Strongly interacting, dynamically disordered and with no small parameter, liquids took a theoretical status between gases and solids, with the historical tradition of hydrodynamic description as the starting point. We review different approaches to liquids as well as recent experimental and theoretical work, and propose that liquids do not need classifying in terms of their proximity to gases and solids or any categorizing for that matter. Instead, they are a unique system in their own class with a notably mixed dynamical state in contrast to pure dynamical states of solids and gases. We start with explaining how the first-principles approach to liquids is an intractable, exponentially complex problem of coupled non-linear oscillators with bifurcations. This is followed by a reduction of the problem based on liquid relaxation time τ representing non-perturbative treatment of strong interactions. On the basis of τ , solid-like high-frequency modes are predicted and we review related recent experiments. We demonstrate how the propagation of these modes can be derived by generalizing either hydrodynamic or elasticity equations. We comment on the historical trend to approach liquids using hydrodynamics and compare it to an alternative solid-like approach. We subsequently discuss how collective modes evolve with temperature and how this evolution affects liquid energy and heat capacity as well as other properties such as fast sound. Here, our emphasis is on understanding experimental data in real, rather than model, liquids. Highlighting the dominant role of solid-like high-frequency modes for liquid energy and heat capacity, we review a wide range of liquids: subcritical low-viscous liquids, supercritical state with two different dynamical and thermodynamic regimes separated by the Frenkel line, highly-viscous liquids in the glass transformation range and liquid-glass transition. We subsequently discuss the fairly recent area of liquid-liquid phase transitions, the area where the solid-like properties of liquids have become further apparent. We then discuss gas-like and solid-like approaches to quantum liquids and theoretical issues that are similar to the classical case. Finally, we summarize the emergent view of liquids as a unique system in a mixed dynamical state, and list several areas where interesting insights may appear and continue the extraordinary liquid story.

It is generally agreed that the supercritical region of a liquid consists of one single state (supercritical fluid). On the other hand, we show here that liquids in this region exist in two qualitatively different states: "rigid" and "nonrigid" liquids. Rigid to nonrigid transition corresponds to the condition τ≈τ(0), where τ is the liquid relaxation time and τ(0) is the minimal period of transverse quasiharmonic waves. This condition defines a new dynamic crossover line on the phase diagram and corresponds to the loss of shear stiffness of a liquid at all available frequencies and, consequently, to the qualitative change in many important liquid properties. We analyze this line theoretically as well as in real and model fluids and show that the transition corresponds to the disappearance of high-frequency sound, to the disappearance of roton minima, qualitative changes in the temperature dependencies of sound velocity, diffusion, viscous flow, and thermal conductivity, an increase in particle thermal speed to half the speed of sound, and a reduction in the constant volume specific heat to 2k(B) per particle. In contrast to the Widom line that exists near the critical point only, the new dynamic line is universal: It separates two liquid states at arbitrarily high pressure and temperature and exists in systems where liquid-gas transition and the critical point are absent altogether. We propose to call the new dynamic line on the phase diagram "Frenkel line".

Heat capacity of matter is considered to be its most important property because it holds information about system's degrees of freedom as well as the regime in which the system operates, classical or quantum. Heat capacity is well understood in gases and solids but not in the third main state of matter, liquids, and is not discussed in physics textbooks as a result. The perceived difficulty is that interactions in a liquid are both strong and system-specific, implying that the energy strongly depends on the liquid type and that, therefore, liquid energy can not be calculated in general form. Here, we develop a phonon theory of liquids where this problem is avoided. The theory covers both classical and quantum regimes. We demonstrate good agreement of calculated and experimental heat capacity of 21 liquids, including noble, metallic, molecular and hydrogen-bonded network liquids in a wide range of temperature and pressure.

Recently, we have proposed a new dynamic line on the phase diagram in the supercritical region. Crossing this line corresponds to the radical changes of the fluid properties. Here, we focus on the dynamics of model Lennard-Jones and Soft-Sphere fluids. We show that the change of the dynamics from the liquid-like to gas-like can be established on the basis of the velocity autocorrelation function and mean-square displacement. Using the rigorous criterion, we show that the crossover of particle dynamics and key liquid properties occurs at the same line. We further show that positive sound dispersion disappears in the vicinity of this line in both kinds of systems. The dynamic line bears no relationship to the existence of the critical point. We find that the region of existence of liquid-like dynamics narrows with the increase of the exponent of the repulsive part of inter-particle potential.

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Important properties of a particle, wave or a statistical system depend on the form of a dispersion relation (DR). Two commonly-discussed dispersion relations are the gapless phonon-like DR and the DR with the energy or frequency gap. More recently, the third and intriguing type of DR has been emerging in different areas of physics: the DR with the gap in momentum, or k-space. It has been increasingly appreciated that gapped momentum states (GMS) have important implications for dynamical and thermodynamic properties of the system. Here, we review the origin of this phenomenon in a range of physical systems, starting from ordinary liquids to holographic models. We observe how GMS emerge in the Maxwell-Frenkel approach to liquid viscoelasticity, relate the k-gap to dissipation and observe how the gaps in DR can continuously change from the energy to momentum space and vice versa. We subsequently discuss how GMS emerge in the two-field description which is analogous to the quantum formulation of dissipation in the Keldysh-Schwinger approach. We discuss experimental evidence for GMS, including the direct evidence of gapped Email addresses: matteo.baggioli@uam.es (Matteo Baggioli), professorvasin@gmail.com (Mikhail Vasin), brazhkin@hppi.troitsk.ru (Vadim Brazhkin), k.trachenko@qmul.ac.uk (Kostya Trachenko)DR coming from strongly-coupled plasma. We also discuss GMS in electromagnetic waves and non-linear Sine-Gordon model. We then move on to discuss the recently developed quasihydrodynamic framework which relates the k-gap with the presence of a softly broken global symmetry and its applications. Finally, we review recent discussions of GMS in relativistic hydrodynamics and holographic models. Throughout the review, we point out essential physical ingredients required by GMS to emerge and make links between different areas of physics, with the view that new and deeper understanding will benefit from studying the GMS in seemingly disparate fields and from clarifying the origin of potentially similar underlying physical ideas and equations. Keywords:12 Dispersion relations resulting from solving (56) for different τ c : (a) τ c = 5, (b) τ c = 0.58, (c) τ c = 0.54, (d) τ c = 0.51, (e) τ c = 0.4

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