In chemistry, a common method of calculating solute and solvent reaction rates in a liquid is through using the quantum mechanical descriptions of inter and intramolecular bonds. One of the intermolecular descriptions is that of Van der Waals interactions - that of the sum of electrical, quantized, yet weak bonding forces. These may include permanent dipole-dipole interactions (Keesom force), dipole-induced dipole interactions (Debye force), and spontaneous dipole interactions (London dispersion forces). Generally Van der Waals forces omit that of ionic bonds between molecules.
As far as I know though, there isn't a model describing supercritical liquid/gas phases, such as that of CO2 which is often used to decaffeinate coffee beans. I know there are some models built to describe plasmas, but these are generally models designed for cross-section analysis used in fusion/fission reactors - they don't describe allowed energy levels in the same manner as say, a solid state semiconductor would.
I'm not quite sure how to tackle this kind of problem. In a field theoretic condensed matter picture (or even many-body statistical Schrodinger equation) solids are generally described by phonon modes; quasiparticle states are evaluated with ladder operators, after setting up the problem with electron & ion density. Sometimes metallic conductors can be described by an electron gas - at sufficiently low temperatures, this is a Fermi liquid.
Fermi liquids have energy levels described by momentum degeneracy and the Pauli exclusion principle. I assume something similar must apply to a gas, but there would be an absurd number of tightly packed available energy levels, and in terms of the Schrodinger equation, most particles would have a Hamiltonian of that similar to a free particle; bumping around other gases though, on the large scale, it's almost a classical description - and in fact, classical descriptions work pretty well. Is it just because the energy levels involved are so high that it's in the classical limit?
Anyways. I can't seem to find any literature on this - all of the above is just my thinking on it. It's mostly a curiosity of mine :)
I am not sure if I understand your question correctly but here is an answer, and two papers which may help you further.
For simulations of the different states of matter (gas, liquid, super critical states) the potential between the gas particles does not change. The density and temperature determine what the state is. So, for instance in the attached paper by Qin et al., CO2 - CO2 interaction is still described by a Lennard-Jones (vdWaals) potential together with electrostatic interaction between well-placed charges in the molecule. In another paper, (I can only attach one, so that one is in the next post) other, maybe somewhat more sophisticated, potentials are used, but still these are supposed to work for all situations, not just in the super critical state.
Here is the second paper.
Maybe I should add to this that it was one of the fundamental problems of statistical mechanics to understand how a single interaction can lead to phase transitions and critical points since at a critical point, or at a phase transition the interaction potentials don't change, only the state of the macroscopic system.
A good answer! I might add that, for dense states of matter, pair potentials are not sufficient. There are three-body forces (among them, triple induced dipole contributions) and worse. If one insists on using pair potentials only, one has to use effective pair potentials, and these are to some extent density-dependent. You may wish to look up the work of Richard Sadus on the subject.
Hey Gert Van der Zwan and Ulrich Deiters! I apologize for the response time, I've had a lot of medical stuff going on which has prevented the use of my right hand.
I will dig into the work of Richard Sadus as well as the papers you've linked, Gert. The explanations given to me in basic chemistry didn't go into much detail as to why the critical points were where they were, and so this should be very revealing :)
Dear Sirs,
It looks like you try to consider a gas as a unifiorm medium. But it is an essentially heterogeneous medium consisting of monomers and clusters. At a critical density we have a mixture of zones with gas-like structure and with liquid-like structure. The closer we are to the critical point, the larger are clusters in the gas-like medium and pores in the liquid-like medium. At the critical density we see an analog of the Curie-Weiss theory for dimensions of clusters falling with growing distance from the critical temperature.
http://dx.doi.org/10.4236/ijamsc.2013.12013
Boris - that is pretty much exactly the sort of data I've been looking for. Thank you for the analysis of it, and physical picture you've drawn; I hope to build a model for phase transitions of this sort that is entirely a many-body schrodinger equation without gravity involved. I have found some research into the micro black holes spawned in the empty space around a black hole - the Reissner-Nordström geometry - and I believe there's some similar physics going on there, but with more data-like entanglement patterns in the "liquid"/"gas" structure.
I'm a little new to the statistical physics, so I'll have to read more into the Cure-Weiss theory...as a related question if you wouldn't mind, what do you mean when you say "dimensions of clusters"?
If it is that aspect you are interested in, you should look into the modern theory of critical phenomena as it was developed in the 1960's and 1970's. Keywords are scaling, renormalization group, universality, and block Hamiltonian.
A good introduction to statistical mechanics of critical phenomena is H.Eugene Stanley's book: Introduction to Phase Transitions and Critical Phenomena. I attached a paper by that same author. Other names to look for are Wilson, Griffiths, and Kadanoff. But there are quite a few more important contributors, the acknowledgement of the attached paper has a nice list.
Henry, I agree with you, the critical and supercritical phenomena in different media (fluids, magnets and Space) have similar features. I am approaching to this problem from the most investigated - thermophysical properties of fluids. There is a huge amount of precise experimental data, collected in electronic databases. To understand the hidden mechanisms under the thermophysical properties of fluids it is needed to solve inverse problems that diverge at smallest distortions in initial data. So, it is important to base on the most precise and regularized experimental data, as the data from the NIST database, It was interesting and educative for me to noyice the parallel between the Curie-Weiss curve for magnets and for heat capacity at a supercritical temperatures. I hope something similar may be found in clusters of galactics.
Henry, in a pure real gas the dimension of a cluster is the number of particles bound together in the cluster. In galactics statistic this definition does not work, because masses of particles differ.
I wish you success in your investigations!
Henry and all participants of this discussion. I have a new paper on this subject:
http://www.scirp.org/Journal/PaperInformation.aspx?PaperID=61716
Best regards, Boris
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