… not even Alabama (a kid’s joke).

One of the first courses I created after landing in the Physics department in Lille, around 2006, was a class of condensed matter physics. It seemed amazing to me, but what was taught as ordinary program of 3rd year physics in Rome, was practically non-existent in the local five-year curriculum. The basic material for that course was based on two books, the Ashcroft and Mermin’s all-time classic Solid State Physics, for the theory of electrons and phonons; and the wonderful States of Matter by David Goldstein, for everything else. This latter book should be in every physicist’s bookshelf, as a daily guide to how applied physics should be taught: a comprehensive view from the atoms up, rather than from the nucleus down, with emphasis on those results of fundamental research whose fruits are likely to be applicable elsewhere. I see this as the way to reach perfection, pretty much the equivalent of the Imitation of Christ for Western medieval scholars, or the Bardo Thodol for the Oriental ones. As you may remember, Chapter 1 starts with the famous quote, “Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906 by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously.”  

Everybody knows what a state of matter is. Or at least, so we think. We learn in high school about solids, liquids, and gases, different states that the same atoms or molecules could occupy, based on the strength of their chemical bonds. Becoming physics students, we also added plasma to our short list. The strong bonds in a solid keep the material rigid, but when we heat it up those bonds start breaking and softening. The softer bonds allow the particles to move around and past each other, however they are still bound together, and we have a liquid. Heat if further, and even such soft bonds break, and the particles float freely almost without interacting with each other, in the gas state. And if we keep heating this gas, electrons and nuclei come apart, and the system enters the plasma state, in which the two species of particles largely behave as independent gases. From the point of view of atoms and molecules, it all boils (ehm… ehm) down to a competition between kinetic energy of motion, that is temperature, and potential energy of the bonding forces. But at a finer scale, quarks inside the protons and neutrons are part of the matter, and we may ask whether their condition depends on the state of aggregation of the nuclei they make up? And at the other end, granular particles display properties that would be quite different to derive just by looking at their constituent atoms. Moreover, recent claims in the popular science press brought up new states of matter, such as time crystals, nuclear matter, neutronium… That is quite a lot to put on the plate.

Look at Goldstein’s book, and changes in temperature and pressure seem the key to describe a “state of matter”, and in fact we have P-T phase diagrams that are powerful tools for understanding the behavior of matter under different thermodynamics conditions. Already phase diagrams show that things are indeed a little more complicated than just solid, liquid, gas and plasma. For example, at T and P above the critical point the distinction between liquid and gas disappears, and we have a supercritical fluid that looks a bit like both a liquid and a gas. In fact, T and P are not properties of atoms and molecules: a molecule does not have a temperature, it rather has a velocity, compounding translational, rotational and vibrational motions. It is only when we look at an ensemble of molecules that we can define what is a temperature and a pressure. A “state of matter” is actually a definition of how these statistical, average properties such as T and P are related to each other, as summarized in its equation of state.

But when we think of different states of the same material, for example water, we have in mind very different properties with respect to what both the phase diagrams and equations of state can describe. The “wetness” of water, or the waves in the sea, for example, are special properties that are not associated with the behavior of each one H2O molecule, and neither with the average behavior of a large amount of H2O molecules. A crystalline solid is rigid, meaning that its constituents cannot move past each other in the lattice, and we say that its viscosity is practically infinite; the same atoms in the liquid state are viscous, but practically incompressible; the same atoms in the gas state have practically zero viscosity, and are easily compressible. A metal is a good conductor of electricity because its electrons live in a quasi-free state inside the solid, but a superconductor is a perfect conductor because its electrons get bound in pairs, which display a completely different behavior from that of a simple sum of two ordinary electrons. Wetness, waves, viscosity, compressibility, conductivity, superconductivity are non-thermodynamic properties that characterize the macroscopic features of the matter in that particular state: they are emergent properties. A state of matter seems to be much better characterized in terms of its emergent properties, than of its thermodynamics.

In fact, we can make states of matter from things other than atoms and molecules. Let us take a plasma, already a very hot guy, and keep increasing its temperature further. At some point, we can destroy the integrity of protons and neutrons, and form a new state of plasma in which the elementary constituents are quarks and gluons. To get there we must touch a temperature of about 5 1012 K, the so-called “Hagedorn temperature”. Is this plasma anywhere comparable to the lower temperature gas-like plasma of electrons and ions? Actually no: it should be more similar to a liquid, because of the strong residual interactions brought by gluons. But then, it is not a stupid question to ask whether such a “liquid” could also… freeze? Well, yes: we may call its “frozen form” the objects we ordinarily observe as protons and neutrons, and in general any elementary particles made up of quarks, the “hadrons”. A hadron is therefore a crystal of the quark-gluon plasma, and atomic nuclei are polycrystals of this plasma: quark-gluon snowflakes, if you like. The quark-gluon matter has its own phase diagram, with the ordinary protons and neutrons being its low-temperature, low-density phase, the plasma state at high temperature a reminder of the conditions at the Big Bang, and a superconducting phase at higher density that should be found in the innermost core of neutron stars.

In 2012, Nobel laureate Frank Wilczek was teaching a course about solid state physics, and out of serendipity he shifted the subject to crystals in more than three dimensions. When he took time as a fourth dimension, he invented the new state of matter today known as “time crystal”, an idea that immediately took the center stage. Imagine to put ice cubes in a glass of water at T>273 K: the ice cubes will melt in a few minutes, while lowering a little bit the water temperature. It would be surprising to observe the ice cubes to reform after another few minutes, and melt again, and so forth, at constant temperature. This would be a time crystal: an arrangement of matter whose lowest-energy state is periodic not only in space, like an ordinary crystal, but also in time. After some initial but healthy skepticism, physical systems conforming to this condition have actually been fabricated in laboratory, and even simulated on Google’s Sycamore quantum computer. The discussion about the peculiar properties of this new state of matter, whether its ground state is an equilibrium one (it is not), or if it violates the conservation of energy (it doesn’t), or the second principle of thermodynamics (it doesn’t), and so on, are very interesting but it would take too much space to be discussed in this brief letter. I am citing such objects just as a further example of exotic state of matter, and if you are interested (and I do hope you are) you could for example take a look at K. Sacha’s review paper (Rep. Prog. Phys. 81 (2017) 16401), or his longer book on the subject Time crystals (Springer 2020).

So, different states of matter can live together, yet in the same thermodynamic conditions. A liquid of protons is actually an assembly of solids of quarks, different states nested within each other. One may think that in this case the peculiarity of quantum mechanics may have something to do with this confusion of states. But states of matter can involve objects much larger than quarks, atoms and molecules. In sand, for example, each grain is a solid but grains do not interact with each other except for gravity. If air is flown in the sand, the whole structure starts behaving like a liquid, and larger objects dispersed in the sand can float on its surface. And yet, sand grains remain solid, and air remains a gas. Such “fluid sand beds” are used in many technological applications, such as the recycling of metals. Granular materials make up probably about 90 per cent of the world around us, but we are so little interested, and sparsely knowledgeable about them. The physics of such interesting state of matter is rarely taught in university physics, while the engineers confront it from an eminently practical point of view. Notably, temperature plays no role here, and interactions between grains are dissipative, because of the existence of static friction and the inelasticity of collisions. The relevant energy scale for a grain of mass mand diameter d is its potential energy mgd. For a typical sand grain this energy is at least 1012 times kBT at room temperature, and ordinary thermodynamic arguments become useless. For example, it is our daily experience that vibrating a bucket of sand makes particles of different sizes to separate. This separation appears to violate the principle of entropy increase, which normally favors mixing instead of separation. However, the fact that here kBT is practically zero implies that entropy considerations are outweighed by dynamical effects, which now become of paramount importance. Mmm… that’s very interesting… maybe I will dedicate a Sunday letter to granular materials.

Speaking of different states of matter that coexist together, can you think of a system in which you mix a liquid and gas and get a solid, without any chemical reactions or changes in temperature or pressure? The answer is… whipped cream. The sneaky part is that for cream at refrigerator temperature, the milk’s colloidal fat is a frozen emulsion of oils and water; when you whip air into it, those solid particles stabilize the air bubbles; when enough air bubbles are introduced, a continuous structure of interfacial solid network of fat droplets, stabilizes the cream as a foamy solid. But if you keep whipping, the fat droplets aggregate further into clumps and the air bubbles are no longer stable: the fat droplets aggregate enough to phase separate, while the air is lost, as nothing is stabilizing the bubbles any more. You are left with just a fat phase, which is what we call “butter”, and an aqueous phase, that is the “buttermilk”. By following these simple steps, anybody can make their own butter at home. And talking of cream and matter: ice cream is another interesting combination of many different states of matter. In fact, it is a frozen emulsion-stabilized foam, embedded in a gel of ice crystals, themselves embedded in a liquid sugar syrup. A delicious state of matter. Maybe I will dedicate another Sunday letter to the physics of colloids…

There is no smartest state

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