There has been a long-standing problem in understanding basic states of matter: solids, liquids and gases. These states are understood on the basis of the pressure-temperature phase diagram where different states are separated from each other by phase transition lines. Two of these lines, the solid-gas and liquid-gas lines, were understood and their functional form was worked out long ago. The third line, the solid-liquid melting line, has remained completely mysterious. It is interesting that in this age of scientific and technological development, something as basic as theoretical understanding of the pressure-temperature dependence of the melting line was still absent.

There are at least two reasons why it is important to solve the melting problem. The first is related to fundamental science and fundamental understanding of the basic states of matter. We can’t understand these states if we don’t know where the melting line is on the phase diagram. The second reason is related to applications: predicting phase diagrams is the key area in materials science. There is an active search for new functional and high-performance materials (eg “Materials genome” project in the US), but in order to predict the stability range of a new material we need to be able to predict its melting point and its melting line. This includes new advanced materials with tailored properties in areas such as drug development and areas where high-performance materials (used in, for example, energy, environmental and space applications) are subject to high pressure and temperature. Knowing the functional form of the melting line immediately tells us up to what temperature the system remains as a solid before it melts.

In order to solve the melting problem, one needs to solve the Clausius-Clapeyron (CC) equation written in 1834 and about 200 years ago. This was done for the solid-gas and liquid-gas lines long ago because solid-gas and liquid-gas transitions involve the gas state which is uncondensed and has no cohesion energy. This greatly simplifies the solution of the CC equation because the latent heat involved in the solid-gas and liquid-gas transitions is related to the energy of cohesion in a condensed phase (solid or liquid). This simplification does not apply to the melting line because both solids and liquids are condensed states. So in order to solve the CC equation, we need to explicitly know the equation governing liquid thermodynamic properties in general form as well as the liquid equation of state. And this problem of liquid theory was exactly the reason why the melting problem remained unsolved for 200 years.

Many physics luminaries (Brillouin, Frenkel etc) worked on the melting problem, including Nobel-prize laureates such as Born. In order to solve the CC equation, one needs to know another equation: the equation for liquid thermodynamic properties such as entropy. And another group of physics luminaries and Nobel laureates among them (Landau, Lifshitz and Pitaevskii) have stated that deriving this equation for liquid properties in general form is impossible. If this is true then solving the melting problem is also impossible.

Over a few years, me and my collaborators thought hard and developed a new understanding of the liquid state. This has given us an advantage in solving the melting problem. I realised that differentiating the CC equation and using the similarity of specific heats of solids and liquids close to melting (this similarity is backed up by our recent theory of liquids and numerous experiments) gives a new melting equation that can be solved.

The solution (Physical Review E 2024) gives the first important result: the analytical function of the melting line turns out to have a simple quadratic, parabolic, form: pressure depends on temperature quadratically along the melting line. Second, it turns out that the parameters of this parabola are governed by fundamental physical constants: the Planck constant and electron charge and mass. Therefore, all melting lines have this fundamental unity of properties. These two results are confirmed by experimental melting lines in several different types of liquids, including noble, molecular, network and metallic.

Properties of condensed matter and fundamental physical constants

Liquid viscosity is strongly system-dependent and also depends on temperature and pressure. It is not possible to calculate liquid viscosity in general form due to strong interactions, the same problem that exists for liquid thermodynamics (see below). It turns out that liquid viscosity at its minimum is universal. We have recently shown how fundamental physical constants provide the lower bound of liquid viscosity (Science Advances 2020). This minimal viscosity is a new quantum property, featuring the Planck constant.

Interestingly, EM Purcell noted that "the viscosities have a big range but they stop at the same place" (Am. J. Phys., 1977), adding that he does not understand this and that Weisskopf has explained it to him (we didn't find this Weisskopf's explanation). Our results provide the answer to this question (Physics Today 2021).

It turns out that the same theoretical minimum for viscosity in terms of fundamental constants applies to a very different property: thermal conductivity and thermal diffusivity (Physical Review B 2021). Fundamental constants also govern the maximal velocity gradient that can be set up by a biochemical machine in cells using chemical energy (Science Advances 2023).

We have similarly applied our ideas to relate another property, the speed of sound in condensed matter phases to fundamental physical constants. We have found (Science Advances 2020) that fundamental constants give the upper bound to the speed of sound in solids and liquids of about 36 km/s. The ratio of this bound to the speed of light depends only on two important dimensionless constants, the fine structure constant and the proton-to-electron mass ratio.

Having realised that several other important properties of condensed matter are governed by fundamental physical constants, I have written a review of this topic (Advances in Physics 2023).

One of the triumphs of physics has been the theory of solid state developed at the beginning of the last century. This provided a fundamental understanding of the basic properties of solid matter such as the ability to absorb and transfer heat. Solid state theory was preceded by the equally successful theory of gases. But when we turn to the third main phase of matter, the liquid phase, most textbooks remain silent about the most basic liquid properties such as heat capacity. Paradoxical though it may seem in this age of scientific and technological advancement, we still do not understand the most basic aspects of liquid behaviour. The combination of strong interactions with large atomic rearrangements has proved to be the ultimate obstacle to developing a theory of liquids. This was famously summarized as the "no small parameter" problem (eg by Pitaevskii). Landau and Lifshitz repeatedly assert that liquid energy and heat capacity can not be calculated in general form, contrary to solids and gases (Statistical Physics, Pergamon Press, 1969, par. 66 and par. 74). I have proposed that contrary to this statement, liquid energy and heat capacity can in fact be calculated and understood.

The argument of Landau, Lifshitz and Pitaevskii concerns the approach to liquids in which the interaction energy is calculated in addition to the gas kinetic energy, as an integral over interatomic potentials and correlations functions. This general approach formed the basis for previous theories of liquids. Apart from the problem of being system-specific as discussed by L&L, there are other important disadvantages of this approach: (a) interatomic potentials and correlations functions are not generally available, consequently, apart from very simple and exotic liquids such as those composed of noble elements, it has not been possible to calculate liquid energy and heat capacity; (b) calculations become difficult beyond simple cases and approximations difficult to control; (c) it is not easy to see how the results are consistent with large experimental decrease of liquid heat capacity over a wide temperature range.

Together with collaborators, I have developed theory of liquids based on excitations, phonons. Differently from solids where the number of phonons is fixed to 3N, this number is variable in liquids and reduces with temperature. This addresses the problem set out by Landau, Lifshitz and Pitaevskii: there is in fact a small parameter in the liquid theory, but this parameter operates in a variable phase space. This gives the key to liquid theory, as discussed in my book (Cambridge University Press, 2023). This theory for the first time explains a wide range of experimental data in many real liquids, including their specific heats.

"The most detailed and rigorous test of the phonon theory of thermodynamics" was undertaken (J Proctor, Physics of Fluids 2020 and book on liquid and supercritical states 2020 and Proctor's book ). Proctor notes that the theory is falsifiable but tests did not falsify it.

According to previous textbook understanding, no differences can be made between a gas and a liquid above the critical point, an abrupt terminus of the liquid-gas coexisting line. Recently, we have discovered that this is not the case, and that the phase diagram of matter should be modified. We have proposed that a new line ("Frenkel line") exists on the phase diagram above the critical point at arbitrarily high pressure and temperature, and which separates two physically distinct states of matter. Crossing the line corresponds to qualitative changes of physical properties of the system.

The Frenkel line has been since experimentally confirmed in supercritical Ne, CH

I have proposed a Lagrangian theory of liquid dynamics involving two scalar fields. Dissipation was believed to be hard to describe by a Lagrangian, however in my theory the energy dissipated by one field is taken up by the second field (Physics Reports 2020, Physical Review D 2020). The theory results in gapped momentum states (GMS), the effect which we now realize extends beyond liquids and into other areas such as plasma, holographic models, relativistic hydrodynamics and so on.

The problem of glass transition was considered as one of the deepest and most interesting challenges in physics. The problem consists of two parts, dynamic and thermodynamic. The thermodynamic part of the problem of glass transition is related to two widely observed experimental facts: (a) heat capacity changes with a jump at the glass transition temperature Tg, and (b) no distinct solid glass phase has been identified experimentally or suggested theoretically. Therefore, the problem is to explain the jump of heat capacity without asserting the existence of a distinct solid glass phase. This problem is also common to other disordered systems, including spin glasses. I have proposed that if, as is the case, Tg is defined as the temperature at which the liquid stops relaxing at the experimental time scale, the jump of heat capacity at Tg follows as a necessary consequence due to the change of system's elastic, vibrational and thermal properties. The resulting equations have no fitting parameters, and compare well with experimental results. This theory explains widely observed time-dependent effects of glass transition, including logarithmic increase of Tg with the quench rate, and identifies three distinct regimes of relaxation (Physical Review B, 2011)

The dynamic part of the problem of glass transition is to explain the physical origin of several famous relaxation laws and dynamic effects that liquids show in the glass transformation range: the Vogel-Fulcher-Tammann law, stretched-exponential relaxation, dynamic crossovers etc. These phenomena are universal and are seen in other systems as well (e.g. spin glasses), yet their physical origin is not understood. I have proposed a solution to this problem based on elastic interactions in a liquid. The solution is based on a non-trivial way in which high-frequency elastic waves propagate in a liquid. Crucially, the propagation range of these waves increases with liquid viscosity. This is in contrast to frequently discussed hydrodynamic waves, whose propagation range decreases with viscosity.

Radiation damage is a large and important area in science and technology, and has far-reaching consequences for materials performance in a wide range of applications, including immobilization of nuclear waste and future fusion reactors. This performance crucially depends on whether a material is amorphizable by radiation damage or is able to recover back to crystal. The phenomenon of resistance to amorphization by radiation damage remains largely not understood from theoretical standpoint. I elucidated the phenomenon of resistance to amorphization by radiation damage using massive parallel molecular dynamics simulations. This was supported by quantum-mechanical calculations of electronic structure.

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