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22

Aug

2023

Aug

2023

I had a memorable library day trying to find an answer to a question that is simple to formulate: what is a theoretical value of energy and heat capacity of a classical liquid? I looked through all textbooks dedicated to liquids as well as statistical physics and condensed matter textbooks in the Rayleigh Library at the Cavendish Laboratory in Cambridge. To my surprise, they had either very little or nothing to say about the matter. I then took a short walk to Cambridge University Library and did a thorough search there. This returned the same result.

My surprise quickly grew closer to an astonishment, for two reasons. First, the heat capacity is one of the central properties in physics. Constant-volume heat capacity is the temperature derivative of the system energy, the foremost property in physics including statistical physics. Heat capacity informs us about the system’s degrees of freedom degrees of freedom and regimes in which the system is in, classical or quantum. It is also a common indicator of phase transitions, their types and so on. Understanding energy and heat capacity of solids and gases is a central and fundamental part of theories of these two phases. Thermodynamic properties such as energy and heat capacity are also related to important kinetic and transport properties such as thermal conductivity. Not having this understanding in liquids, the third basic state of matter, is a glaring gap in our theories. This is especially so in view of enormous progress in condensed matter research in the last century.

The second reason for my surprise was that the textbooks did not mention the absence of discussion of liquid heat capacity as an issue. It is harder to solve a problem if we don’t know it exists.

The available textbooks are mostly concerned with liquid structure and dynamics. They do not discuss most basic thermodynamic properties such as liquid energy and heat capacity capacity or explain whether the absence of this discussion is related to a fundamental theoretical problem. Textbooks where we might expect to find this discussion but don’t, include those dedicated to liquids, advanced condensed matter texts and statistical physics textbooks. This list has a notable outlier: the Statistical Physics textbook by Landau and Lifshitz. Landau and Lifshitz discuss general thermodynamic properties of liquids and explain why they *can not* be calculated, contrary to solids and gases. The reason is the combination of strong interactions and dynamical disorder. Strong interactions mean that gas theories are inapplicable to liquids. Dynamical disorder and the absence of fixed positions imply that the harmonic approximation used in the solid state theory is seemingly inapplicable to liquids even as a rough approximation. This precludes using our well-developed theories designed for gases and solids or their modifications including the perturbation theory. Therefore, Landau and Lifshitz conclude, general results for thermodynamic properties of liquids are impossible to derive. As aptly summarised by Pitaevskii, liquids do not have a small parameter (differently to solids and gases).

The problems listed by Landau, Lifshitz and Pitaevskii are fundamental. They explain why liquids have not been understood at the level nowhere near close to solids and gases. In view of enormous progress of condensed matter research, it is perhaps striking to realise that we do not have a basic understanding of liquids as the third state of matter and certainly not on par with solids and gases. This was one of the reasons I decided to look into this problem.

A set of new results have emerged in the last few decades related to collective excitations in liquids, phonons. It has taken a combination of experiment, theory and modelling to understand phonons in liquids well enough to connect them to liquid thermodynamic properties. Recall that this connection between phonons and thermodynamic properties was the basis of the Einstein and Debye approach to solids which laid the foundations of the modern solid state theory. The upshot is that the liquid theory can still be constructed on the basis of phonons, but the key point is that, differently from solids, the phase space available to phonons is not fixed but is variable instead. In particular, this phase space reduces with temperature. This reduction quantitatively explains the experimental liquid data and in particular the reduction of liquid specific heat from the solidlike to the ideal gas value with temperature. It has taken another decade to obtain an independent verification of this theory.

The small parameter in the liquid theory is therefore the same as in the solid state theory: small phonon displacements. However, in important difference to solids, this small parameter operates in a variable phase space. This addresses the problems stated by Landau, Lifshitz and Pitaevskii above.

Once looked from a longer-term perspective, the history of liquid theory and in particular the history of collective excitations in liquids reveals a fascinating story which involves physics luminaries and includes milestone contributions from Maxwell in 1867, followed by Frenkel and Landau. A separate and largely unknown line of enquiry aiming to connect phonons to liquid thermodynamics involved the work of Sommerfeld and Brillouin over 100 years ago. This was around the same time when the papers by Einstein and Debye were published and laid the foundations of the modern solid state theory based on phonons. Developing this line of inquiry in liquids had largely stopped soon after, and theories of liquids and solids diverged at the point of a fundamental approach. Whereas the solid state theory continued to be developed on the basis of phonons, theories of liquids started to use the approach based on interatomic interactions and correlation functions. As discussed in this book in detail, this approach faced several inherent limitations and fundamental problems.

Mathematics needed to discuss liquid theory is similarly interesting. For example, when we get to the equation written (but not solved) by Frenkel to describe phonons in liquids, we find that this equation was introduced by Kirchhoff in 1857 and discussed by Heaviside and Poincare.

This book reviews this research, starting with the early work by Sommerfeld and Brillouin and ending with recent independent verifications of the liquid theory. I follow the variation of the phase space in liquids in a wide range of parameters on the phase diagram, from low-temperature liquids to high-temperature supercritical fluids. I then come back to low-temperature viscous liquids approaching liquid-glass transition.

I also show how developments in liquid theory resulted in new unexpected insights. For example, the variation of the phase space available to phonons is related to liquid viscosity which quantifies the ability to flow. Viscosity has a minimum related to the crossover of particles dynamics from liquidlike to gaslike. It turns out that this minimum is governed by fundamental physical constants including the Planck constant. I show how this provides an answer to the question asked by Purcell and considered by Weisskopf in the 1970s, namely why viscosity never drops below a certain value comparable to that of water? I also show that viscosity minimum set by fundamental constants implies that liquid-based life (water-based life in our world) is well attuned to fundamental physical constants including the degree of quantumness of the physical world.

The liquid theory and its independent verifications discussed in this book focus on real liquids and their experimental properties rather than on model systems (such as hard-spheres and van der Waals models). This importantly differentiates this book from others.

The selection of topics in this book is helpfully aided and narrowed down by adopting a well-established approach in physics where an interacting system is fundamentally understood on the basis of its excitations. Consequently, a large part of this book discusses collective excitations in liquids and their relation to basic liquid properties throughout the history of liquid research. This shows how earlier and more recent ideas physically link to each other and in ways not previously considered.

In his well-known book “Gases, liquids and solids”, Tabor calls liquids “neglected step-child of physical scientists” and “Cinderella of modern physics” as compared to solids and gases. Although this observation was made nearly 30 years ago, Tabor would have reached the same conclusion regarding liquid thermodynamics on the basis of more recent literature. An important aim of this book is to make liquids a full family member on par with the other two states of matter, if only more sophisticated due to the liquid ability to sustain a variable phase space.

This book reaches out to scientists at any stage of their career who are interested in the states of matter and a history of a long-standing problem of understanding liquids theoretically. The second group are researchers and graduate students working in the area of liquids and related areas such as soft condensed matter physics and systems with strong dynamical disorder. The third group are lecturers looking to include liquids in the undergraduate and graduate courses such as statistical or condensed matter physics as well as students who can use this book as a reference.

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