requirement that information has a physical representation for information-theoretic entropy. Landauer's result assumes that it does have a physical representation, and derives some physical consequences.
A simple example where information-theoretic entropy is used where there is no physical representation: Video codecs.
I'm guessing you've heard of mpeg and h.264 encoding. Which one encodes a movie better? One way of answering this question is to ask: Which codec added less entropy (perhaps for the same compression)?
For that matter: Before Shannon's information entropy, one might wonder if there is a way (another codec) perhaps recovering the information after mpeg coding and decoding. However, now you know that information-entropy can only increase or stay the same, which tells you that subsequent "correction codec" cannot remove entropy introduced by mpeg codec.
Thanks. I don't really like that example with codecs, because I could always argue that any codec can only be a physical system, operating in some environment at temperature T and will be constrained by Landauer's, ets.
Either way, I think we've digressed. I'm actually very happy with yours: "Adding energy to a system with negative temperature reduces the number of microstates.", because this is clear and unambiguous.
I think his argument, is that I have no right to talk about bits [edit: in the context of stat mech], before defining physical representation of these bits - that is a physical system, states and transitions between states.
> is that I have no right to talk about bits, before defining physical representation of these bits
I'm saying something slightly different: You can talk about bits in a system without a physical representation; That system can have an information-entropy associated with it. Once you implement a physical system representing those bits, then Landauer's comment applies.
>> A system's entropy in statisical mechanics is k log W, where k is Boltzmann's constant and W is the number of microstates. The microstate is a configuration of, say, electrons in energy levels.
We are not freshman, in natural units (pretty much any system of natural units) Boltzmann's constant is 1. And entropy is measured in bits ('nats', but ln 2 is also equal to 1) ;) . For a system with two microstates entropy would be 1 bit.
Thank you for cutting through all the "science journalism" or whatever bullshit, and actually bothering to put the _science_ in context (with some humor too!).
I love the analogies! "What happens when a negative temperature object meets a positive temperature object? To find out, imagine that the Dalai Lama meets Warren Buffett."
Very interesting.
Too complicated. IANAP, but it sounds like they just created a gas in which particles are more likely to be found at higher (absolute) temperatures than lower temperatures, which is the opposite of what usually happens (http://en.wikipedia.org/wiki/Boltzmann_distribution). In fact, from the abstract of the paper itself:
"Absolute temperature is usually bound to be positive. Under special conditions, however, negative temperatures—in which high-energy states are more occupied than low-energy states—are also possible..."
But "more particles are high-energy than low-energy" is pretty meaningless. It doesn't explain at all why a system so close to absolute zero would give away energy to boiling water, for example.
The answer is perhaps somewhat technical, but hopefully this shows the argument why heat flows from negative to positive temperature.
Entropy is essentially the (logarithm of) the number of states a system can be in, without changing the macroscopic observables. These states have all the same probability. The second law of thermodynamics is then simply a consequence of the number of allowed transitions of the system. And temperature is the change of the number of states if energy is added to the system. That heat flows from the lower (positive) energy to the higher is then a consequence of calculating the probabilities, as is the observation that heat flows always from a negative to a positive temperature system.
Perhaps a example will make this somewhat clearer:
Think of a chain of 20 capacitors, each can be charged or uncharged and I call the 10 left capacitors my left subsystem, and the 10 on the right the right subsystem. Initially there is 1 charged capacitor in the left and 3 in the right. Then the probability that after one charge moves there are two charged capacitors in each subsystem is 9/16 ( since 9 of the uncharged capacitors are on the left). In this case the temperature is positive in both subsystems. (The number of possible configurations of the left 10 capacitors is higher for two charged ones ( 5*4) than for one charged ( 5).
The negative temperature case would in this analogy be, if in one subsystem there are more than 5 charged capacitors. Then there are more charged than uncharged capacitors, the number of allowed states would decrease if I add additional charge. ( I can distribute 9 charges in 10 different ways, but 10 charges just in one way.) But nothing happens about the argument of transition probabilities. If there are 7 charged capacitors on the left and 2 on the right, then after moving a charge the probability that there are 6 and 3 charged ones is 8/11 ( 8 of the 11 uncharged capacitors are on the right).
In the case of the boiling water and the negative (close to zero) system it is the same, it is about counting possible states. And since in one the number of states increases if I add energy ( the temperature is positive) and in the other the number of states increases if I remove energy ( the temperature is negative), both have a preference for transferring energy from the negative energy system to the positive one.
I read through the article and the comments and felt I almost understood what was going on but was still missing something. Your example then made it all click for me in one of those "Aha!" moments.
I've been coding too much. Once I read that that objects with negative temperature behave as if they were hotter than objects that are at any positive temperature, my first thought was along the lines of "well yes, overflow". I should get out more.
"In fact, as I’ll try to explain, objects at a negative temperature actually behave as if they’re HOTTER than objects that are at any positive temperature."
Another way to look at it is that the important quantity is not temperature, but its reciprocal. Absolute zero becomes infinity under this transformation, hence unattainable. Heat energy will flow from lower 1/T to higher 1/T, and more heat energy added to a system lowers the 1/T quantity of that system.
