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I have a particle system of seven protons and seven (or sometimes eight) neutrons (each formed by their appropriate quarks, etc.) bound together in a state that can be macroscopically described as a nucleus. If relevant, there are also about seven electrons that are bound to this arrangement. These particle systems are usually found in pairs, bound to eachother.

Macroscopically, this can be modeled as the elemental Nitrogen ($N_2$), and in other disciplines (such as chemistry), it is treated as a basic unit.

We know that at a certain level of thermal energy, this system of elementary particles exist inert and packed together in what can be macroscopically described as a "liquid". We know that this is this temperature is about 77.36 Kelvin (measured experimentally) at the most. Any higher and they start repelling each other and behave as a macroscopic gas.

Is there any way, from simply analyzing the particles that make up this system (the quarks making up fourteen protons and 14-16 neutrons, the electrons) and their interactions due to the current model of particles (is this The Standard Model?), to find this temperature 77.36 Kelvin?

Can we "derive" 77.36 K from only knowing the particles and their interactions with each other, in the strong nuclear force and electromagnetic force and weak nuclear force?

If so, what is this derivation?

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2 Answers

Theoretically, yes it should be possible to derive the boiling point of diatomic nitrogen from fundamental forces. In fact, you don't even need to involve the strong force or weak force (or the strong nuclear force, which is sort of different). The strong forces bind the quarks together into nucleons and the nucleons together into nuclei, but they have essentially no effect on distance scales much larger than that of an atomic nucleus. So, for purposes of calculating the boiling point of nitrogen, you can treat the nucleus as basically a point charge. The only force that is relevant to calculating a boiling point is the electromagnetic force.

Now the bad news: even something as simple as calculating the energy levels of helium, with 2 electrons, is impossible to do analytically. To analyze the behavior of even just those two electrons (and He nucleus) in detail, you need to use either perturbation theory or a numerical simulation, or both. And of course, the complexity increases with the number of particles, so simulating the 14 electrons and 2 nuclei of a nitrogen molecule is absurdly complicated. Perhaps it's been done, but I'm not a condensed matter physicist so I wouldn't know where to look for a reference. Maybe someone else can provide you with that information.

If you were to calculate the boiling point of nitrogen, I believe the main effect that you'd take into account would be the instantaneous dipole interaction. According to the Wikipedia article, it gives an interaction energy in terms of the polarizabilities and ionization energies of the molecules. Those are the quantities that you would have to extract from your simulation and/or perturbative calculation of the dynamics of the nitrogen molecule, if you wanted to calculate the effect from first principles.

$$E_{AB}^{\rm disp} \approx -{3 \alpha^A \alpha^B I_A I_B\over 4(I_A + I_B)} R^{-6}$$

(that formula is actually for monatomic noble gases, it may not apply to diatomic molecules)

Once you get the interaction energy as a function of intermolecular separation $R$, you would then have to do either another numerical simulation, or a rather complicated calculation, to show that a large pool of nitrogen molecules subject to the given intermolecular force undergoes a phase transition at 77.36K (at standard pressure, I assume). There are various thermodynamic models you could use, some more accurate than others, but of course the more accuracy you want, the more computation power you'll need. I suspect that in order to get within a few degrees of the actual temperature, you would need to do something more computationally intensive than would be possible by hand.

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Given infinite computational power, could 77.36k be arrived at using only the first principles you speak of? –  Justin L. Nov 4 '10 at 7:52
@Justin Certainly you could bind to the point where you could see it is not a real constant (it is slightly dependent of the microstate of the system). But this requires literally almost infinite power. –  mbq Nov 4 '10 at 9:58
+1 Good overview of the problem, and not unnecessarily technical! –  Noldorin Nov 5 '10 at 16:25
To calculate Helium energy levels is not too bad variationally, the wavefunction is only 6 dimensional, and reduces to 4 under symmetry, so this is not intractable. The intractability sets in around Nitrogen, when the number of electronic dimensions goes to 100. For large atoms, the system is completely intractable. I find Hartree Fock to be a miracle. –  Ron Maimon Aug 26 '11 at 16:23
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Apologies in advance if the first part of this comes off a bit argumentative, but I think there is an important point about physical theory that should be made. This point is also implicit in David Zaslavsky's answer as well.

Rant on effective theories

Actually trying to calculate macroscopic properties like "chemistry" from fundamental theories like QCD and QED, or string theory, etc. seems to me to be missing the point of much of modern physics. In particular, there's enough of a separation of length, energy, and time scales that the "physics" that goes into the boiling point of Nitrogen is almost completely disjoint from the physics of quarks and and the nucleus.

Indeed, if to do any physics we had to start from the very bottom every time, science in a form usable and comprehensible by humans could not exist! One of the deepest ideas of the 20th century is the notion of "effective field theory and renormalization" which to me makes clear why we can treat all the effects coming from high-energy physics as "just" couplings going into our vastly more useful phenomenological theories, i.e. all we get out of particle physics is some numbers such as the mass of various particles, or effective interaction potentials such as the Coulomb interaction, etc. and these numbers are just as useful if you measure them from doing experiments as if you were to calculate them. (And as calculating many of these (with some notable exceptions) involves intractable many-body problems, at this point in time it's actually much more accurate to measure them).

Breakdown of scales

In this particular case of determining boiling points, let me lay out what I think the relevant scales are from the bottom up and perhaps real experts can come by and correct me.

QCD and "below"

First of all, effects from nuclear physics and below are totally negligible, as pointed out by David Zaslavsky already. So we can just treat the nuclei (and electrons) as single particles and take the masses as something given from measurement. If you wanted to calculate these masses, it's my impression that there are pretty good models for the masses of nuclei if you have the masses of protons and neutrons. However, the masses of protons and neutrons have not been (directly) calculated to any reasonable accuracy up until fairly recently, if at all. And these models assume certain values for quark masses, which I think nobody really knows how to calculate at all. In any case, you could certainly dig deeper and deeper with respect to calculating the masses of things, but other effects won't be important for the boiling point of nitrogen.

QED and quantum mechanics

Second, higher corrections from QED and indeed, even quantum effects between molecules will probably be ignorable too, since for Nitrogen at its boiling temperature, the molecules will be far away from quantum coherence (I'm less sure of this, but a quick check by estimating the de Broglie wavelength seems to work out; note that this will not be true for liquid Helium or Hydrogen). What this means is that you can probably get away with treating the Nitrogen molecules as classical (dumbbell-shaped) dipoles with an effective pair interaction coming, as David Zaslavsky pointed out, from a single-molecule quantum ab initio calculation. I'm not sure if current techniques allow for this to be done with any reasonable accuracy, but with 14 electrons, it is a fairly serious problem not too far away from the current frontiers in the field of quantum chemistry.

Classical molecular dynamics

Now we get to scales which are to me, more interesting, and where the physics of boiling finally appears.

As David Zaslavsky and mbq have pointed out, boiling point is a thermodynamic property, thus it is something which only exists for a macroscopic number of molecules. That said, one won't need to attempt a simulation with 10^23 Nitrogen molecules to get a reasonable estimate; one of the triumphs of statistical mechanics is an understanding of how to estimate "finite-size effects"; however, it should be emphasized that this is yet another many-body problem (read as, impossible to solve analytically, requires difficult computer simulation to get right).

So finally, to get the boiling point, you could model Nitrogen molecules as some kind of pair-interacting dipolar gas in a classical molecular dynamics simulation at fixed pressure, say, and tune the temperature up until the molecules in your simulation begin to behave in a gaslike way rather than liquidlike (this "behavior" can be made more precise with the notion of pair correlation functions, for instance). Note a few things here. The greatest error in your estimation will be coming from your ability to put in a bunch of particles here in a controlled fashion in your simulation. This is going to vastly outweigh any error from the steps "further up" other than having a good effective potential. This explains why scientists who actually do molecular dynamics simulation don't spend that much time worrying about QCD and particle physics.

Places to start looking and references

OK, that all written, I've looked around for some references that would be useful on molecular dynamics simulation of liquid nitrogen.

Dominique Levesque and Jean Jacques Weis write in the 1992 book The Monte Carlo Method in Condensed Matter Physics a chapter "Recent progress in the simulation of classical fluids" which goes over the main computational techniques commonly used now. This contains a short section on Nitrogen, as well.

Javier Carrero-Mantilla and Mario Llano-Restrepo, Fluid Phase Equilibria, Volume 208, Issues 1-2, 15 June 2003, Pages 155-169 writes on simulation of liquid Nitrogen and binary mixtures with some other simple liquids. I've attached a figure from the paper which shows the liquid-vapor coexistence curve as a function of density, from which you could read off boiling point if you wanted. The agreement is pretty amazing, and they do use the strategy I outlined above - simulate Nitrogen as a classical fluid with a tuned intermolecular potential; their source for the potential is apparently C.S. Murthy, K. Singer, M.L. Klein and I.R. McDonald, Mol. Phys. 41 (1980), pp. 1387–1399..

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I could dig around more for newer references, but if you're really interested, you can use google scholar and some of the search terms like "ab initio", "liquid-vapor coexistence", "Liquid nitrogen", "molecular dynamics" to find more...

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