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It seems that we have quantum description of solid and gas, but there seem to be few quantum models of liquid, with the exception of Liquid Helium or perhaps Fermi Liquid.

For solids, we often study crystal, which is easier due to the lattice symmetry. For gas, particles often interact weakly so it's easy to formulate a non-interacting model. In this sense, I can see why people choose to study these systems first.

But are there any models that make use of quantum mechanics to explore the liquid phase? Perhaps a quantum mechanical description of water. Maybe we can combine Shrodinger Equation and Navier Stokes equation in some way. Of course Navier Stokes equation itself is already hard enough, so perhaps we need to go to certain limits.

Qmechanic
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Liuke LYU
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  • As you're correctly observing, the combination of strong(er) interactions and high entropy of liquids makes a full quantum description very difficult; see https://www.pnas.org/content/114/41/10846.short for one example model – Al Nejati Jan 05 '22 at 01:34

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Fermi liquid, Luttinger liquid, Bose-Einstein condensate, Superconductivity and similar states of matter are known as quantum fluids or quantum liquids, and there are books written on the subject, see, e.g., Interactions in quantum fluids. The word liquid/fluid is however somewhat ambiguous, as it may mean different (although not mutually exclusive) things:

  • Fluid/liquid as a phase of matter is studied in the theory of phase-transitions / critical-phenomena. Water is a well-known and well-studied example of a sysmtem exhibiting a phas etransition,a nd it shows many interesting properties in its liquid phase (see, e.g., Superheated water and Supercooling). However, beyond the derivation of the intermolecular potentials (the subject of quantum chemistry), quantum effects are of real relevance to critical phenomena (see, e.g., Lectures On Phase Transitions And The Renormalization Group).
  • Mechanical properties of liquids, such as studied using Navier-Dtokes equation. Superfluid Helium and atomic Bose-Einstein condensates do fall into this category.
  • Quantum liquids are often not really liquids in the above sense, as matter phases or mechanical entities (although one could stretch the analogy rather far - e.g., if we consider Wigner cristals as a solid phase of electron liquid). Fermi liquid was called this way as an extension of Fermi gas description, because it accounts for the interactions between particles - the same distinction as exist between the ideal gas of atoms and a liquid in statistical physics. The analogy was thus grounded in the analogy of the mathematical/physical description, rather than physical similarities. Luttinger liquid then received a separate name, because it is different from the Fermi liquid, while also being a liquid of electrons. It is also common to speak more generally about non-Fermi-liquid behavior, meaning the electronic properties. Similarly, Atomic Bose-Einstein condensates may exhibit physical properties similar to liquids in conventional everyday sense, but their close brothers - Bose-Einstein condensates of photons (in lasers and masers), excitons or the superconducting state have physical properties very different from water or orange juice.
Roger V.
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    This answer doesn't answer the question of the OP. – Jun Seo-He Jan 06 '22 at 19:58
  • @JunSeo-He I think it does. FYI, I didn't downvote your answer. – Roger V. Jan 07 '22 at 06:22
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    It doesnt try to give a quantum description to the nature of liquids like my answer does. – Jun Seo-He Jan 07 '22 at 06:32
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    @JunSeo-He It does. However, it doesn't make sense to talk about quantum description of liquids before we agree upon about what liquid is - the OP is confused about it. I don't see how your answer give "a quantum description fo liquids": chemical potential is not a quantum property (first paragraph), while Fermi and Dirac distributions are not specific to liquids (second paragraph). In fact, Fermi distribution does not hold for Fermi liquid and many other "quantum liquids". – Roger V. Jan 07 '22 at 07:41
  • 1)Yet some macroscopic quantities like temperature and chemical potential appear in the distribution of quantum particles over energy.2)The OP clearly means liquids such as water and oil at room at room temperature at 1atm pressure. – Jun Seo-He Jan 07 '22 at 21:29
  • @JunSeo-He chemical potential and temperature are the parameters of the grand vanonical distribution, whether quantum or not, whatever is the substance/material described. – Roger V. Jan 08 '22 at 06:08
  • @RogerVadim "it doesn't make sense to talk about quantum description of liquids before we agree upon about what liquid is - the OP is confused about it.". I think OP is asking about quantum description of water at normal pressure and room temperature. There is no ambiguity or confusion from OP's side. I don't think you have addressed the question at all. An actual answer to this question may involve a description of the hilbert space for liquid water, an environment at temperature T=300K surrounding it and possibly decoherence etc. – Prem Nov 01 '22 at 14:02
  • @Prem I respect your opinion, but disagree with it: the very first sentence in the OP mentions quantum liquids, which the author used to confuse with quantum description of liquid. Furthermore, since they accepted my answer, it probably clarified something for them. If you have a different view on the subject, write your own answer. – Roger V. Nov 02 '22 at 07:09
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    No hardfeelings. Your answer is certainly very informative. I don't know what OP intended to ask, but one of the questions that they ended up asking was about quantum description of water in our everyday life. I cannot answer this question at this moment because I am also looking for an answer, just like the OP had been. By the way, @Al Nejati 's link in his comment to the original question seems to be useful. – Prem Nov 02 '22 at 08:58
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In a gas the chemical potential is 0 because interaction between the molecules are considered negligible .However in a liquid this is not the case so the chemical potential will be different from 0.

For a big number of molecules the distribution of molecules over energy will be Dirac-Fermi or Bose-Einstein distribution depending on the molecule and you just apply the formulas for these distributions.

Jun Seo-He
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    Do you have a reason to downvote this answer? – Jun Seo-He Jan 05 '22 at 07:24
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    I wasn't one of the downvoters, but this answer (like many of your recent answers) would need a lot more detail to get an upvote from me. It's also not clear to me that your answer is actually correct—can we really use the gas picture when interactions are non-negligible?—but that's where more detail in your answer will help. – Michael Seifert Jan 05 '22 at 21:38
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    I agree with Michael. The QM description of liquids is completely different from the way gases are described. The strong correlation between molecules makes it a completely different system. – John Rennie Jan 06 '22 at 08:35
  • @JohnRennie in a gas the chemical potential is 0 because interaction between the molecules are considered negligible .However in a liquid this is not the case so the chemical potential will be different from 0. – Jun Seo-He Jan 06 '22 at 18:01