Wikipedia:Reference desk/Archives/Science/2018 April 26

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April 26

Mass of entropy, and so on

Entropy is a genuine source of energy. For example, the proton gradient in the mitochondria really does power the cell and keep us alive. But at a simple level, it seems very mysterious...

• Suppose I have two tiny evacuated chambers, linked by a pore, that contain a total of three helium atoms. There is only one way for all three to be on the left, but there are three ways for one to be on the right and two on the left. So there is an entropy of k_B T ln 1 in the first case = 0 and an entropy of k_B T ln 3 in the second case = (0.0000862 eV/K)(300K)(1.10) = 0.0284 eV at 300 K. By analogy with a mitochondrion, this gradient could be tapped at a gate in the pore to transfer that energy to some chemical reaction or mechanical process.

• I think the entropy there relates to Landauer's principle in that one bit of information has a negative energy value of k_B T ln 2 = 0.0179 eV. Putting one of the three atoms on the far side can be done three ways, generating, it would seem, (k_B T ln 3)/(k_B T ln 2) bits of information. I'll admit, I would have thought the choice of 1-in-3 would be 1.5 bits (1 bit if you choose A, 2 bits if you choose B-or-C, then B or C) but this works out to 1.59 bits, not sure why. It would appear that generating information creates energy out of thin air, though I would assume the total mass of the entire set of evacuated chambers and contents never varies.

• A basic Carnot heat engine would appear to work by equilibrating the lone helium atom at T_H, which then crosses the pore to generate usable energy of k_B T_H ln 3; it then is cooled to T_L before it crosses back to consume usable energy of k_B T_L ln 3. Heating and cooling the actual atom will transfer 3/2 k_B (T_H - T_L) of heat energy. The efficiency, therefore, is ... erm, well, I'm getting (2 ln 3)/3, which surely isn't right. That calculation seems maddeningly close to making sense.

• But if we have no gate in the pore, where does the energy go? The atoms will be on the same side 25% of the time, and opposite sides 75% of the time. Being at 300 K, each atom on average has 1/2 kT = 0.0129 eV of energy per degree of freedom, so I take it they have 0.0388 eV of total kinetic energy on average. If the entropy of being on opposite sides adds to the kinetic energy of the atoms, that would imply that they are moving much faster whenever they are apart. Does that make sense? Is there some force that can be described as?

• But how small does the pore have to be? Could the three atoms be in a simple spherical chamber where you might simply measure which side each is closest to? Do they still have higher energy when they happen not to be on the same side? Can you do that dividing top-vs-bottom, left-vs-right, front-vs-rear??? Since the energy depends on information I suppose it depends on which way(s) you are actually measuring the atoms, and this necessarily perturbs them?

• Now the amount of entropy from a single bit of information at 300 K is a whopping 0.0179 eV, which may seem small, but is huge compared to the possible masses of neutrinos at 10-4 eV or less. Simply knowing a neutrino exists, therefore, should create a bit of information. Is that factored into the neutrino's mass already? Does learning of the existence of a neutrino add to its momentum like those helium atoms above, causing it to massively accelerate? Can you do this with, say, an entangled neutrino by learning about the other particle created by a reaction? Is the "temperature" for this entropy determination going to be determined solely by the neutrino's KE, or something else, i.e. is there a way to erase information about a stationary neutrino and leave it with a negative overall KE?

Any low-level example of this sort you might care to raise should be useful for understanding... Wnt (talk) 07:04, 26 April 2018 (UTC)

I think the important point to make is that information is not a substance, any more than caloric is. Reducing the disorder of a system can be described as "cooling" or "adding information", but in neither case is anything physically added or removed from the system - only the energy distribution of the system's components is changed. See Gibbs free energy as a starting point for the actual calculations required in these cases. Tevildo (talk) 07:50, 26 April 2018 (UTC)

That's what I was doing above - taking T delta S to figure the delta H. Caveat being that at the individual particle level "T" is the energy in each degree of freedom of one particle, and "H" is energy that could be extracted at a gate, if one existed. (True, I was simply dumping that work back into the particle as heat in the 4th point above) Wnt (talk) 11:44, 26 April 2018 (UTC)

• Sorry, but all that is not even wrong. A careful reading of our article entropy might or might not clear what confusions you may have (that is one of our best articles in the topic, but it is still quite a hard read).

Pay attention in particular to the fact that temperature and entropy are statistical concepts: they apply only to systems with a large number of atoms. You can compute an "entropy by atom" or similar by dividing the system entropy by the number of atoms but it does not mean that that entropy is held by a single atom in any meaningful sense. Tigraan^{Click here to contact me} 12:18, 26 April 2018 (UTC)

Per Tigraan, entropy is an emergent property of a system; the entropy of a system of atoms divided by the number of atoms does give you a number, but that value is different than the entropy of a single atom in isolation. The number of distinct states of a system depends on the relationships of particles in that system. Similarly, the temperature of 1 atom is a meaningless concept. Temperature is a mass property of a system of a large number of particles. A single particle has no meaningful temperature. Also, the OP is confusing thermodymic entropy with information theory entropy. Entropy in thermodynamics and information theory discusses the difference, at a very abstract level, the limit of informational entropy is thermodynamic entropy, but to say that the information about a particle adds mass to it is beyond silly. It is broadly true that information-as-negative-entropy has some thermodynamic meaning in that somewhere, there must be some system that you remove entropy from to store information, with the caveat that information is not stored in the same system as the system it is describing (that is, information about a neutrino is not stored in a neutrino!) Also, entropy mass, as such, is just the mass of heat energy; that is the energy a system has from its random thermal motions, if you cool a system down it loses mass; that lost mass is the "mass of entropy". --Jayron32 12:42, 26 April 2018 (UTC)

@Jayron32: I'm sorry, but what I take from Entropy in thermodynamics and information theory seems almost directly opposite to what you say. There is an ln 2 conversion factor between the Shannon entropy h and the physical entropy S, but that's in the formulae I use above. They give an example there of a single particle in a box with partition being used to provide work to two pistons, which therefore must acquire relativistic mass.

I will admit that there is something to this "emergent property" bit that I don't necessarily understand, which is the meaning of the temperature of a system. If I have three particles at a given temperature, does that mean that the average energy per degree of freedom works out to yield that temperature by the Boltzmann constant, or does it mean that they are in equilibrium with that distribution of energies so that their summed KE's are necessarily not conserved? I'm not sure I can have information about an isolated system, because if information is energy, my knowledge of it means the system is not isolated???? Wnt (talk) 14:24, 26 April 2018 (UTC)

I found a paper on a single-particle Carnot cycle at [1]. But it is ... formidable. I am not sure how much of the complexity is truly essential to understanding, as I certainly do not at this point. I note very superficially that it has a graph of temperature for anywhere from 0 to 1000 particles in a simulated system. Wnt (talk) 14:40, 26 April 2018 (UTC)

There are two very different types of questions that need to be answered here, and I feel we're at cross purposes. The two questions are 1) Can you calculate X and 2) Does X have any useful meaning? Clearly, you already know the answer to 1) You've made calculations and gotten numbers. What Tigraan and I are trying to tell you is that 2) is more important here. Any of these values can trivially be reduced to 1 particle systems. But what does a temperature of a 1 particle system tell you about that system? What real, physical, actual thing is such a system supposed to model, or what application do I have for it? That's what Tigraan meant above when he noted "You can compute an "entropy by atom" or similar by dividing the system entropy by the number of atoms but it does not mean that that entropy is held by a single atom in any meaningful sense." The entire system of thermodynamics breaks down when you try to consider 1-particle systems. A 1-particle system isn't a thermodynamics problem, it's a dynamics, problem. That is, we've reduced our system to where we can analyze it's behavior as we can with any other single object (or small number of objects) which are colliding and interacting. The whole point of thermodynamics is that large systems of interacting objects produce behaviors which are predictable and modelable without reference to the individual motions of the particles, and that one can understand how the system works without ever needing to understand how any one particle works. Once you've gotten down to looking at the one-particle situation, sure, you can trivially do thermodynamic calculations on it, and its a fun academic exercise, but so what. --Jayron32 14:58, 26 April 2018 (UTC)

This is a learning exercise. I don't understand entropy well enough, so I was thinking if I can work out a direct correspondence, I'll get a better knowledge of what it means. I still don't understand where the energy would come from if splitting up the three particles on one side of a chamber actually does release it due to entropy. I still don't understand whether creating a lot of new information can create mass-energy from nothing like some new variant of steady state theory, or whether it is conceivable to move information from some high-temperature source into a very light structure that would have negative mass and so on. So if this is a "trivial" exercise, please, by all means lead on! Wnt (talk) 23:50, 26 April 2018 (UTC)

If you're trying to understand entropy in a general sense, take the numbers out for a minute. Just look at it qualitatively: consider that you have 2 particles and are trying to use the 2 particles to move a 3rd particle to the right. Moving an object in physics is what is called work (physics). Now, imagine the 2 particles are moving at the third from the same side, and both hit it to the right. Now, that third particle will start moving in that direction. Your two particles have done work on the third. HOWEVER, in order to do that, you need to assure that the striking particles are always hitting the target from the same side. If the two particles are moving about randomly, in all sorts of directions, with no overall motion in any one direction, the sum total of all of their hits will result in no net motion of the target particle. The total energy of the system (that is the energy of all of the particles) is the same in situation 1 (concerted motion produces work) as in situatiomn 2 (random motion does not produce work). The difference between 1 and 2 is called "entropy". All entropy is is the difference between the total energy of a system and the actual energy used to do work. Of course, being energy that energy has an associated mass, but entropy is inherent to the system and because of that, you can't take the entropy out to find its mass effect on the system; you can calculate that number, but it has no physical meaning. The mass of the total energy in situation 1 is still the same as in situation 2. Entropy is not contained in any of the particles, and cannot be parted out physically, it is just a way to represent the notion that there can be energy in a system which is unavailable to do work. --Jayron32 14:24, 27 April 2018 (UTC)

@Wnt: Be sure to read Jayron32's post above, as that is a concise yet almost accurate of what entropy represents. I have a small nitpick about it (hence the "almost"), but be sure to understand the post before reading the nitpick.

Ok, here we go. The problem is that the sum total of all of their hits will result in no net motion of the target particle is only true in an average sense. Statistical physics only apply to large number of particles; in a three-particle system, entropy (as defined by information theory formulas) can increase by random deviations and you can extract work from a single-temperature source cycle (the average over many cycles will have nonpositive work, but you will see single-cycles where work is positive); however, the probability of such random deviations decreases exponentially with the number of particles involved. The second principle is basically the assumption that we never see improbable deviations in many-particle systems (under certain tedious hypotheses that are not worth mentioning here). Tigraan^{Click here to contact me} 15:11, 27 April 2018 (UTC)

Yes, I knew that inaccuracy, but was trying to contruct a simple, visual model to display entropy. Thanks for expanding on it with the corrections! --Jayron32 16:36, 27 April 2018 (UTC)

You should take serious what Jayron and Tigraan are saying here about how thermodynamics arises. If you are very careful about deriving the thermodynamic formulas of entropy of simple systems, see e.g. the book by F. Reif, you'll see that it depends on the level of coarse graining that you need to introduce. For an isolated system we introduce a function $\Omega (E)$ which counts the number of energy eigenstates between energies $E$ and $E+\Delta E$. Then $\Omega (E)$ will be proportional to the parameter $\Delta E$, but entropy needs to be defined as being proportional to $\log \left(\Omega (E)\right)$. In the thermodynamic limit where we consider the limit of infinitely large systems and consider the specific entropy, the choice of $\Delta E$ doesn't affect the result.

But for finite systems, the entropy will get a contribution proportional to $\log \left(\Delta E\right)$, which is still utterly negligible in practice, of course. This term that arises as a result of having to choose a coarse graining scale to define the entropy will make increasing negative contributions the more fine grained view of the system you take. In the limit where you can resolve the exact physical state, this term will exactly cancel out the ordinary entropy of the system. When you know the exact physical state, $\log \left(\Omega (E)\right)=0$, and therefore the entropy is zero. If you work at this fine grained level where the entropy is zero, then the laws of physics will ensure that it stays zero as they forbid erasure of information. If you work at a coarse grained level, then systems will tend to lose information that's visible at the coarse grained level to a more fine grained level (due to statistics). This is how the second law that says that entropy can only increase, arises. Count Iblis (talk) 02:43, 27 April 2018 (UTC)

Silkie chickens

Why do Silkie chickens have 5 toes on each foot instead of the usual 4?? Georgia guy (talk) 15:18, 26 April 2018 (UTC)

As described in our article on polydactyly § other animals, this is a genetic mutation that is now standardized and encouraged by the humans who breed this type of chicken.

I searched the online database of the Poultry Science library at North Carolina State University, and found this paper: Genomic Regions Associated with Dermal Hyperpigmentation, Polydactyly and Other Morphological Traits in the Silkie Chicken (2010). If you're interested in more technical reasoning for the multiple toes: "A single SNP in a highly conserved cis-regulatory region of Sonic Hedgehog was significantly associated with polydactyly..." which is biology-ese that means the multiple toes are caused by a single mutation, with lots of technicalities.

Nimur (talk) 15:41, 26 April 2018 (UTC)