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Why did you only show the E(T) function for positive temperatures?
This is a good point. The negative side gives good intuition for the "negative temperatures are hotter than any positive temperature" argument.
I admit that some definitions can be better than others. A whale lives underwater, but that's about the only thing it has in common with a fish, and it has everything else in common with a whale. You could still make a word to mean "animal that lives underwater". There are cases where where it lives is so important that that alone is sufficient to make a word for it. If you met someone who used the word "fish" to mean "animal that lives underwater", and used it in contexts where it was clear what it meant (like among other people who also used it that way), you might be able to convince them to change their definition, but you'd need a better argument than "my definition is always true, whereas yours is only true in the special case that the fish is not a mammal".
The distinction here goes deeper than calling a whale a fish (I do agree with the content of the linked essay).
If a layperson asks me what temperature is, I'll say something like, "It has to do with how energetic something is" or even "something's tendency to burn you". But I would never say "It's the average kinetic energy of the translational degrees of freedom of the system" because they don't know what most of those words mean. That latter definition is almost always used in the context of, essentially, undergraduate problem sets as a convenient fiction for approximating the real temperature of monatomic ideal gases--which, again, is usually a stepping stone to the thermodynamic definition of temperature as a partial derivative of entropy.
Alternatively, we could just have temperature(lay person) and temperature(precise). I will always insist on temperature(precise) being the entropic definition. And I have no problem with people choosing whatever definition they want for temperature(lay person) if it helps someone's intuition along.
Why is one definition more fundamental than another? Why is only one definition "actual"?
Because one is true in all circumstances and the other isn't? What are you actually objecting to? That physical theories can be more fundamental than each other?
Average kinetic energy always corresponds to average kinetic energy, and the amount of energy it takes to create a marginal amount of entropy always corresponds to the amount of energy it takes to create a marginal amount of entropy. Each definition corresponds perfectly to itself all of the time, and applies to the other in the case of idealized objects. How is one more general?
I just mean as definitions of temperature. There's temperature(from kinetic energy) and temperature(from entropy). Temperature(from entropy) is a fundamental definition of temperature. Temperature(from kinetic energy) only tells you the actual temperature in certain circumstances.
What do you mean by "true"? They both can be expressed for any object. They are both equal for idealized objects.
Only one of them actually corresponds with temperature for all objects. They are both equal for one subclass of idealized objects, in which case the "average kinetic energy" definition follows from the the entropic definition, not the other way around. All I'm saying is that it's worth emphasizing that one definition is strictly more general than the other.
In response to
Entropy and Temperature
This is a good article making a valuable point. But this —
Temperature is sometimes taught as, "a measure of the average kinetic energy of the particles," because for an ideal gas U/N = (3/2) kBT. This is wrong, for the same reason that the ideal gas entropy isn't the definition of entropy.
— is a confusing way to speak. There is such a thing as "the average kinetic energy of the particles", and one measure of this thing is called "temperature" in some contexts. There is nothing wrong with this as long as you are clear about what context you are in.
If you fall into the sun, your atoms will be strewn far and wide, and it won't be because of something "in the mind". There is a long and perfectly valid convention of calling the relevant feature of the sun its "temperature".
I think more precisely, there is such a thing as "the average kinetic energy of the particles", and this agrees with the more general definition of temperature "1 / (derivative of entropy with respect to energy)" in very specific contexts.
That there is a more general definition of temperature which is always true is worth emphasizing.
I think his "in the mind" is correct in his context, because in the model of entropy he is discussing, temperature_entropy is dependent on entropy, is dependent on your knowledge of the states of the system.
I'll repeat what I said earlier in the context of the discussion of different theories of time.
Me, I think the people who identify exists_everydaymode with exists_spacetimemodel are just conceptually confused by their high falutin ideas. Exists_everydaymode didn't cease to exist when we got our fancy new spacetime model to play with, and it's relevance and functionality didn't cease to exist either. "I have cancer" is really distinguishable in important ways to us from "I had cancer."
New physics didn't make old ideas useless. Temperature_kineticenergy is probably more relevant in most situations.
because they don't know what temperature is
The OP makes his mistake by identifying temperature_entropy with temperature_kineticenergy.
I'm don't see the issue in saying [you don't know what temperature really is] to someone working with the definition [T = average kinetic energy]. One definition of temperature is always true. The other is only true for idealized objects.
In response to
Stupid Questions December 2014
Is there any plausible way the earth could be moved away from the sun and into an orbit which would keep the earth habitable when the sun becomes a red giant?
According to http://arxiv.org/abs/astro-ph/0503520 we would need to be able to boost our current orbital radius to about 7 AU.
This would correspond to a change in specific orbital energy of 132712440018/(2(1 AU)) to 132712440018 / (2(7 AU)). (where the 12-digit constant is the standard gravitational parameter of the sun. This is like 5.6 * 10^10 in Joules / Kilogram, or about 3.4 * 10^34 Joules when we restore the reduced mass of the earth/sun (which I'm approximating as just the mass of the earth).
Wolframalpha helpfully supplies that this is 28 times the total energy released by the sun in 1 year.
Or, if you like, it's equivalent to the total mass energy of ~3.7 * 10^18 Kilograms of matter (about 1.5% the mass of the asteroid Vespa).
So until we're able to harness and control energy on the order of magnitude of the total energetic output of the sun for multiple years, we won't be able to do this any time soon.
There might be an exceedingly clever way to do this by playing with orbits of nearby asteroids to perturb the orbit of the earth over long timescales, but the change in energy we're talking about here is pretty huge.
In response to
Why is the A-Theory of Time Attractive?
I feel like this is just a really obnoxious argument about definitions.
I especially feel like this is a really obnoxious argument about definitions when the wiki article quotes things like:
"Take the supposed illusion of change. This must mean that something, X, appears to change when in fact it does not change at all. That may be true about X; but how could the illusion occur unless there were change somewhere? If there is no change in X, there must be a change in the deluded mind that contemplates X. The illusion of change is actually a changing illusion. Thus the illusion of change implies the reality of some change. Change, therefore, is invincible in its stubbornness; for no one can deny the appearance of change."
So, to taboo a bunch of words, and to try and state my take on the actual issue as I understand it (including some snark):
B theory: Let there be this thing called spacetime which encodes all moments of time (past,present, future) and space (i.e., the universe). The phenomenal experience of existence is akin to tracking a very particular slice of spacetime move along at the speed that time inches forward, as observed by me.
A theory: My mind is the fundamental metaphysical object, and moments of "time" can only be oriented with respect to my immediate phenomenal experience of reality. Trying to say something about a grand catalog of time (including the future) robs me of this phenomenal experience because I know what I'm feeling, and I'm feeling the phenomenal experience of existing right now, dammit! Point to that on your fancy spacetime chart!
Read this way, I suppose the most succinct objection of the A-theorist is: "If all of spacetime exists, all reference frames are equivalent, etc. etc., why am I, in this moment, existing right now?" To which, I imagine, a B-theorist would respond by saying, "Because you're right here," and would then point to their location on the spacetime chart.
But this isn't actually an argument about what time is like. It's an argument about how whether or not we should privilege the phenomenal experience of existing--of experiencing the now. That is, does me experiencing life right at this moment mean that this moment is special?
I suppose I can see why people that aren't computationalists would be bothered by the B theory, because it does rob you of that special-ness.
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Here's a discussion of the paper by the authors. For a sort of critical discussion of the result, see the comments in this blog post.