Collective modes and thermodynamics of the liquid state

Strongly interacting, dynamically disordered and with no small parameter, liquids took a theoretical status between gases and solids. We review different approaches to liquids and propose that liquids do not need classifying in terms of their proximity to gases and solids. 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 $\tau$ representing non-perturbative treatment of strong interactions. On the basis of $\tau$, solid-like high-frequency modes are predicted and we review related recent experiments. We demonstrate how 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 affects liquid energy and other properties such as fast sound. Here, our emphasis is on real, rather than model, liquids. Highlighting the dominant role of high-frequency modes for liquid energy, 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 and liquid-glass transition. We also discuss liquid-liquid phase transitions where the solid-like properties of liquids have become further apparent. We then discuss gas-like and solid-like approaches to quantum liquids and persisting theoretical problems. We list areas where interesting insights may appear and continue the extraordinary liquid story.