Collective modes and thermodynamics of the liquid state
CCollective modes and thermodynamics of the liquid state
K. Trachenko and V. V. Brazhkin School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, UK and Institute for High Pressure Physics, RAS, 142190, Moscow, Russia
Strongly interacting, dynamically disordered and with no small parameter, liquids took a theo-retical status between gases and solids, with the historical tradition of hydrodynamic descriptionas the starting point. We review different approaches to liquids as well as recent experimental andtheoretical work, and propose that liquids do not need classifying in terms of their proximity togases and solids or any categorizing for that matter. Instead, they are a unique system in their ownclass 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, exponentiallycomplex problem of coupled non-linear oscillators with bifurcations. This is followed by a reduc-tion of the problem based on liquid relaxation time τ representing non-perturbative treatment ofstrong interactions. On the basis of τ , solid-like high-frequency modes are predicted and we reviewrelated recent experiments. We demonstrate how the propagation of these modes can be derivedby generalizing either hydrodynamic or elasticity equations. We comment on the historical trendto approach liquids using hydrodynamics and compare it to an alternative solid-like approach. Wesubsequently discuss how collective modes evolve with temperature and how this evolution affectsliquid energy and heat capacity as well as other properties such as fast sound. Here, our emphasis ison understanding experimental data in real, rather than model, liquids. Highlighting the dominantrole of solid-like high-frequency modes for liquid energy and heat capacity, we review a wide rangeof liquids: subcritical low-viscous liquids, supercritical state with two different dynamical and ther-modynamic regimes separated by the Frenkel line, highly-viscous liquids in the glass transformationrange and liquid-glass transition. We subsequently discuss the fairly recent area of liquid-liquidphase 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 thatare similar to the classical case. Finally, we summarize the emergent view of liquids as a uniquesystem in a mixed dynamical state, and list several areas where interesting insights may appear andcontinue the extraordinary liquid story. Contents
Introduction First-principles approach and its failure Relaxation time and phonon states in liquids:Frenkel’s reduction
Continuity of solid and liquid states andliquid-glass transition Hydrodynamic and solid-like elastic regimes ofwave propagation
Experimental evidence for high-frequencycollective modes in liquids Fast sound Generalized hydrodynamics Comment on the hydrodynamic approach toliquids Phonon theory of liquid thermodynamics
Heat capacity of supercritical fluids
Heat capacity of liquids and system’sfundamental length Evolution of collective modes in liquids:summary Viscous liquids
Liquid-glass transition a r X i v : . [ c ond - m a t . s o f t ] D ec Temperature
P r e s s u r es o l i dl i q u i d p c r p t r T t r T c r g a s FIG. 1: Colour online. Phase diagram of matter with tripleand critical points shown. Schematic illustration.
Phase transitions in liquids Quantum liquids: solid-like and gas-likeapproaches Mixed and pure dynamical states: liquids,solids, gases Conclusions and outlook References INTRODUCTION
Condensed matter physics as a term originated fromadding liquids to the then-existing field of solid statephysics. Proposals to do so precede what is oftenthought, and date back to the 1930s when J. Frenkelproposed to develop liquid theory as a generalization ofsolid state theory and unify the two states under the term“condensed bodies” [1]. At the same time, the seemingsimilarity of liquids and gases in terms of their abilityto flow has led to the unified term “fluids”. Such a dualclassification of liquids is more than just semantics: it hasgiven rise to two fundamentally different ways of describ-ing liquids theoretically in hydrodynamic and solid-likeapproaches. The phase diagram of matter in Figure 1highlights the intermediate location of liquids betweensolids and gases and hints at the duality of their physicalproperties that will come out in our detailed analysis.It is the intermediate state of liquids which has ulti-mately resulted in great difficulties when developing liq-uid theory because well-developed theoretical tools forthe two limiting states of gases and solids failed. It isalso the intermediate state of liquids and the combina- tion of solid-like and gas-like properties which continuesto be remarkably intriguing for theorists. According toFigure 1, one can start in the gas state above the criticalpoint, move to the liquid state and end up in the solidglass state (if crystallization is avoided) in a seeminglycontinuous way and without any qualitative changes ofphysical properties. This is a surprising observation froma theoretical point of view and signifies the intermediatestate of liquids and the duality of their physical proper-ties.At the end of this review, we will see that liquids neednot be thought of in terms of their proximity to solids orgases and do not require any other categorization: theyare self-contained systems with interesting, unique andrich dynamical and thermodynamic properties. In fact,understanding this richness helps better understand theproperties of gases and solids by delineating them as twolimiting states of matter in terms of dynamics and ther-modynamics.The long and extraordinary history of liquid researchincludes several notable discouraging assertions. One ofthe most important properties crucial to properly under-standing liquids is that they are strongly-interacting sys-tems. Particles in liquids are close enough to be withinthe reach of interatomic forces as in solids, resulting inthe condensed liquid state. The energy of a system with N particles and pair-wise interaction energy U ( r ) can bewritten as E = 32 N k B T + N ρ (cid:90) U ( r ) g ( r ) dV (1)where ρ is number density and g ( r ) is pair distributionfunction. U ( r ) is strong and system-dependent; consequently, E or other thermodynamic properties of the liquid arestrongly system-dependent. For this reason, Landau andLifshitz assert [2] (twice, in paragraphs 66 and 74) thatit is impossible to derive any general equation describ-ing liquid properties or their temperature dependence.Whatever approximation scheme or method used, anyapproach aimed at deriving a generally applicable resultusing Eq. (1), or evaluating the configurational part ofthe partition function, is destined to fail.The above problem does not originate in strongly-interacting solids because the smallness of atomic vi-brations around the fixed reference lattice, crystalline oramorphous, enables expansion of the potential energy inTaylor series. The harmonic term in this expansion, com-bined with the kinetic term, gives the phonon energy ofthe solid consistent with experimental heat capacities.These can be corrected by the next-order terms in theTaylor series for potential energy. Traditionally, this ap-proach is deemed inapplicable to liquids due to the ab-sence of fixed reference points around which an expan-sion can be made. The problem also does not originatein weakly-interacting gases: they have no fixed referencepoints but interactions are small so that the perturbationtheory is warranted.Liquids have neither the small displacements of solidsnor the small interactions of gases. Summarized aptly byLandau, liquids have no small parameter.For this reason, we are seemingly compelled to treatliquids as general strongly-interacting disordered sys-tems, where disorder is both static and dynamic, withno simplifying assumptions. In this spirit, large amountof work was aimed at elucidating the structure and dy-namics of liquids. In comparison, the discussion of liquidthermodynamic properties such as heat capacity is nearlynon-existent. Indeed, physics textbooks have very lit-tle, if anything, to say about liquid specific heat, includ-ing textbooks dedicated to liquids [2–12]. In an amusingstory about his teaching experience in the University ofIllinois (UIUC), Granato recalls living in fear about a po-tential student question about liquid heat capacity [13].Observing that the question was never asked by a total of10000 students, Granato proposes that “...an importantdeficiency in our standard teaching method is a failureto mention sufficiently the unsolved problems in physics.Indeed, there is nothing said about liquids [heat capac-ity] in the standard introductory textbooks, and little ornothing in advanced texts as well. In fact, there is littlegeneral awareness even of what the basic experimentalfacts to be explained are.” It is probably fair to say thatthe question of liquid heat capacity would be out of thecomfort zone not only for general condensed matter prac-titioners but also for many working in the area related tothe liquid state such as soft matter.Historically, thermodynamic properties of liquids havebeen approached from the gas state, a seemingly appro-priate approach in view of liquid fluidity. For example,common approaches start with the kinetic energy of thegas and aim to calculate the potential energy using theperturbation theory. The dynamical properties of liquidsare discussed on the basis of hydrodynamic theory wherethe elements of solid-like behaviour are introduced as asubsequent correction [4–12, 14]. This is in interestingcontrast to experiments informing us that liquids not farfrom melting points are close to solids in terms of density,bulk moduli, heat capacity and other main properties,but are very different from gases.The focus of this review is on understanding liquidthermodynamic properties such as heat capacity andtheir relationship to collective modes. To be more spe-cific and set the stage early, we show the experimentalspecific heat of liquid mercury in Figure 2. We observethat c v starts from around 3 k B just above the meltingpoint and decreases to about 2 k B at high temperature.As discussed below, this effect is very common and op-erates in over 20 different liquids we analyzed, includ-ing metallic, noble, molecular and network liquids, andis present in complex liquids. The decrease of c v inter- T/T m cv(kB) FIG. 2: Experimental specific heat of liquid mercury in k B units [21, 22]. The x -axis is in the relative temperature unitswhere T m is the melting temperature. estingly contrasts the temperature dependence of c v insolids which is either constant in the classical harmoniccase or increases due to anharmonicity or due to phononexcitations at low temperature. We also observe that liq-uid c v is significantly larger than the gas value of k B ina wide temperature range in Figure 2.Notably, the commonly discussed Van Der Waalsmodel of liquids gives c v = k B [2], the ideal gas value.The same result holds for another commonly discussedmodel of liquids, hard spheres, as well as for several othermore elaborate models. Clearly, real liquids have an im-portant mechanism at operation that significantly affectstheir c v and that is missed by several common liquidmodels.Notwithstanding the theoretical difficulties involved intreating liquids, we rely on the known result that low-energy states of a strongly-interacting system are collec-tive excitations or modes (throughout this review, we useterms “phonons”, “modes” and “collective excitations”interchangeably depending on context and common us-age). In solids, collective modes, phonons, play a centralrole in the theory, including the theory of thermodynamicproperties. Can collective modes in liquids play the samerole? It is from this perspective that we review collectivemodes in liquids. In our review, we emphasize the maindifferent approaches to collective modes in liquids andlist starting equations in each approach. We do not dis-cuss details of how the field has branched out over time;that formidable task is outside the scope of this paper.To a large extent, this was done in earlier textbooks andreviews [4–12, 14].We focus on real rather than model liquids, measurableeffects and take a pragmatic approach to understand themain experimental properties of liquids such as heat ca-pacity and provide relationships between different physi-cal properties. Throughout this review, we seek to makeconnections between different areas of physics that helpunderstand the problem. We are not trying to be com-pletely comprehensive, focusing instead on providing apedagogical introduction, interpreting previous basic re-sults and fundamental equations and explaining recentadvances. Our discussion includes original work not re-ported previously as well as results from our publishedwork.As already mentioned, the long and extraordinary his-tory of liquid research is related to problems of theoret-ical description. The fundamental problem of the first-principles description of liquids is not generally discussed,so we start with explaining that this problem is due tothe intractability of the exponential complexity of find-ing bifurcations and stationary points in the system ofcoupled non-linear oscillators. We then discuss how theproblem can be reduced using Frenkel’s idea of liquid re-laxation time. On this basis, several important assertionscan be made regarding the continuity of liquid and solidstates and the propagation of solid-like collective modesin liquids. We subsequently review how collective modescan be studied by either incorporating elastic effects inhydrodynamic equations or viscous effects in elasticityequations. We find the same results in both approaches,supporting the view that the historical hydrodynamic de-scription of liquids is not unique and that a solid-likedescription is equally justified. This assertion becomesmore specific when we review and comment on general-ized hydrodynamics. As far as liquid thermodynamics isconcerned, it turns out that the solid-like elastic regimeis the relevant one because high-frequency solid-like col-lective modes contribute most to the energy.We then proceed to review recent experimental ev-idence for high-frequency solid-like collective modes inliquids and discuss their similarity to those in solids.We subsequently discuss how the evolution of collectivemodes in liquids can be related to liquid energy and heatcapacity in widely different liquid regimes: low-viscoussubcritical liquids; high-temperature supercritical gas-like fluids; highly-viscous liquids in the glass transfor-mation range; and systems at the liquid-glass transition.In all cases, high-frequency modes govern the main ther-modynamic properties of liquids such as energy and heatcapacity and affect other interesting effects such as fastsound.The solid-like properties of liquids have additionallybecome apparent in the recently accumulated and re-viewed data on liquid-liquid phase transitions. We finallydiscuss the gas-like and solid-like approach in quantumliquids and interesting issues regarding the operation ofBose-Einstein condensates in real liquids.At the end of this review we will see that most im-portant properties of liquids and supercritical fluids canbe consistently understood in the picture in which these systems are in notably mixed dynamical state. Therefore,the emergent picture of liquids is that they do not needclassifying on the basis of their proximity to fluid gases orsolids, or any other compartmentalizing for that matter.Instead, they should be considered as distinct systemsin the mixed state of particle dynamics, the state thatshould serve as a starting point for liquid description.Moreover, we will see that appreciating the mixed stateof particle dynamics in liquids helps understand gasesand solids better as two limiting and dynamically pure states. This point is particularly useful for understand-ing the supercritical matter.We conclude with possible future work which maybring new understanding and advance the remarkable liq-uid story.Before we start, we comment on several terms used inthis review. Traditionally, the term “liquids” is used forsubcritical conditions on the phase diagram. The systemsabove the critical point are often referred to as “supercrit-ical fluids”. We continue to use these terms in our reviewwhere we also propose that the supercritical system infact consists of two states in terms of particle dynamicsand physical properties: a “rigid liquid-like” state be-low the Frenkel line and a “non-rigid gas-like fluid” stateabove the line. We use the term “glass” to commonly de-note a very viscous liquid which stops flowing at the typ-ical experimental time scale. The term “viscous liquid”commonly refers to liquids in the glass transformationrange, implying viscosity considerably higher than thatin, for example, water at ambient conditions. The termis quantitatively defined at the beginning of the section“Viscous liquids”. FIRST-PRINCIPLES APPROACH AND ITSFAILURE
The absence of a small parameter in liquids pointed outin the Introduction, is one perceived reason that makesthe theoretical description of liquids difficult. It tells uswhy perturbation-based approaches that are successfulin solids and gases do not work in liquids. Yet it is in-teresting to explore the actual reason for the difficultyof constructing a first-principles theory of liquids usingthe same microscopic approach as in the solid theory. Asfar as we know, this point is not discussed in textbooks[1–12].Below we show that the challenge for the first-principles description of liquids can be well formulatedin the language of non-linear theory where it acquires aspecific meaning. In this language, the challenge is re-lated to the intractability of the exponentially complexproblem involved in solving a large number of couplednon-linear equations.First-principles treatment of collective modes in a solidis based on solving coupled Newton equations of motionfor N atoms. We assume that the atoms oscillate aroundfixed lattice points q i , and introduce atomic coordinates q i and displacements x i = q i − q i . The potential energyis expanded in series as far as quadratic terms: U = U + 12 (cid:88) ij k ij x j x k (2)Writing the equations of motion as (cid:88) i m i ¨ x i + k ji x i = 0 (3)and seeking the solutions as x k = b k exp( iωt ) gives thecharacteristic equation for the eigenfrequencies (cid:12)(cid:12) k ij − ω m i (cid:12)(cid:12) = 0 (4)Eq. (4) gives most detailed information about collec-tive modes in the system, and returns 3 N eigenfrequen-cies, ranging from the lowest frequency set by the sys-tem size to the largest frequency in the system, often re-ferred to as Debye frequency (note that Debye frequencyis the result of quadratic approximation to the energyspectrum, and is somewhat lower than the maximal fre-quency of the real spectrum). Each atomic coordinatecan be expressed as a superposition of normal coordi-nates as x k = (cid:88) α ∆ kα Θ α (5)where Θ α = Re ( C α exp( iω α t )) are normal coordinates, C α are arbitrary complex constants and ∆ kα are minorsof the determinant (4) [15]. This result is central for thedevelopment of many areas in the solid state theory.Note that the above treatment does not assume acrystalline lattice. Crystallinity, if present, is the nextstep in the treatment enabling to write the solution asa set of plane waves with x ∝ exp( ikan ), where k is thewavenumber and a is the shortest interatomic separation,and derive dispersion curves for model systems.To continue to use the first-principles description ofliquids, we need to account for particle rearrangementsin liquids. As discussed in the next section, particle dy-namics in the liquid consists of small solid-like oscillationsaround quasi-equilibrium positions and diffusive jumpsto new neighbouring locations. This corresponds to po-tential energy of the double-well form shown in Figure3 which endows particles with both oscillatory motionand thermally-induced jumps between different minima.Note that in an equilibrium liquid, each diffusing par-ticle visits many minima, hence the potential energy ismulti-well, however the minima and energy profiles can V x FIG. 3: Double-well potential describing the particle mo-tion in liquids and involving jumps between different quasi-equilibrium positions. be assumed to be close to their averages in a homoge-neous system so that the double-well potential in Figure3 suffices.To model the double-well energy, the harmonic expan-sion (2) needs to be extended to include higher terms, atwhich point the equations of motion become non-linear.The simpler form often considered includes the third andfourth powers of x i , “ − x + x ”: U = U + 12 (cid:88) ij k ij x j x k + (cid:88) ijl k ijl x i x j x l + (cid:88) ijlm k ijl x i x j x l x m + ... (6)or, if a symmetric form of U is preferred, the higher-order potential can be written in “ − x + x ” or similarlysymmetric form as in Figure 3.At small enough energy or temperature of particlesmotion, Eq. (6) is used to describe the effects of anhar-monicity of atomic motion in solids using the perturba-tion theory. The main results include the correction tothe Dulong-Petit result of solids, thermal expansion andmodification of the phonon spectrum, phonon scatteringand so on. Unfortunately, the quantitative evaluation ofanharmonicity effects has remained a challenge, with thefrequent result that the accuracy of leading-order anhar-monic perturbation theory is unknown and the magni-tude of anharmonic terms is challenging to justify [23–27]. Experimental data such as phonon lifetimes andfrequency shifts can provide quantitative estimates foranharmonicity effects and expansion coefficients in par-ticular, although this involves complications and limitsthe predictive power of the theory [23].The real problem is at higher energy where the anhar-monicity in Eq. (6) is not small and jumps between dif-ferent minima in Fig. 3 become operative, as they do inliquids. Here, the perturbation approach does not apply,and we enter the realm of non-linear physics [28, 29]. Theillustrative example is the simplest system of two coupledDuffing oscillators with the energy (see, e.g. [28]): E = (cid:88) i =1 (cid:18)
12 ˙ x i + α x i − β x i (cid:19) + (cid:15) x − x ) (7)and the equations of motion¨ x + αx + (cid:15) ( x − x ) − βx = 0¨ x + αx + (cid:15) ( x − x ) − βx = 0 (8)where (cid:15) is the coupling strength.This model is not integrable, and can not be solvedanalytically but using approximations only. However, asimpler model can written in terms of variables ψ n = √ ω (cid:0) ω x n + i dx n dt (cid:1) , where ω is the frequency of the un-coupled oscillator: i dψ dt = ω ψ + Ω2 ( ψ − ψ ) − α | ψ | ψ i dψ dt = ω ψ + Ω2 ( ψ − ψ ) − α | ψ | ψ (9)where the last terms represent the non-linearity and Ω = (cid:15) ω .Eqs. (9) is known as the discrete self-trapping (DST)model, and is one of the rare examples in non-linearphysics that are exactly solvable analytically. The im-portant results can be summarized as follows. At lowenergy, the stationary points on the map ( x, ˙ x ) (or onthe map of two other independent dynamical variables)do not change, and the motion remains oscillatory andsimilar to the linear case. The character of oscillationsqualitatively changes at a certain energy that depends on Ω α : the old stationary point becomes an unstable saddlepoint, and a new pair of stable stationary points emerge,separated by the energy barrier [28]. This corresponds tothe bifurcation point, the emergence of new solutions as aresult of changes of parameters in the dynamical system.This is illustrated in Figure 4.An accompanying interesting insight is that contraryto the linear harmonic case, the energy is not equallypartitioned between the oscillating points but can local-ize at one point, reflecting the more general insight thatthe superposition principle no longer works in non-linearsystems in general.The importance of the above result is that it demon-strates that the first-principles treatment of the non-linear equations of motion gives rise, via the bifurcation b uuvacc b FIG. 4: Colour online. The phase portrait of the discretetrapping model (9) at two different energies. Independentvariables u and v are functions of ψ and ψ in Eq. (9),and describe the trajectories of two coupled non-linear oscil-lators. At low energy (top), two stationary points “a” and“b” remain unchanged as in the case of two coupled linearoscillators, corresponding to weak non-linearity. At high en-ergy (bottom), the bifurcation takes place: point “a” becomesan unstable saddle point and two new stationary points “c”appear. Schematic illustration, adapted from Ref. [28]. at high energy, a new qualitatively different solution: in-stead of oscillating around a fixed position at low energyas in a solid, a particle starts to move between two sta-ble stationary points at high energy, corresponding tothe liquid-like motion of particles between two minimain Figure 3. It proves that in the most simple non-linearsystem, the liquid-like motion emerges as a bifurcation ofthe solid-like solution.The DST model (9) is not identical to the originalsimple system of coupled Duffing oscillators (8). Thedifference with the DST model is that, due to the non-integrability of (8), islands of chaotic dynamics appearon the phase map and grow with the system energy. Theexcitations in the original model (8) can only be foundusing approximate techniques. However, the DST modelis close to (8) for small oscillation amplitudes and smallcouplings (cid:15) . This proximity between the two models isused to assert the same qualitative result, the emergenceof the bifurcation of solutions.We note the result from this discussion to which wereturn below: the bifurcation in the original model (8)emerges at energies E ∝ (cid:15) or amplitudes x ∝ (cid:15) , theresult which is not unexpected: the energy of couplingneeds to be surmounted in order to break away from thelow-energy solid-like solution.The real problem appears when the number of non-linear oscillators, N , increases. The analysis of N = 3non-linear coupled oscillators is complicated from theoutset by the fact that the corresponding DST modelis non-integrable to begin with. The approximations in-volved in the increasingly complicated analysis of station-ary states, new bifurcations emerging from these statesand corresponding collective modes become harder tocontrol. The results from computer modeling indicatethe emergence of many unanticipated modes and chaoticbehaviour at higher energy. The problem significantlyincreases for N = 4, including finding new stationarypoints and related collective modes, analyzing non-trivialbranching of next-generation bifurcations and so on. Forlarger N , only approximate qualitative observations canbe made regarding the energy spectrum, energy localiza-tion and emerging collective modes. This is done on thebasis of approximations and insights from N = 2 − N . The problem offinding stationary states, bifurcations, collective modesand their evolution with the system’s energy is exponen-tially complex and intractable for arbitrary N [28].Therefore, the failure of the first-principles treatmentof liquids at the same level as Eq. (3) for solids has itsorigin in the intractability of the exponentially complexproblem of calculating bifurcations, stationary points andcollective modes in a large system of coupled non-linearequations. RELAXATION TIME AND PHONON STATES INLIQUIDS: FRENKEL’S REDUCTION
It is fitting to discuss terms such as “collective modes”,“phonons” and other quasi-particles in relation to J.Frenkel’s work because he was involved in coining anddisseminating these terms. For example, the term“phonon”, attributed to Tamm, first appeared in printin Frenkel’s 1932 publication [30].Frenkel’s ideas occupy a significant part of our discus-sion. This might appear unusual to the reader, in viewthat this is not the case in other liquid textbooks [2–12].Frenkel’s work is not unknown but why would we wantto delve into it in detail now? We find that many discus-sions of liquids either do not mention Frenkel’s work (see,e.g., Refs. [16–20]) or mention it in an irrelevant con-text, yet they develop many ideas which, when strippedof details, are essentially due to Frenkel to a large extent.This will become apparent in this review. More impor-tantly, we find that, combined with recent experimentalevidence, Frenkel’s work related to collective modes inliquids gives a constructive tool to develop a predictive thermodynamic theory of liquids.Frenkel proposed a number of new ideas of how to un-derstand liquids emphasizing their “gas-like” and “solid-like” properties [1]. Some of the ideas such as the “holetheory” of liquids were not followed or developed, per-haps for the reason that the picture was qualitative andwithout links to experimental data. It should be notedthat the experimental data on liquids at the time wasonly very basic so Frenkel’s theoretical work was trulypioneering. However, other ideas discussed in Frenkelbook and his earlier papers on liquids transformed thefield in a way which is not fully appreciated even today.This transformation proceeded slowly and sporadicallyover the last 80–90 years since Frenkel’s work, duringwhich alternative approaches to liquids were developingand Frenkel’s ideas forgotten and surfaced anew morethan once (see, e.g., Refs. [16, 18, 20]). In our view,Frenkel was too ahead of his time. A transformative idea,proposed and experimentally confirmed within a gener-ation of scientists has a larger chance of succeeding ascompared to the Frenkel’s case where the new idea wasproposed long before its confirmation. For example, hisproposal that liquids are able to support solid-like longi-tudinal and transverse modes with frequencies extendingto the highest Debye frequency implies that liquids arejust like solids (solid glasses) in terms of their ability tosustain collective modes. Therefore, main liquid proper-ties such as energy and heat capacity can be describedusing the same first-principles approach based on collec-tive modes as solids - an assertion that is considered veryunusual. The evidence for this has come only recently be-cause liquids turned out to be too hard to probe experi-mentally. The evidence has started to mount only afterpowerful X-ray synchrotrons were deployed, and morethan 80 years after Frenkel’s first published paper on thesubject.Frenkel’s work on liquids is interestingly described bySir N. F. Mott [31]:“Frenkel was a theoretical physicist. By this I amstressing that he was primarily and most of all interestedin what is happening in real systems, and the mathe-matics he used served his physics and not otherwise as issometimes the case for the modern generation of scien-tists... He asks: ‘what is really happening and how canthis be explained?’ ”
Liquid relaxation time and phonon states
Throughout this paper, we are using terms such ascollective modes and phonons inter-changeably. Theirmeaning will be clarified in the later sections where wewill also comment on the issue of dissipation of harmonicwaves in disordered systems including glasses and liquids.Dating to 1926 [32] and developed in his later book [1],main ideas of Frenkel on liquids preceded the advance of
FIG. 5: Colour online. Illustration of a particle jump betweentwo quasi-equilibrium positions in a liquid. These jumps takeplace with a period of τ on average. the non-linear theory discussed earlier. Frenkel’s discus-sion includes many important ideas, of which we reviewonly those relevant to understanding collective modesand different regimes of wave propagation in liquids.Frenkel was naturally led to liquid dynamics by hiswork on defect migration in solids, and viewed the twoprocesses as sharing important qualitative properties.The migration rate of defects in a solid (crystalline oramorphous) is governed the potential energy barrier U set by the surrounding atoms. At fixed volume of the“cage” formed by the nearest neighbours, U is very largefor the diffusion event to occur in any reasonable time.However, the cage thermally oscillates and periodicallyopens up fast local diffusion pathways. If ∆ r is the in-crease of the cage radius required for the atom to jumpfrom its case (see Figure 5), U is [1]: U = 8 πGr ∆ r (10)where r is the cage radius and G is shear modulus. Notethat when a sphere expands in a static elastic medium, nocompression takes place at any point. Instead, the systemexpands by the amount equal to the increase of the spherevolume [1], resulting in a pure shear deformation. Thestrain components u from an expanding sphere (notingthat u → r → ∞ ) are u rr = − b/r , u θθ = u φφ = b/r [33], giving pure shear u ii = 0. As a result, theenergy to statically expand the sphere depends on shearmodulus G only.Frenkel considered the above picture applicable to liq-uids as well as solids, and introduced liquid relaxationtime τ as the average time between particle jumps at onepoint in space in a liquid.The range of τ is bound by two important values. Ifcrystallization is avoided, τ increases at low tempera-ture until it reaches the value at which the liquid stopsflowing at the experimental time scale, corresponding to τ = 10 − s and the liquid-glass transition [34, 35]. Athigh temperature, τ approaches its minimal value givenby Debye vibration period, τ D ≈ . τ , no parti-cle rearrangements take place. Hence, the system is asolid glass describable by Eqs. (3,4) and supports onelongitudinal mode and two transverse modes. At timeslonger than τ , the system is a flowing liquid, and hencedoes not support shear stress or shear modes but one lon-gitudinal mode only as any elastic medium (in a denseliquid, the wavelength of this mode extends to the short-est wavelength comparable to interatomic separations asdiscussed below). This is equivalent to asserting that theonly difference between a liquid and a solid glass is thatthe liquid does not support all transverse modes as thesolid, but only those above the Frenkel frequency ω F : ω > ω F = 1 τ (11)where we omit the factor of 2 π in ω = πτ for brevityand for the reason that in liquids the range of τ spans16 orders of magnitude, making a small constant factorunimportant.Eq. (11) implies that liquids have solid-like ability tosupport shear stress, with the only difference that thisability exists not at zero frequency as in solids but atfrequency larger than ω F (below we often use the term“solid-like” to denote the property in (11)). This was anunexpected insight at the time, and took many decades toprove experimentally as discussed below. It also posed afundamental question about the difference between solidsand liquids: liquids are different from solids by the valueof ω F only which is a quantitative difference rather thana qualitative one. In Frenkel’s view, this reflected thecontinuity of liquid and solid states, the question that isstill debated in the context of the problem of liquid-glasstransition. We will discuss this in the next sections.The longitudinal mode remains propagating in theFrenkel’s picture based on τ : density fluctuations existin any interacting system. We will see below that inreal dense liquids, experiments have ascertained that thelongitudinal vibrations extend to the largest Debye fre-quency as in solids. However, the presence of relaxationprocess and τ differently affects the propagation of thelongitudinal collective modes in different regimes ω > τ and ω < τ , as discussed in the next sections.We note that the separation of particle motion in theliquid into oscillatory and diffusive jump motion workswell for liquids with large τ (or viscosity, see next sec-tion). For smaller τ at high temperature, jumps canbecome less pronounced and oscillations increasingly an-harmonic. The disappearance of oscillatory componentof particle motion can be related to the Frenkel line dis-cussed in the later section.We also note that the concept of τ implies averagerelaxation time. In real liquids, there is a distribution ofrelaxation times as is widely established in experimentssuch as dielectric spectroscopy (see, e.g., Ref. [37]). Relationship to Maxwell relaxation theory
Here, we discuss the important relationship betweenthe analysis of Frenkel and Maxwell. Maxwell proposedthat a body is generally capable of both elastic and vis-cous deformation and, under external perturbation suchas shear stress, the total strain is the sum of viscous andelastic strains [38]. The y − gradient of horizontal velocity v x due to viscous deformation is ∂v x ∂y = P xy η , where P xy is shear stress and η is viscosity. The gradient of velocitydue to elastic deformation is ∂v x ∂y = G dP xy dt where G isshear modulus. When both viscous and elastic deforma-tions are present, the velocity of a layer v x is the sum ofthe two velocities, giving: ∂v x ∂y = 1 G dP xy dt + P xy η (12)The presence of both viscous and elastic response hasbeen subsequently called “viscoelastic” response, and iscommonly used at present.When external perturbation stops and v x = 0, Eq.(12) gives P xy = P exp (cid:18) − tτ M (cid:19) τ M = ηG (13)where τ M is Maxwell relaxation time.Frenkel has proposed that the time constant in Eq.(13), τ M , is related to liquid relaxation time τ he intro-duced (the time between particle rearrangements), andconcluded that τ M ≈ τ . Then, relaxation of shear stressin a viscoelastic liquid is exponential with Frenkel’s liquidrelaxation time τ : P xy = P exp (cid:18) − tτ (cid:19) (14)The second equation in (13) where τ is used instead of τ M is often called the Maxwell relationship: η = G ∞ τ (15)Here, G ∞ is the “instantaneous” shear modulus. G ∞ is understood to be the shear modulus at high frequency at which the liquid supports shear stress. In practice,this frequency can be taken as the maximal frequency ofshear waves present in the liquid, comparable to Debyefrequency [39].The activation energy for particle jumps in the liquidcan be calculated using Eq. (10), but with the provisothat G is the shear modulus at high-frequency.Experimentally, shear stress and various correlationsin viscous liquids (liquids where τ (cid:29) τ D ; see Section“Viscous liquids” for more detailed discussion below) andglasses decay according to the stretched-exponential re-laxation (SER) law rather than pure exponential as inEq. (14): f ∝ exp (cid:32) − (cid:18) tτ (cid:19) β (cid:33) (16)where f is a decaying function such as P xy in (14) and β is the stretching parameter conforming to 0 < β < τ in both (14) and (16). Frenkel reduction
It is interesting to discuss the meaning of Frenkel’s the-ory from the point of view of a first-principles descriptionof liquids. This theory is not a first-principles descriptionat level (3) and (4) but, as discussed in the earlier section,the first-principles treatment of liquid collective modes isexponentially complex and not tractable. Instead, thisapproach singles out the main physical property of liq-uids ( τ , or viscosity, see Eq. (15)) which governs therelative contributions of oscillatory and diffusive motionand which ultimately controls the phonon states in theliquid. This reduces the exponentially large problem toone physically relevant parameter. We call it the “Frenkelreduction” [45].Implicit in this reduction is a physically reasonableassumption that quasi-equilibrium states and the localparticle surroundings of jumping atoms in a homoge-neous liquid are equivalent, and that fluctuations in astatistically large system can be ignored [2]. In the lan-guage of non-linear theory, the reduction lies in assumingthat emerging new bifurcations and stationary states at0all generations produce physically equivalent states onaverage. This implies that as temperature (or energy)increases, the conditions governing particle jumps canbe considered approximately the same everywhere in thesystem. Therefore, particle dynamics is governed by thetemperature-activated jumps as the dynamics of pointdefects in solids: τ = τ D exp (cid:18) UT (cid:19) (17)where U is given by (10).It is generally agreed that τ and viscosity in liq-uids are indeed governed by the temperature-activatedprocess, with a caveat that U can include an addi-tional temperature-dependent term due to cooperativityof molecular relaxation, in which case τ grows faster-thanArrhenius (“super-Arrhenius”) as discussed below. Thiscooperative process is of the same nature as the one gov-erning the non-exponentiality of relaxation in (16).We recall the result from the non-linear theory that abifurcation emerges when the energy of the particle be-comes comparable to the coupling energy between twonon-linear oscillators. Noting that this result is derivedapproximately, we can relate the coupling energy to theactivation energy given in (10). Indeed, the coupling en-ergy in the system of non-linear equations is the energythat a particle needs to escape a bound state with an-other particle. This energy is of the same nature andorder of magnitude as that needed to break the atomiccage shown in Figure 5. Therefore, the approximation inthe Frenkel theory is of the same nature as the one in thenon-linear theory.In our discussion of generalized hydrodynamics below,we will see that the introduction of relaxation process andsolid-like features in the hydrodynamic equations is doneat the same level as in Eq. (14) in the Frenkel theory, byassuming the exponential decay of different correlationfunctions with the decay time τ .We emphasize that τ is readily measured using sev-eral well-established experiments including dielectric re-laxation experiments, NMR, positron annihilation spec-troscopy and so on. τ can also be derived from viscositymeasurements using Eq. (15) using widely available tech-niques including the classic Stokes experiments applica-ble to many types of liquids including at high pressureand temperature [47]. τ can also be calculated in molec-ular dynamics simulations as, for example, time decayof various correlation functions. In Figure 6 we show τ measured in salol over many orders of magnitude as anexample, and comment on it in the next section.In essence, Frenkel reduction introduces a cutoff fre-quency ω F (see 11) above which the liquid can be de-scribed by the same first-principles equations of motionas the solid in Eqs. (3) and (4). Therefore, liquid col-lective modes include both longitudinal and transverse - 1 ) log[ t ( s)] FIG. 6: Relaxation time of salol measured in dielectric relax-ation experiments [56]. modes with frequency above ω F in the solid-like elasticregime and one longitudinal hydrodynamic mode withfrequency below ω F (shear mode is non-propagating be-low frequency ω F as discussed below).Recall Landau’s assertion that a thermodynamic the-ory of liquids can not be developed because liquids haveno small parameter. How is this fundamental problemaddressed here? According to Frenkel reduction, liquidsbehave like solids with small oscillating particle displace-ments serving as a small parameter. Large-amplitude dif-fusive particle jumps continue to play an important role,but do not destroy the existence of the small parame-ter. Instead, the jumps serve to modify the phonon spec-trum: their frequency, ω F , sets the minimal frequencyabove which the small-parameter description applies andsolid-like modes propagate.This approach is therefore a method of non-perturbative treatment of strong interactions, the centralproblem in field theories and other areas of physics [45].It is markedly different from any other method of treat-ing strong interactions contemplated in areas outside ofliquids. CONTINUITY OF SOLID AND LIQUID STATESAND LIQUID-GLASS TRANSITION
The picture of liquid based on relaxation time τ has anotable consequence for liquid-solid transitions. In 1935,Frenkel published an article in Nature entitled, “Conti-nuity of the solid and the liquid states”, [48] where heproposed and later developed [1] an argument that liq-uids and solids are qualitatively the same. This followsfrom the concept of τ : as τ increases on lowering the tem-1perature beyond the experimental time frame, the liquidbecomes frozen glass, and supports shear modes at allfrequencies including at zero frequency. Hence, liquidsand solids are different in terms of τ only, i.e. quantita-tively, but not qualitatively. Frenkel subsequently statedthat “classification of condensed bodied into solids andliquids [has] a relative meaning convenient for practicalpurposes but devoid of scientific value” [1], an assertionthat many would find unusual today let alone then.This idea was quickly criticized by Landau [49, 50]on the basis that the liquid-crystal transition involvessymmetry changes and therefore can not be continu-ous according to the phase transitions theory. This de-bate unfortunately reflected a misunderstanding becauseFrenkel was emphasizing supercooled liquids that be-comes glasses on cooling, rather than crystals [1].Remarkably, essentially the same debate is still contin-uing in the area of liquid-glass transition where one of themain discussed questions is whether a phase transition isinvolved [34, 35, 51–55]? According to the large set of ex-perimental data, liquids and glasses are structurally iden-tical, and liquid-glass transition does not involve struc-tural changes. Yet at the glass transition temperature T g the heat capacity changes with a jump, seemingly provid-ing support to the thermodynamic signature of the glasstransition. Here, T g is defined as temperature at which τ exceeds the experimental time frame, τ = 10 − s, corresponding to liquid becoming frozen in terms ofparticle rearrangements during the observation period.We will return to the question of heat capacity jumpat T g when we discuss thermodynamic properties of vis-cous liquids. Here we note that although few considerthe jump of heat capacity at T g as a phase transition,versatile proposals were related to a possible phase tran-sition at lower temperature T < T g [34, 35, 51–55]. Thepossibility of this was suggested by the Vogel-Fulcher-Tammann (VFT) temperature dependence of τ : τ ∝ exp (cid:18) AT − T (cid:19) (18)where A and T are constants. τ in the VFT law diverges at T , and the same appliesto viscosity η according to Eq. (15). This led to pro-posals that the “ideal” glass transition takes place at T to the ideal glass state. The transition and the state areostensibly not seen because its observation is suppressedby very slow relaxation process below T g , and remain tobe of unknown nature [34, 35, 51–55].An example of the super-Arrhenius behaviour is shownin Figure 6 for a commonly measured glass-forming sys-tem, salol [56]. Here and in other cases, τ is described bythe VFT dependence fairly well, although a more carefulexperimental analysis revealed that on lowering the tem-perature, τ crosses over from the VFT to Arrhenius (ornearly Arrhenius) behavior [57–62]. This takes place at about midway of the glass transformation range where τ ≈ − s, i.e. above T g and hence well above T .Known more widely in the experimental community ascompared to theorists, the crossover removes the basis forconsidering divergences and a possible thermodynamicphase transition at T .An interesting question is what causes the crossoverfrom the VFT law at high temperature to nearly Ar-rhenius at low. A useful insight comes from the obser-vation that a sudden local jump event such as the oneshown in Figure 5 induces an elastic wave with a wave-length comparable to interatomic separation and cagesize. This wave propagates in the system and affectsrelaxation of other events, setting the cooperativity ofmolecular relaxation. As discussed in the next section,being a high-frequency wave, it propagates in the solid-like elastic regime with the propagation length given byEq. (26): d = λωτ ≈ cτ (19)where c is the speed of sound, ω is frequency and λ iswavelength. As discussed in the next section, d increaseswith τ in this regime, in contrast to the propagationlength of the commonly considered hydrodynamic waves[3].At high temperature when τ ≈ τ D , d = cτ D ≈ a ,where a is interatomic separation. This means that thewave does not propagate beyond the nearest neighborsand that the relaxation is non-cooperative (independent)and is Arrhenius and exponential as a result. Impor-tantly, d increases on lowering the temperature because τ increases. This increases the cooperativity of molecu-lar relaxation [41] but only until d reaches system size L .Therefore, the crossover from the VFT law to Arrheniustakes place at τ = Lc , in quantitative agreement withexperiments [44]. HYDRODYNAMIC AND SOLID-LIKE ELASTICREGIMES OF WAVE PROPAGATION
As discussed above, liquids behave differently depend-ing on observation time or frequency. Frequencies ω > ω F and ω < ω F correspond to solid-like elastic regime ( ωτ >
1) and hydrodynamic regime ( ωτ < P = A exp( iωt ), giving (cid:18) G dP xy dt + P xy η (cid:19) exp( iωt ) = 1 η (1 + iωτ ) P (20)where we used η = Gτ .2For ωτ >
1, (20) gives η ( iωτ ) P = PG iω = G dPdt , orpurely elastic response. For ωτ <
1, (20) returns purelyviscous response, Pη .To discuss liquid’s ability to operate in both regimesdepending on ω F , we can either start with hydrodynamicequations and introduce the solid-like elastic response orstart with elasticity equations and introduce the hydro-dynamic response. The first method has received mostattention in the history of liquid research, and generallyforms the basis for a variety of approaches collectivelyknown as “Generalized Hydrodynamics” discussed in thelater section. The second method is not commonly dis-cussed and its implications are less understood.Below we consider important examples of the differencein which collective modes operate in the hydrodynamicand solid-like elastic regimes, and start with the secondmethod. Modifying elasticity: including hydrodynamics inelasticity equations
Condition (11), ωτ > ωτ (cid:29) ωτ > ωτ <
1. We willsee that dissipation length, the length over which an in-duced wave is dissipated due to viscous effects, behavesqualitatively differently in the two regimes.We consider both elastic and viscous response in theform equivalent to Eq. (12) dsdt = P η + 12 G dPdt (21)where s is shear strain and introduce the operator A = 1 + τ ddt (22)where τ = ηG from (15). Then, Eq. (21) can be writtenas dsdt = 12 η AP (23)If A − is the reciprocal operator to A , P = 2 ηA − dsdt .Because ddt = A − τ from Eq. (22), P = 2 G (1 − A − ) s .Comparing this with P = 2 Gs , we find that the presenceof relaxation process is equivalent to the substitution of G by the operator M = G (1 − A − ). The above constitutes the modification of the con-stituent elasticity equations by introducing the relaxationprocess in the liquid and τ , i.e. approach to liquids fromthe solid elastic state: P = 2 Gs → P = 2 G (1 − A − ) s (24)Let us now consider the propagation of the wave of P and s with time dependence exp( iωt ). Differentiationgives multiplication by iω . Then, A = 1 + iωτ , and M is: M = G iωτ (25)If M = R exp( iφ ), the inverse complex velocity is v = (cid:112) ρM = (cid:112) ρR (cos φ − i sin φ ), where ρ is den-sity. P and s depend on time and position x as f = exp( iω ( t − x/v )). Using the above expressionfor v , f = exp( iωt ) exp( − ikx ) exp( − βx ), where k = ω (cid:112) ρR cos φ and absorbtion coefficient β = ω (cid:112) ρR sin φ .Combining the last two expressions for k and β , we write β = π tan φ λ , where λ = πk is the wavelength.From Eq. (25), tan φ = ωτ . For high-frequency waves ωτ (cid:29)
1, tan φ ≈ φ = ωτ , giving β = πλωτ . Lets introducethe propagation length d = 1 /β so that f ∝ exp( − x/d ).Then, d = λωτπ . Therefore, this theory gives propagatingshear waves in the solid-like elastic regime ωτ (cid:29)
1, withthe propagation length d ≈ λ · ωτ ( ωτ (cid:29)
1) (26)We note that this result is derived for plane waves, andit approximately holds in disordered systems for wave-lengths that are large enough. At smaller wavelengthscomparable to structural inhomogeneities, d is reduceddue to the dissipation of plane waves in the disorderedmedium. The dissipation is related to how well the eigen-states of the disordered system can be approximated byplane waves (for more detailed discussion, see Ref. ([46]).In the hydrodynamic regime ωτ (cid:28)
1, we find φ = π and d = λ π . Different from the high-frequency case, thismeans that low-frequency shear waves are not propagat-ing (because they are dissipated over the distance com-parable to the wavelength), a result that is also knownfrom hydrodynamics [3].The consideration of the propagation velocity of lon-gitudinal waves involves the bulk modulus which can bewritten in the form L = K + K iωτ containing the non-zero static part as well as the frequency-dependent partas in (25). Repeating the same steps as above, the prop-agation length in the solid-like elastic regime ωτ (cid:29) ωτ (cid:28)
1, the propagation length becomes d ≈ λωτ ( ωτ (cid:28)
1) (27)Comparing Eqs. (26) and (27), we see that the twodifferent regimes give qualitatively different character ofwaves dissipation: the propagation length increases with τ and viscosity in the former, but decreases with τ andviscosity in the latter.The decrease of the propagation length with liquid vis-cosity in the commonly discussed hydrodynamic regimeis a familiar result from fluid mechanics [3]. On the otherhand, the increase of propagation length in the solid-likeelastic regime is less known.An important insight from this discussion is that thetwo regimes of waves propagation are different from thephysical point of view and yield qualitatively differentresults, including directly opposite results for the propa-gation length. This implies that essential physics in thehydrodynamic regime and its underlying equations cannot be extrapolated to the solid-like elastic regime (andvice versa). By extrapolating here we mean extendingthe hydrodynamic regime to large k and ω while keepingthe underlying physics and associated equations qualita-tively the same. We will return to this point below whenwe discuss the approach to liquids based on generalizedhydrodynamics.Our second case study is related to the crossover be-tween two regimes of propagation. In the solid-like elasticregime, the propagation velocity in the isotropic mediumis v = (cid:113) B + Gρ [33], where B and G are bulk and shearmoduli, respectively. This is the case for solids as well asliquids in the solid-like elastic regime where shear wavesabove ω F are propagating. In the hydrodynamic regimewhere no shear waves propagate as discussed above, thepropagation speed is v = (cid:113) Bρ , corresponding to G = 0.Therefore, Frenkel argued, the transition between the tworegimes results in the noticeable increase of the propaga-tion speed by a factor (cid:113) GB . The transition can beachieved by either changing τ at a given frequency by al-tering temperature or pressure, or by changing frequencyat fixed temperature and pressure.In the later section, “Fast sound”, we will revisit thiseffect on the basis of recent experimental results. Modifying hydrodynamics: including elasticity inhydrodynamic equations
Eqs. (22)-(24) modify (generalize) elasticity equationsby including relaxation and viscous effects in the liq-uid in the form of viscous flow at times longer than τ . Equally, Frenkel argued [1], one can generalize hy-drodynamic equations by endowing the system with thesolid-like property to sustain shear stress at times shorterthan τ . This idea is generally similar in its spirit to theapproach of Generalized Hydrodynamics that appearedlater (see “Generalized Hydrodynamics” section below),although Frenkel implemented the idea differently. Apartfrom the general interest, this implementation deservesattention because it is not discussed in traditional gener-alized hydrodynamics approaches [5, 7, 8].Lets write the Navier-Stokes equation as ∇ v = 1 η (cid:18) ρ d v dt + ∇ p (cid:19) (28)where v is velocity, p is pressure, η is shear viscosity, ρ isdensity and the full derivative is ddt = ∂∂t + v ∇ .Eqs. (22)-(24) account for both long-time viscosityand short-time elasticity. From (21)-(23), we see thataccounting for both effects is equivalent to making thesubstitution η → η + G ddt . Using η = Gτ from Eq. (15),the substitution becomes:1 η → η (cid:18) τ ddt (cid:19) (29)Using (29) in Eq. (28) gives η ∇ v = (cid:18) τ ddt (cid:19) (cid:18) ρ d v dt + ∇ p (cid:19) (30)Having proposed Eq. (30), Frenkel did not analyze itor its solutions. We do it below.We consider the absence of external forces, p = 0 andthe slowly-flowing fluid so that ddt = ∂∂t . Then, Eq. (30)reads η ∂ v∂x = ρτ ∂ v∂t + ρ ∂v∂t (31)where v can be y or z velocity components perpendicularto x .In contrast to the Navier-Stokes equation, the gener-alized hydrodynamic equation Eq. (31) contains the sec-ond time derivative of v and hence allows for propagatingwaves. Indeed, Eq. (31) without the last term reducesto the wave equation for propagating shear waves withvelocity c s = (cid:113) ητρ = (cid:113) Gρ . The last term represents dis-sipation. Using η = Gτ = ρc s τ , we re-write Eq. (31)as c s ∂ v∂x = ∂ v∂t + 1 τ ∂v∂t (32)4Seeking the solution of (32) as v = v exp ( i ( kx − Ω t ))gives the quadratic equation for Ω:Ω + Ω iτ − c s k = 0 (33)Equation (33) has purely imaginary roots if c s k < τ , approximately corresponding to the hydrodynamicregime ωτ <
1. Therefore, we find that shear waves arenot propagating in the hydrodynamic regime ωτ < c s k > τ (corresponding to the solid-like elasticregime ωτ > − i τ ± (cid:113) c s k − τ ,and we find v ∝ exp (cid:18) − t τ (cid:19) exp( iωt ) ω = (cid:114) c s k − τ (34)Eq. (34) describes propagating shear waves, contraryto the original Navier-Stokes equation. We therefore findthat shear waves are propagating in the solid-like elasticregime ωτ >
1, the same result we derived in the previ-ous section where elasticity equations were modified toincorporate fluidity.According to Eq. (34), the increase of τ or viscos-ity gives smaller wave dissipation (larger lifetime) in thesolid-like elastic regime ωτ >
1, contrary to the hydro-dynamic regime [3]. This is the same effect that we havediscussed in the previous section where we found the in-crease of the propagation length of shear waves with τ and viscosity (see Eq. (26)).We note that for large τ or viscosity, ω in Eq. (34)becomes ω = c s k as in the case of shear waves withno dissipation at all. These are solid-like elastic waveswith wavelengths extending to the shortest interatomicseparations and frequencies up to the highest Debye fre-quency as predicted in the solid-like elastic approach byEq. (11).We also note that ω of shear waves in Eq. (34) doesnot increase from 0 to its linear branch ω = c s k in ajump-like manner as follows from (11). Instead, startingfrom about ω = ω F = τ , ω gradually increases from thesquare-root dependence to the linear dependence ω = c s k at large τ . This is consistent with the experimentalresult showing a gradual increase of the speed of soundand shear rigidity with the wave frequency [85]. We willrevisit this point when we discuss the phonon approachto liquid thermodynamics.To derive the propagation of longitudinal waves, weneed to include the longitudinal viscosity in the Navier-Stokes equations and modify it similarly to (29), remem-bering that bulk viscosity is related to the bulk modulus which, in addition to frequency-depending term, alwayshas non-zero static term [1]. This will give propagatinglongitudinal waves in both solid-like elastic regime andin the hydrodynamic regime, in agreement with the re-sults in the previous section. We will not pursue thisderivation here.Therefore, we find that as far as wave propagation isconcerned, equations of hydrodynamics modified (gener-alized) to include solid-like elastic effects give the sameresults as equations of elasticity modified to include vis-cous effects.Interestingly, it is the approach of “Generalized hydro-dynamics” which historically received wide attention anddevelopment and has become a distinct area of research[5, 7, 8]. We will discuss this approach in the later sec-tion. This reflects the historical trend we alluded to inthe introduction: the community largely viewed liquidsas systems conforming to the hydrodynamic equation atthe fundamental level, with possible solid-like elastic ef-fects to be introduced, if needed, on top. To some ex-tent, this view was consistent with existing experimentsat the time that mostly probed low-energy properties ofliquids. As discussed in the next Section, high-energy ex-periments uncovering solid-like properties of liquids haveemerged relatively recently.It can be argued that the approach to liquids start-ing with the solid-like elastic description contains moreinformation about structure and dynamics and, there-fore, is more suited to discuss high-frequency dynamicsof liquids. This becomes particularly important for con-structing the phonon theory of liquid thermodynamicswhere high-frequency modes govern system’s energy andheat capacity as discussed in the later section. EXPERIMENTAL EVIDENCE FORHIGH-FREQUENCY COLLECTIVE MODES INLIQUIDS
Low-frequency collective modes, including familiarsound waves, are well understood in liquids. Yet thesemodes make a negligible contribution to liquid energyand heat capacity. Indeed, the liquid energy is almostentirely governed by high-frequency modes due to theapproximately quadratic density of phonon states. How-ever, the prediction of high-frequency solid-like modes inliquids in the regime ωτ > ωτ < glass [81, 82] for comparison. In Figure 8, we showthe dispersion curves recently measured in liquid Sn [72],Fe, Cu and Zn using the experimental setup to studyliquids with high melting points [73].In Figure 7, we observe a striking similarity betweenliquids and their polycrystalline and crystalline coun-terparts in terms of longitudinal and transverse disper-sion curves. We further note the similarity of disper-sion curves in liquids and solid glasses. Overall, Fig-ures 7 and 8 present an important experimental evidenceregarding collective excitations in liquids. We observethat despite topological and dynamical disorder, solid-like quasi-linear dispersion curves exist in liquids in awide range of k and up to the largest k correspondingto interatomic separations, as is the case in solids. No-tably, this includes both high-frequency longitudinal andtransverse modes.We comment on damping of collective modes in liquids.A conservative system, crystalline or amorphous, has itseigenmodes which are non-decaying. Indeed, Eq. (4)does not require system’s crystallinity. For a disorderedstructure, Eq. (4) gives eigenstates and eigenfrequen-cies corresponding to collective non-decaying excitations.For long wavelengths and small energies, these states aresimilar to harmonic plane waves and their damping indisordered systems is small. For short wavelengths, theeigenstates of the disordered system are different from theplane waves, and so damping of short-wavelength planewaves becomes appreciable. Yet the experimental dis-persion curves obtained by harmonic probes such as X-rays or neutrons show that high-frequency plane waves G a n (THz) q / q ( c )N a n (THz) q / q ( b ) S i O 2 g l a s s
E(meV) k ( n m - 1 ) ( a ) FIG. 7: Colour online. Experimental dispersion curves. (a):longitudinal (filled black bullets) and transverse (filled redbullets) dispersion curves in SiO glass [81]. (b): longitudinal(filled black bullets) and transverse (filled red bullets) excita-tions in liquid Na. Open diamonds correspond to longitudi-nal (black) and transverse (red) excitations in polycrystallineNa, and dashed-dotted lines to longitudinal (black) and trans-verse (red) branches along [111] direction in Na single crystal[70]. (c): longitudinal (black bullets) and transverse (red bul-lets) excitation in liquid Ga. The bullets are bracketed bythe highest and lowest frequency branches measured in bulkcrystalline β -Ga along high symmetry directions, with blackand red dashed-dotted lines corresponding to longitudinal andtransverse excitations, respectively [71]. Dispersion curves inNa and Ga are reported in reduced zone units. S n( a )
E(meV)E(meV)E(meV)E(meV)
F e( b ) C u( c ) k ( n m - 1 )k ( n m - 1 )k ( n m - 1 )Z n( d ) k ( n m - 1 ) FIG. 8: Colour online. Longitudinal (black and blue crosses)and transverse (filled red bullets) dispersion curves in (a) liq-uid Sn [72], (b) liquid Fe, (c) liquid Cu and (d) liquid Zn [73].Red filled triangles in (a) are the results from ab initio simu-lations [72]. Blue and black crosses correspond to recent andearlier experiments, respectively. Dashed lines indicate theslope corresponding to the hydrodynamic sound in the limitof low k . are propagating in liquids, as witnessed by the data inFigures 7 and 8. From the physical point of view, thisfollows from the fact that despite long-range disorder,a well-defined short-range order exists in liquids, glassesand other disordered systems, as is seen from the peaks ofpair distribution functions in the short as well as mediumrange. Therefore, high-frequency harmonic plane waves,even though damped, are able to propagate at least thedistance comparable to the typical length of the short-range order. We will find below that this length, theinteratomic separation, which is also the fundamentallength of the system, plays a profound role in govern-ing the thermodynamic properties of liquids.We have noted the similarity of vibrational propertiesbetween disordered liquids and their crystalline counter-parts. Interestingly, similarity (and the lack thereof) be-tween disordered glasses and their parent crystals havealso been widely discussed. The widely discussed “Bo-son” peak in the low-frequency range has been longthought to be present in glasses only but not in crys-tals and to originate from disorder. However, later work[83, 84] has demonstrated that similar vibrational fea-tures are present in crystals as well, provided glasses andcrystals have similar density. FAST SOUND
It is now good time to revisit the origin of fast soundmentioned earlier using detailed experimental data dis-cussed in the previous section.Starting from larger k -values, the measured speed ofsound often exceeds the hydrodynamic value. This isseen in Figure 8 where the hydrodynamic speed of soundis shown as a dashed line. The increase of the measuredspeed of sound over its hydrodynamic value is often calledas “fast sound” or “positive sound dispersion” (PSD).We recall Frenkel prediction discussed earlier: at highfrequency where liquid’s shear modulus becomes non-zero, the propagation velocity crosses over from its hy-drodynamic value v = (cid:113) Bρ to the solid-like elastic value v = (cid:113) B + Gρ [33, 34], where B and G are bulk and shearmoduli, respectively.The physical origin of the fast sound has remained con-troversial, including understanding relative contributionsof the above mechanism and other effects such as dis-order. Experimentally, the crossover of the longitudinalsound velocity from its hydrodynamic to solid-like elasticvalue has been been well-studied in viscous liquids wherethe system starts sustaining rigidity at MHz frequencies(see, e.g., Ref. [85], where fast sound is seen at fairlylarge wavelengths at which the liquid can be consideredas a homogeneous medium). It is generally agreed thatin this range of frequencies, fast sound originates fromthis mechanism [85].7At smaller wavelengths approaching the length ofmedium and short-range order, the wave feels structuralinhomogeneities, and disorder of liquids and glasses startsto affect the dispersion relationship. PSD, with the rela-tive magnitude of few per cent, was observed in a modelharmonic glass and attributed to the “instantaneous re-laxation” due to fast decay and dissipation of short-wavelength phonons in a disordered system [86]. Laterwork demonstrated that starting from mesoscopic wave-lengths, the effective speed of the longitudinal sound canalso decrease [87–89]. Different mechanisms and contri-butions to PSD were subsequently discussed [76, 90]. Theinstantaneous relaxation is likely to be significant closeto the zone boundary [81] (or the first Brillouin pseudo-zone, related to the short-range order in disordered sys-tems [76]), although large PSD in silica glass may berelated to the effect of mixing with the low-lying opticmodes. In water, fast sound was discussed on the basisof coupling between the longitudinal and transverse ex-citations, and it was found that the onset of transverseexcitations coincides with the inverse of liquid relaxationtime [79, 80], as predicted by (11).Recent detailed experimental data discussed in the pre-vious section enable us to directly address the origin ofthe fast sound and its magnitude. Combining v h = (cid:113) Bρ , v t = (cid:113) Gρ and v l = (cid:113) B + Gρ (see, e.g., Ref. [34]),where v h is the velocity of the low-frequency hydrody-namic sound, v t is the transverse sound velocity and v l is the longitudinal velocity from the measured dispersioncurves, we write v l = v h + 43 v t (35)We note that the expression v l = Bρ + v t is the iden-tity for isotropic solids, and also applies to liquids inwhich the longitudinal speed of sound changes from thehydrodynamic to solid-like elastic value due to the onsetof shear rigidity.Using the data from Refs. [72] and [73], we have taken v l and v t from the dispersion curves for Fe, Cu, Zn andSn shown in Figure 8 at k points where the observed PSDis maximal and where ω ( k ) ( E ( k )) is in the quasi-linearregime before starting to curve at large k . For Fe, Cu, Zn,we use the new data shown in blue in Figure 8 and con-sider the following k points: k = 7 . − (first point onthe transverse branch in Figure 8), k = 7 . − (secondpoint on the transverse branch) and k = 8 nm − (secondpoint on the transverse branch), respectively. For Sn,large PSD is seen at about k = 3 . − correspondingto the second point on the longitudinal branch in Figure8a. To find v t at this k , we extrapolated the higher-lyingtransverse points to lower k while keeping them parallelto the simulation points, yielding v t = 1220 ±
150 m/s. Using experimental v h and v t , we have calculated v l using Eq. (35). We show calculated and experimental v l in Table I below. v h v t v l (experimental) v l (calculated)[m/s] [m/s] [m/s] [m/s]Fe 3800 1870 ±
50 4370 ±
30 4370 ± ±
50 3890 ±
30 3875 ± ±
50 3330 ±
30 3350 ± ±
150 2890 ±
30 2820 ± v l and v l calculatedon the basis of v h and v t using Eq. (35) as discussed in thetext. The data for v l , v t and v h is from Refs. [72] and [73]. We observe in Table 1 that the calculated and experi-mental v l agree with each other very well. We thereforefind that the mechanism of fast sound based on the onsetof shear rigidity quantitatively accounts for the experi-mental data of real liquids in the wide range of k spanningmore than half of the first Brillouin pseudo-zone.It is interesting to discuss pressure and temperatureconditions at which the fast sound operates in this pic-ture. The above mechanism implies that the fast sounddisappears when the system loses shear resistance andtransverse modes at all available frequencies. As dis-cussed later, this takes place above the Frenkel line whichdemarcates liquid-like and gas-like properties at hightemperature including in the supercritical region.As already mentioned, other effects contributing toPSD can be operative, including the effects due to disor-der at large k . GENERALIZED HYDRODYNAMICS
In the earlier section, we have discussed modify-ing (generalizing) hydrodynamic equations by includingsolid-like elastic effects as one way to describe both elasticand hydrodynamic response of the liquid. “Generalizedhydrodynamics” as a distinct term refers to a number ofproposals seeking to achieve essentially the same result byusing a number of different phenomenological approaches[5, 7, 8]. One starts with hydrodynamic equations ini-tially applicable to low ω and k , and introduces a way toextend them to include the range of large ω and k .From the point of view of thermodynamics, account-ing for modes with high ω is important because thesemodes make the largest contribution to the system en-ergy. The contribution of hydrodynamic modes is negli-gible by comparison.Generalized hydrodynamics is a large field (see, e. g.,[5, 7, 8, 12]) which we can only discuss briefly emphasiz-ing key starting equations and schemes of their modifica-tion to include higher-energy effects, with the aim to offerreaders a feel for methods used and physics discussed.8The hydrodynamic description starts with viewing theliquid as a continuous homogeneous medium and con-straining it with continuity equation and conservationlaws such as energy and momentum conservation. Ac-counting for thermal conductivity and viscous dissipationusing the Navier-Stokes equation, the system of equationscan be linearized and solved. This gives several dissipa-tive modes, from which the evaluation of the density-density correlation function gives the structure factor S ( k, ω ) in the Landau-Placzek form which includes sev-eral Lorentzians: S ( k, ω ) ∝ γ − γ χk ω + ( χk ) +1 γ (cid:32) Γ k ( ω + ck ) + (Γ k ) + Γ k ( ω − ck ) + (Γ k ) (cid:33) (36)where χ is thermal diffusivity, γ = C p C v and dissipation Γdepends on χ , γ , viscosity and density.The first term corresponds to the central Rayleigh peakand thermal diffusivity mode. The second two terms cor-respond to the Brillouin-Mandelstam peaks, and describeacoustic modes with the adiabatic speed of sound c . Theratio between the intensity of the Rayleigh peak, I R ,and the Brillouin-Mandelstam peak, I BM , is the Landau-Placzek ratio: I R I BM = γ −
1. Applied originally to lightscattering experiments, Eq. (36) is also viewed as aconvenient fit to high-energy experiments probing non-hydrodynamic processes where the fit that may includeseveral Lorentzians or their modifications.Generalizing hydrodynamic equations and extendingthem to large k and ω is often done in terms of cor-relation functions. Solving the hydrodynamic Navier-Stokes equation for the transverse current correlationfunction J t ( k, t ), ∂∂t J t ( k, t ) = − νk J t ( k, t ), where ν is kinematic viscosity, gives for the Fourier transform J t ( k, ω ) a Lorentzian form similar to (36): J t ( k, ω ) = 2 v νk ω + ( νk ) (37)where ν is kinematic viscosity and v = J t ( k, t = 0).The generalization is done in terms of the memoryfunction K t ( k, t ) defined in the equation for J t ( k, ω ) as ∂∂t J t ( k, ω ) = − k t (cid:90) K t ( k, t − t (cid:48) ) J t ( k, t (cid:48) ) dt (cid:48) (38)where K t ( k, t − t (cid:48) ) is the shear viscosity function or thememory function for J t ( k, ω ) which describes its timedependence (“memory”). Introducing ˜ J t ( k, s ) as the Laplace transform J t ( k, ω ) = 2Re[ ˜ J t ( k, s )] s = iω and taking the Laplacetransform of (38) gives˜ J t ( k, s ) = v s + k ˜ K t ( k, s ) (39)The generalization introduces the dependence k and ω by writing ˜ K t ( k, s ) as the sum of real and imaginaryparts [ ˜ K t ( k, s )] s = iω = K (cid:48) t ( k, ω ) + iK (cid:48)(cid:48) t ( k, ω ). Then, J t ( k, ω ) = 2 v k K (cid:48) t ( k, ω )( ω + k K (cid:48)(cid:48) t ( k, ω )) + ( k K (cid:48) t ( k, ω )) (40)giving the generalized hydrodynamic description of thetransverse current correlation function with a resonancespectrum.Further analysis depends on the form of K t ( k, t ), whichis often postulated as K t ( k, t ) = K t ( k,
0) exp (cid:18) − tτ ( k ) (cid:19) (41)Eq. (41) decays with time relaxation time τ , and werecognize that this is essentially the same behavior de-scribed by earlier Eqs. (14) or (16), except the postulatedform also assumes k -dependence of τ . In generalized hy-drodynamics, Eq. (41) is used not only for K but also forseveral types of correlation and memory functions. Theseoften include modifications such as including more expo-nentials with different decay times in order to improvethe fit to experimental or simulation data.Mode-coupling schemes consider correlation functionsfor density and current density, factorise higher-ordercorrelation functions by expressing them as the prod-uct of two time correlation functions with coupling co-efficients in the form of static correlation functions, andgive a better agreement for the relaxation function ascompared to the single exponential decay model.Neglecting k -dependence of τ for the moment, tak-ing the Laplace transforms of (41) to find K (cid:48) t ( k, ω ) and K (cid:48)(cid:48) t ( k, ω ) and using them in (40) gives J t ( k, ω ) as [5] J t ( k, ω ) ∝ (cid:0) ω − (cid:0) k K t ( k, − τ (cid:1)(cid:1) + f ( τ, K t ( k, f is the non-essential function of τ and K t ( k, k K t ( k, > τ .This condition defines the high-frequency regime of wavepropagation in the solid-like elastic medium. Impor-tantly, this condition is essentially the same as the onewe derived from the generalized hydrodynamic equation9(32), as follows from the discussion between Eqs. (32)and (34).Similar expressions can be derived for the longitudinalcurrent correlation function which also includes a statictime-independent term which does not decay. This termcorresponds to non-zero bulk modulus which gives prop-agating longitudinal waves in the hydrodynamic regime,as discussed in the previous section.An alternative approach to generalize hydrodynamicsis to make a phenomenological assumption that a dynam-ical variable in the liquid is described by the generalizedLangevin equation: ∂a ( t ) ∂t + i Ω a ( t ) + t (cid:90) a ( t (cid:48) ) K ( t − t (cid:48) ) dt (cid:48) = f ( t ) (43)where the first two terms reflect the possibility of prop-agating modes, the third term plays the role of frictionwith the memory function K and f is the random force.This approach proceeds by treating a ( t ) not as a singlevariable but as a collection of variables of choice so that a ( t ) becomes a vector including, in its simplest forms,conserved density, current density and energy variables.These variables are further generalized to include theirdependence on wavenumber k . This gives a set of cou-pled equations solved in the matrix form. The set of dy-namical variables can be extended to include the stresstensor and heat currents. In this case, the generalizedviscosity is found to have the same exponential decayas in (41) once the stress tensor is explicitly introducedas a dynamical variable, the assumption is made regard-ing stress correlation function and a number of approxi-mations are made. Then, similar viscoelastic effects arefound as in the previous approach [5].Propagation of shear and longitudinal modes is alsodiscussed in the mode-coupling theories mentionedabove. The theory seeks to take a more general ap-proach in the following sense. Considering that corre-lation functions are due to density and current densitycorrelators, the theory represents ˜ K t ( k, s ) in (39) by thesecond-order memory functions M t ( k, t ) and M l ( k, t ) fortransverse and longitudinal currents, so that the trans-verse function ˜ J t ( k, s ) and longitudinal function ˜ J l ( k, s )acquire the forms of damped oscillators. ˜ J l ( k, s ) differsfrom ˜ J t ( k, s ) by the presence of non-zero static term,giving a finite static restoring force for the longitudinalmode. As in the previous considerations, this gives prop-agating longitudinal modes in the hydrodynamic regime.Rather than postulating the relaxation functions M t ( k, t )and M l ( k, t ) as in (41), the mode-coupling theory consid-ers higher-order correlation functions and approximatesthem by the products of two-time correlation functions.Memory functions can then be calculated using the re-sults from molecular dynamics simulations such as staticcorrelation functions and other parameters required as the input. For simple systems, the onset of shear wavepropagation can be related to certain shoulder-like fea-tures in the calculated memory function.The amount of current research in generalized hydro-dynamics has markedly decreased as compared to severaldecades ago [5]. Interestingly, the steer towards going be-yond the hydrodynamic description and generalized hy-drodynamics came from the experimental, and not the-oretical, community after the solid-like properties of liq-uids were discovered and problems related to the hydro-dynamic description of those properties became apparent[66, 70]. Some of the more recent examples include ex-ploring how hydrodynamic description gives rise to a sin-gle underlying relaxation process and accounting for theviscoelastic effects using several first frequency moments(see [91–93] and references therein). Other approachesassume ad hoc that more dynamical variables and theirsecond and third derivatives are involved in extrapolat-ing the hydrodynamic regime to high k and ω [94] and,following earlier proposals [95], use the generalized collec-tive modes schemes where the sum of exponentials suchas (41) is assumed to describe the decay of correlations.General disadvantages of this and similar schemes arerelated to the phenomenological and empirical nature ofthe method [66, 70].Continuing interest in generalized hydrodynamics isstimulated by fitting the experimental spectra where, forexample, the second-order memory function is assumedto take the exponential form (41) [76] or as a sum of twoor more exponentials [96]. COMMENT ON THE HYDRODYNAMICAPPROACH TO LIQUIDS
Challenges involved in generalized hydrodynamicswere appreciated by practitioners at the early stages ofdevelopment [5], including often phenomenological andempirical ways involved in extrapolating hydrodynamicdescription into the solid-like elastic regime. We do notreview these here, although we note the following. Gen-eralized hydrodynamics introduces k and ω -dependenciesin the liquid properties such as diffusion, viscosity, ther-mal conductivity, heat capacity and so on, with theaim to calculate and discuss these functions in the non-hydrodynamic regime. It is not entirely clear what isthe physical meaning of concepts such as diffusion or vis-cosity at large ω where the system’s response is elasticrather than viscous. Understanding physical effects atthese frequencies is important because short-wavelengthmodes govern most important system properties such asenergy.We question a more fundamental premise of the hy-drodynamic description of liquids: “The advantage ofapproaching the large ( k , ω ) region by generalizing thehydrodynamic description is that one maintains contact0with the long-wavelength, low-frequency region at allstages of the development. This gives insight to the struc-ture of the resulting equation” [5]. Although being ableto track the evolution of equations may be insightful insome cases, it may not be advantageous in general. Thereis no fundamental reason to designate the hydrodynamicapproach as the universally correct starting point. Thetraditional reason for the hydrodynamic approach to liq-uids is that they are flowing systems and therefore obeyhydrodynamic equations. As we have discussed above,this applies to times t > τ ( ω < ω F ) only whereas for t < τ ( ω > ω F ) the system is solid-like and can be de-scribed by solid-like equations. Furthermore, we haveseen that the same properties of collective modes are ob-tained by either starting with the hydrodynamic equa-tions and incorporating solid-like elastic effects or start-ing with the elasticity equations and incorporating thehydrodynamic fluidity.Instead, we propose that for the purposes of funda-mental microscopic description, liquids should be consid-ered for what they are: systems with molecular dynam-ics of both types, solid-like oscillatory motion and diffu-sive jumps, with relative weights of these motions chang-ing with temperature. As discussed below, these rela-tive weights govern most important system properties.In this approach, the hydrodynamic regime ( ωτ < ωτ >
1) can, and in manycases should, be considered separately and without nec-essarily seeking to extrapolate one regime onto the other.In addition to avoiding problems of ad-hoc extrapolationassumptions often present in generalized hydrodynamics,this approach has the added benefit of rigorously delin-eating different regimes of liquid dynamics where impor-tant properties are qualitatively different. This will be-come particularly apparent when we discuss the changeof dynamics in the supercritical region at the Frenkel line,the effect that the hydrodynamic description misses.The hydrodynamic and solid-like elastic descriptionsof liquids apply in their respective domains. It turns outthat it is the solid-like description that is relevant for con-structing the thermodynamic theory of liquids discussedin the next section. This is because high-frequency modesmake the largest contribution to the system energy dueto quadratic density of states ∝ ω , and propagate in thesolid-like elastic regime ωτ >
1. Importantly, this doesnot require extrapolations involved in the generalized hy-drodynamics approach.
PHONON THEORY OF LIQUIDTHERMODYNAMICSHarmonic theory
We have seen above that collective modes in liquidsinclude one longitudinal mode and two transverse modes propagating at frequency ω > ω F = τ in the solid-likeelastic regime. The energy of these modes is the liq-uid vibrational energy. In addition to oscillating, parti-cles in the liquids undergo diffusive jumps between quasi-equilibrium positions as discussed above. We write thetotal liquid energy as E = K + P l + P t ( ω > ω F ) + P d (44)In Eq. (44), K is the sum of all kinetic terms includingvibrational and diffusional components. In the classicalcase, K = N k B T , and does not depend on how the ki-netic energy partitions into oscillating and diffusive com-ponents. P l and P t ( ω > ω F ) are potential energies ofthe longitudinal mode and transverse phonons with fre-quency ω > ω F , respectively. For now, we tentativelyinclude in Eq. (44) the term P d , related to the energy ofinteraction of diffusing particles with other parts of thesystem. P d is understood to be part of system’s potentialenergy which is not already contained in the potential en-ergy of the phonon terms, P l and P t ( ω > ω F ). P d is smallcompared to other terms in (44) as discussed below.The smallness of P d can be discussed by approachingthe liquid from either gas or solid state. Lets consider adilute interacting gas where system’s potential energy isentirely given by the potential energy of the longitudinalmode, P l , with the available wavelengths that depend onpressure and temperature. The remaining energy in thesystem is the kinetic energy corresponding to the freeparticle motion, giving P d = 0. Density increase (andtemperature decrease) result in decreasing wavelength ofthe longitudinal mode until it reaches values compara-ble to solid-like interatomic separation a (see the earliersection “Experimental evidence for high-frequency col-lective modes in liquids). In this dense gas regime, thesystem’s potential energy is still given by P l , which isthe energy of longitudinal mode but now with the fullsolid-like spectrum of wavelengths ranging from the sys-tem size to a . Further density increase or temperaturedecrease result in the appearance of the solid-like oscilla-tory component of motion. This process is most conve-niently discussed above the critical point where no liquid-gas phase transition intervenes and where the crossoverfrom purely diffusive motion to combined diffusive andsolid-like oscillatory motion takes place at the Frenkelline discussed in later sections. The emergence of solid-like oscillatory component of particle motion is relatedto the emergence of transverse modes with frequency ω > ω F in Eq. (44). The potential energy of transversemodes now contributes to the system’s potential energy,and the remaining energy corresponds to the free particlemotion ( P d = 0 in Eq. (44)) as in the dense gas.We can also approach the liquid from the solid state.In the solid, the potential energy is the sum of potentialcomponents of longitudinal and transverse modes. Theemergence of diffusive motion in the liquid results in the1disappearance of transverse modes with frequency ω <ω F according to (11) and modifies the potential energy oftransverse modes to P t ( ω > ω F ) in Eq. (44). This impliessmallness of low-frequency potential energy of transversemodes: P t ( ω < ω F ) (cid:28) P t ( ω > ω F ), where P t ( ω < ω F ) isthe potential energy of low-frequency transverse modes.Instead of low-frequency transverse vibrations with po-tential energy P t ( ω < ω F ) in a solid, atoms in a liquid“slip” and undergo diffusive motions with frequency ω F and associated potential energy P d , hence P d ≈ P t ( ω <ω F ). Combining this with P t ( ω < ω F ) (cid:28) P t ( ω > ω F ), P d (cid:28) P t ( ω > ω F ) follows. Re-phrasing this, were P d large and comparable to P t ( ω > ω F ), strong restoringforces at low frequency would result, and lead to the ex-istence of low-frequency vibrations instead of diffusion.We also note that because P l ≈ P t , P d (cid:28) P t ( ω > ω F )gives P d (cid:28) P l , further implying that P d can be omittedin Eq. (44).We note that in the regime τ (cid:29) τ D , the justificationfor the smallness of P d in the two previous paragraphs be-comes unnecessary. Indeed, using a rigorous statistical-mechanical argument it is easy to show that the totalenergy of diffusing atoms (the sum of their kinetic andpotential energy) can be ignored to a very good approx-imation if τ (cid:29) τ D . This is explained in the “Viscous liq-uids” section below in detail (see Eqs. 69,70,73 and dis-cussion around them), where we also remark that τ (cid:29) τ D corresponds to almost entire range of τ in which liquidsexist as such.Neglecting small P d in Eq. (44) is the only approxima-tion in the theory; subsequent transformations serve tomake the calculations convenient only. Eq. (44) becomes E = K + P l + P t ( ω > ω F ) (45)Eq. (45) can be re-written using the virial theorem P l = E l and P t ( ω > ω F ) = E t ( ω>ω F )2 (here, P and E refer to their average values) and by additionally notingthat the total kinetic energy K is equal to the value ofthe kinetic energy of a solid and can therefore be written,using the virial theorem, as the sum of kinetic terms re-lated to longitudinal and transverse waves: K = E l + E t ,giving E = E l + E t ( ω > ω F )2 + E t E t can be represented as E t = E t ( ω <ω F ) + E t ( ω > ω F ), liquid energy reads E = E l + E t ( ω > ω F ) + E t ( ω < ω F )2 (47)The first two terms in (47) give the energy of propa-gating phonon states in the liquid. The second term is the energy of two transverse modes which decreases withtemperature. This decrease includes both kinetic and po-tential parts, however the total kinetic energy of the sys-tem stays the same as in Eq. (44). The last term ensuresthat the decrease of the energy of transverse waves doesnot change the total kinetic energy, rather than pointsto the existence of low-frequency transverse waves (theseare non-propagating in liquids).Either (46) or (47) can now be used to calculate theliquid energy. Each term in Eqs. (46) or (47) can becalculated as the phonon energy, E ph : E ph = (cid:90) E ( ω, T ) g ( ω ) dω (48)where g ( ω ) is the phonon density of states.Lets consider Eq. (47) and let Z be the partitionfunction associated with the first two terms in Eq. (47).Then, Z is: Z = (2 π ¯ h ) − N (cid:48) (cid:90) exp (cid:32) − T N (cid:88) i =1 ( p i + ω li q i ) (cid:33) dpdq × (cid:90) exp (cid:32) − T N (cid:88) ω ti >ω F ( p i + ω ti q i ) (cid:33) dpdq (49)where ω F = τ , ω li and ω ti are frequencies of longitudinaland transverse waves, N is the number of atoms and N (cid:48) is the number of phonon states that include longitudinalwaves and transverse waves with frequency ω > ω F . Hereand below, k B = 1.We recall our earlier discussion that the longitudinalmode propagates in two different regimes: hydrodynamicregime ωτ < ωτ >
1. Thisgives different dissipation laws in the two regimes, butthis circumstance is unimportant for calculating the en-ergy. Indeed, (48) makes no reference to dissipation, andincludes the mode energy and the density of states only.These are the same in the two regimes, and hence for thepurposes of calculating the energy, the longitudinal modecan be considered as one single mode with Debye densitystates. This statement is not entirely correct becausethe mode is not well described in the regime ωτ ≈ Z = T N (cid:32) N (cid:89) i =1 ¯ hω li (cid:33) − T N (cid:32) N (cid:89) ω ti >ω ¯ hω si (cid:33) − (50)where N is the number of transverse modes with ω > ω F .2In the harmonic approximation, frequencies ω li and ω ti are considered to be temperature-independent, incontrast to anharmonic case discussed in the next sec-tion. Then, Eq. (50) gives the energy E = T ddT ln Z = N T + N T . N can be calculated using the quadratic density ofstates in the Debye model, as is done in solids [2]. Hereand below, the developed theory is at the same levelof approximation as Debye theory of solids. The den-sity of states of transverse modes is g t ( ω ) = Nω mt ω ,where ω mt is Debye frequency of transverse modes andwe have taken into account that the number of trans-verse modes in the solid-like density of states is 2 N . ω mt can be somewhat different from the longitudinal Debyefrequency; for simplicity we assume ω mt ≈ ω D . Then, N = ω D (cid:82) ω F g t ( ω ) dω = 2 N (cid:18) − (cid:16) ω F ω D (cid:17) (cid:19) .To calculate the last term in Eq. (47), we note thatsimilarly to E t ( ω > ω F ) = N T , E t ( ω < ω F ) can becalculated to be E t ( ω < ω F ) = N T , where N is thenumber of shear modes with ω < ω F . Because N =2 N − N , N = 2 N (cid:16) ω F ω D (cid:17) . The total liquid energy is E = ( N + N + N ) T according to Eq. (47), givingfinally [97]: E = N T (cid:32) − (cid:18) ω F ω D (cid:19) (cid:33) (51)At low temperature where τ (cid:29) τ D , or ω F (cid:28) ω D , Eq.(51) gives c v = N dEdT = 3, the harmonic solid result.At high temperature when τ → τ D and ω F → ω D , Eq.(51) gives c v = 2, consistent with the experimental re-sult in Figure 2. As the number of transverse modeswith frequency above ω F decreases with temperature, c v decreases from about 3 to 2. A quantitative agreement inthe entire temperature range can be studied by using Eq.(15) or ω F = G ∞ η , where η is taken from the independentexperiment. This way, E in Eq. (51) and c v have nofree fitting parameters. The agreement of Eq. (51) withthe experimental c v of liquid Hg is good at this level ofapproximation already [97].In this picture, the decrease of c v with temperatureis due to the evolution of collective modes in the liquid,namely the reduction of the number of transverse modesabove the frequency ω F = τ . We will discuss exper-imental data of c v for several types of liquids in moredetail below, including metallic, noble and molecular liq-uids, and will find that their c v similarly decreases withtemperature as Eq. (51) predicts. The same trend, thedecrease of c v with temperature, has been experimen-tally found in complex liquids, including such systemsas toluene, propane, ether, chloroform, benzene, methylcyclohexane and cyclopentane, hexane, heptane, octaneand so on [106]. We have focused on calculating liquid energy and re-sulting heat capacity that have contributions from collec-tive modes and diffusing atoms. We have not discussedliquid entropy which includes the configurational entropymeasuring the total phase space available to the system,the phase space sampled by diffusive particle jumps. Un-like entropy, the energy is not related to exploring thephase space, and corresponds to the instantaneous stateof the system (in the microcanonical ensemble, or aver-aged over fluctuations in the canonical ensemble). Wewill return to this point below when we discuss thermo-dynamic properties of viscous liquids.We make two remarks related to using the Debyemodel. First, the Debye model is particularly relevantfor disordered isotropic systems such as glasses [2], whichare known to be nearly identical to liquids from the struc-tural point of view [34]. Furthermore, we have seen ear-lier that the dispersion curves in liquids are very similarto those in solids (including crystals, poly-crystals andglasses). Therefore, the Debye model can be used in liq-uids to the same extent as in solids. One important con-sequence of this is that high-frequency modes in liquidsmake the largest contribution to the energy, as they doin solids including disordered solids. This is re-iteratedelsewhere in this paper.Second, recall our earlier observation that ω graduallyincreases from 0 to ω = ck around ω F with a square-root dependence (see Eq. (34) and discussion below).Writing ω D (cid:82) ω F g t ( ω ) dω in the previous paragraph assumes asharp lower frequency cutoff at ω F , and is an approxima-tion in this sense. The approximation is justified becauseit is the highest frequency modes above ω F that makethe most contribution to the liquid energy, and becauseDebye density of states we employ is already an approx-imation to the frequency spectrum, the approximationthat may be larger than the one involved in substitutingthe square-root crossover with a sharper cutoff. Comment on the phonon theory of liquidthermodynamics
We pause for the moment to make several commentsabout Eq. (51) and its relationship to our starting Eq.(1) in the Introduction. g ( r ) and U ( r ) featuring in Eq.(1) are not generally available apart from simple systemssuch as Lennard-Jones liquids. For simple liquids, g ( r )and U ( r ) can be determined from experiments or sim-ulations and subsequently used in Eq. (1). Unfortu-nately, neither g ( r ) nor U ( r ) are available for liquids withany larger degree of complexity of structure or interac-tions. For example, many-body correlations [98, 99] andnetwork effects can be strong in familiar liquid systemssuch as olive oil, SiO , Se, glycerol, or even water [101],resulting in complicated structural correlation functions3that cannot be reduced to the simple two- or even three-body correlations that are often used. As discussed inRef. [6], approximations become difficult to control whenthe order of correlation functions already exceeds three-body correlations. Similarly, it is challenging to extractmultiple correlation functions from the experiment. Thesame problems exist for interatomic interactions, whichcan be equally multibody and complex, and consequentlynot amenable to determination in experiments or simula-tions. On the other hand, ω F ( τ ) is available much morewidely as discussed above, enabling us to calculate andunderstand liquid c v readily.Next, expressing the liquid energy in terms of ω F inEq. (51) represents a more general description of liquidsas compared to Eq. (1). In Eq. (1), the energy stronglydepends on interactions. It was for this reason that Lan-dau and Lifshitz state that the liquid energy is stronglysystem-dependent and therefore cannot be calculated ingeneral form [2]. Let us now consider liquids with verydifferent structural correlations and interatomic interac-tions such as, for example, H O, Hg, AsS, olive oil, andglycerol. As long as ω F of the above liquids is the sameat a certain temperature, Eq. (51) predicts that theirenergy is the same (in molecular liquids, we are referringto the inter-molecular energy as discussed below in moredetail). In this sense, expressing the liquid energy as afunction of ω F only is a more general description because ω F is a uniformly common property for all liquids.Finally, Eq. (51), as well as its modifications below,are simple. This makes it fairly easy to understand andinterpret experimental data as discussed in the later sec-tion.An objection could be raised that, although our ap-proach explains the experimental c v of liquids as dis-cussed below, the approach is based on ω F , the emer-gent property rather than on the ostensibly lower-leveldata such as g ( r ) and U ( r ) in Eq. (1). This brings usto an important question of what we aim to achieve bya physical theory. According to one view, “The pointof any physical theory is to make statements about theoutcomes of future experiments on the basis of resultsfrom the previous experiment” [100]. This emphasizesrelationships between experimental properties. In thissense, Eq. (51) provides a relationship between liquidthermodynamic properties such as energy and c v on oneside and its dynamical and oscillatory properties such as ω F on the other. Including anharmonicity and thermal expansion
In calculating the energy E = T ddT ln Z , we haveassumed that the phonon frequencies are temperature-independent. Generally, the phonon frequencies reducewith temperature. This takes place at both constantpressure and constant volume. At constant volume, re- duction of frequencies is related to inherent anharmonic-ity and increased vibration amplitudes. If frequencies aretemperature-dependent, applying E = T ddT ln Z to Eq.(50) gives E = N T − T N (cid:88) i =1 ω li dω li dT + N T − T N (cid:88) i =1 ω ti dω ti dT (52)where the derivatives are at constant volume. Eq. (52)gives the first two terms in Eq. (47).Using Gr¨uneisen approximation, it is possible to derivea useful approximate relation: ω (cid:0) d ω i d T (cid:1) v = − α , where α is the coefficient of thermal expansion [102, 103]. Usingthis in (52) gives E = ( N + N ) T (cid:18) αT (cid:19) (53)The last term in Eq. (47), E t ( ω< /τ )2 , can be calcu-lated in the same way, giving N T (cid:0) αT (cid:1) , where N is the number of shear modes with ω < ω F calculatedin the previous section. Adding this term to Eq. (53)and using N and N from the previous section gives theanharmonic liquid energy: E = N T (cid:18) αT (cid:19) (cid:32) − (cid:18) ω F ω D (cid:19) (cid:33) (54)which reduces to (51) when α = 0.Eq. (54) has been found to quantitatively describe c v of 5 commonly studied liquid metals in a wider tem-perature range where c v decreases from about 3 aroundthe melting point to 2 at high temperature [104]. Thepresence of the anharmonic term in Eq. (54), (cid:0) αT (cid:1) ,explains why experimental c v of liquids may exceed theDulong-Petit value c v = 3 close to the melting point[21, 22].At low temperature when τ (cid:29) τ D , Eq. (54) gives E = 3 N T (cid:18) αT (cid:19) (55)and c v is c v = 3 (1 + αT ) (56)Eq. (56) is equally applicable to solids and viscousliquids where τ (cid:29) τ D , and has been found consistentwith several simulated crystalline and amorphous sys-tems [102].We note that Eqs. (55) and (56) don’t need to bederived from Eq. (54), and also follow from consideringthe solid as a starting point where all three modes arepresent.4 Including quantum effects
If the temperature range includes low temperaturewhere ¯ hω D T (cid:28) F ph = E + T (cid:88) i ln (cid:18) − exp (cid:18) − ¯ hω i T (cid:19)(cid:19) (57)where E is the energy of zero-point vibrations. In calcu-lating the energy, E ph = F ph − T dF ph dT , we assume dω i dT (cid:54) = 0as in the previous section, giving for the phonon energy E ph = E + ¯ h (cid:88) i ω i − T dω i dT exp (cid:0) ¯ hω i T (cid:1) − d ω i d T = − αω i as before gives E ph = E + (cid:18) αT (cid:19) (cid:88) i ¯ hω i exp (cid:0) ¯ hω i T (cid:1) − (cid:80) , with Debyevibrational density of states for longitudinal phonons, g ( ω ) = Nω ω , where ω D is Debye frequency. Inte-grating from 0 to ω D gives (cid:80) = N T D (cid:0) ¯ hω D T (cid:1) , where D ( x ) = x x (cid:82) z d z exp( z ) − is Debye function [2]. The en-ergy of two transverse modes with frequency ω > ω F ,the second term in Eq. (47), can be similarly calculatedby substituting (cid:80) with density of states g ( ω ) = Nω ω ,where the normalization accounts for the number oftransverse modes of 2 N . Integrating from ω F to ω D gives (cid:80) = 2 N T D (cid:0) ¯ hω D T (cid:1) − N T (cid:16) ω F ω D (cid:17) D (cid:0) ¯ hω F T (cid:1) . Finally, E t ( ω < ω F ) in the last term in Eq. (47) is obtained by in-tegrating (cid:80) from 0 to ω F with the same density of states,giving (cid:80) = 2 N T (cid:16) ω F ω D (cid:17) D (cid:0) ¯ hω F T (cid:1) . Putting all terms inEq. (59) and then Eq. (47) gives finally the liquid energy E = E + N T (cid:18) αT (cid:19) (cid:32) D (cid:18) ¯ hω D T (cid:19) − (cid:18) ω F ω D (cid:19) D (cid:18) ¯ hω F T (cid:19)(cid:33) (60)In general, E is temperature-dependent because it de-pends on ω F and therefore T . However, this becomes im-portant at temperatures of several K only, whereas be-low we deal with significantly higher temperatures where E and its derivative in (60) are small compared to thesecond temperature-dependent term. In the subsequentcomparison of (60) with experimental c v , we therefore donot include E .In the high-temperature classical limit where ¯ hω D T (cid:28) ¯ hω F T (cid:28) ω F < ω D ), Debye functionsbecome 1, and (60) reduces to the energy of the classicalliquid, Eq. (54).For some of the liquids discussed in the next section,the high-temperature classical approximation ¯ hω D T (cid:28) c v . Comparison with experimental data
The most straightforward comparison of the above the-ory to experiments is to calculate the energy using Eqs.(51), (54) or (60) and experimental ω F . This is oftendone by fitting ω F to function such as the VFT law, us-ing it to calculate the energy and differentiating it to find c v and compare it to the experimental data. Eqs. (51),(54) or (60) involve no free fitting parameters, and con-tain parameters related to system properties only. If ω F is calculated from experimental viscosity as ω F = G ∞ η ,Eq. (51) contains G ∞ and τ D which enter as the product G ∞ τ D . Eq. (54) contains parameters G ∞ τ D and α . InEq. (60), G ∞ and τ D feature separately.In the last few years, we have compared theoreticaland experimental c v of over 20 different systems, in-cluding metallic, noble, molecular and network liquids[97, 104, 105]. We aimed to check our theoretical pre-dictions in the widest temperature range possible, andtherefore used the data at pressures exceeding the criticalpressures from the National Institute of Standards andTechnology (NIST) database [107]. As a result, manystudied liquids are supercritical. In Figure 9, we showthe comparison of theoretical and experimental data forseveral representative liquids. We have included threeliquids in each class: metallic, noble and molecular liq-uids.We observe good agreement between experiments andtheoretical predictions in a wide temperature range ofabout 50-1300 K in Figure 9. The agreement supportsthe interpretation of the universal decrease of c v withtemperature: the decrease is due to the reduction of thenumber of transverse modes propagating above frequency τ .We note that Debye model is not a good approxima-tion in molecular and hydrogen-bonded systems wherethe frequency of intra-molecular vibrations considerablyexceeds the rest of frequencies in the system (e.g. 3572 Kin CO and 2260 K in O ). However, the intra-molecularmodes are not excited in the temperature range of ex-5 cv (kB) T ( K )
R b P b
C O A r H g cv (kB)
T ( K ) X eC H K r
FIG. 9: Colour online. Experimental c v (black color) inmetallic, noble and molecular liquids ( k B = 1). Experimental c v are measured on isobars. Theoretical c v (red color) wascalculated using Eq. (60). The data are from Ref. [105]. Thedata for molecular and noble liquids are taken at high pres-sure to increase the temperature range where these systemsexist in the liquid form [107]. We show the data in two graphsto avoid overlapping. perimental c v (see Figure 9). Therefore, the contributionof intra-molecular motion to c v is purely rotational, c rot .The rotational motion is excited in the considered tem-perature range, and is classical, giving c rot = R for linearmolecules such as CO and O and c rot = R for moleculeswith three rotation axes such as CH . Consequently, c v for liquid CO shown in Figure (9) corresponds to the heatcapacity per molecule, with c rot subtracted from the ex-perimental data. In this case, N in Eqs. (51), (54) or(60) refers to the number of molecules. Phonon excitations at low temperature
The number of excited phonon states increases withtemperature. At low temperature, this results in thewell-known increase of c v : c v ∝ T . This increase cancompete with the decrease of c v to the progressive lossof transverse modes discussed above. In practice, all liq-uids solidify at low temperature and room pressure ex-cept helium. In liquid helium under pressure, c v can firstincrease with temperature due to the phonon excitationeffects. This is followed by the decrease of c v at higher temperature, similar to the behavior of classical liquidsin Figure 9. As a result, c v can have a maximum [108].An interesting assertion can be made about the op-eration of transverse modes in a hypothetical liquid inthe limit of zero temperature: transverse modes do notcontribute to liquid’s energy and specific heat in thislimit [97]. Indeed, let us consider a liquid with a cer-tain ω F and calculate the quantum energy of two trans-verse modes with frequency above ω F as E tT ( ω > ω F ) = ω D (cid:82) ω F ¯ hω exp ¯ hωT − g t ( ω ) dω . E tT ( ω > ω F ) can be written as E tT ( ω > ω F ) = ω D (cid:90) ¯ hωg t ( ω ) dω exp ¯ hωT − − ω F (cid:90) ¯ hωg t ( ω ) dω exp ¯ hωT − g t ( ω ) = Nω ω gives: E tT ( ω > ω F ) = 2 N T D (cid:18) ¯ hω D T (cid:19) − N T (cid:18) ω F ω D (cid:19) D (cid:18) ¯ hω F T (cid:19) (62)In the low-temperature limit where D ( x ) = π x , thetwo terms cancel exactly, giving E tT ( ω > ω F ) = 0. Thesame result follows without relying on the Debye modeland from observing that in the low-temperature limit,the upper integration limits in both terms in (61) can beextended to infinity due to fast convergence of integrals.Then, E tT ( ω > ω F ) in (61) is the difference between twoidentical terms and is zero [97].Physically, the reason for E tT ( ω > ω F ) = 0 is that onlyhigh-frequency transverse modes exist in a liquid accord-ing to (11), but these are not excited at low temperature.We will re-visit this result in the later section dis-cussing solid-like approaches to quantum liquids such asliquid helium. HEAT CAPACITY OF SUPERCRITICAL FLUIDS
In the above discussion, c v decreases from about 3 atlow temperature to 2 at high, corresponding to the com-plete loss of solid-like transverse modes. It is interestingto ask how c v changes on further temperature increase.On general grounds, one expects to find the gas-like value c v = at high temperature where the kinetic energydominates.If the system is below the critical point (see Figure1), further temperature increase involves boiling and thefirst-order transition, with c v discontinuously decreasingto in the gas phase. The intervening phase transitionexcludes the state of the liquid where c v can graduallychange from 2 to and where interesting physics oper-ates. However, this becomes possible above the critical6point. This brings us to the interesting discussion of thesupercritical state of matter. Frenkel line
Supercritical fluids started to be widely deployed inmany important industrial processes [109, 110] once theirhigh dissolving and extracting properties were appreci-ated. These properties are unique to supercritical fluidsand primarily result from the combination of high den-sity and high particle mobility. Theoretically, little wasknown about the supercritical state, apart from the gen-eral assertion that supercritical fluids can be thought ofas high-density gases or high-temperature fluids whoseproperties change smoothly with temperature or pres-sure and without qualitative changes of properties. Thisassertion followed from the known absence of a phasetransition above the critical point.We have recently proposed that this picture should bemodified, and that a new line, the Frenkel line (FL), ex-ists above the critical point and separates two states withdistinct properties (see Figure 10) [111–114]. The mainidea of the FL lies in considering how particle dynamicschanges in response to pressure and temperature. Recallthat particle dynamics in the liquid can be separated intosolid-like oscillatory and gas-like diffusive components.This separation applies equally to supercritical fluids asit does to subcritical liquids: increasing temperature re-duces τ , and each particle spends less time oscillating andmore time jumping; increasing pressure reverses this andresults in the increase of time spent oscillating relativeto jumping. Increasing temperature at constant pressure(or decreasing pressure at constant temperature) eventu-ally results in the disappearance of the solid-like oscilla-tory motion of particles; all that remains is the diffusivegas-like motion. This disappearance represents the quali-tative change in particle dynamics and gives the point onthe FL in Figure 10. Notably, the FL exists at arbitrarilyhigh pressure and temperature, as does the melting line.Qualitatively, the FL corresponds to τ → τ D (here, τ D refers to the minimal period of transverse modes),implying that particle motion loses its oscillatory com-ponent. Quantitatively, the FL can be rigorously definedby pressure and temperature at which the minimum ofthe velocity autocorrelation function (VAF) disappears[113]. Above the line defined in such a way, velocitiesof a large number of particles stop changing their signand particles lose the oscillatory component of motion.Above the line, VAF is monotonically decaying as in agas [113].Another criterion for the FL which is important forour discussion of thermodynamic properties and whichcoincides with the VAF criterion is c v = 2 [113]. Indeed, τ = τ D corresponds to the complete loss of two trans-verse modes at all available frequencies (see Eq. (11)). Temperature
P r e s s u r e l i q u i d - l i k e g a s - l i k e
F r e n k e l l i n e
FIG. 10: Colour online. The Frenkel line in the supercriti-cal region. Particle dynamics includes both oscillatory anddiffusive components below the line, and is purely diffusiveabove the line. Below the line, the system is able to sup-port rigidity and transverse modes at high frequency. Abovethe line, particle motion is purely diffusive, and the ability tosupport rigidity and transverse modes is lost at all availablefrequencies. Crossing the Frenkel line from below correspondsto the transition between the “rigid” liquid to the “non-rigid”gas-like fluid.
The ability to support transverse waves is associated withsolid-like rigidity. Therefore, τ = τ D corresponds to thecrossover from the “rigid” liquid to the “non-rigid” gas-like fluid where no transverse modes exist [111–114, 124],corresponding to the qualitative change of the excitationspectrum.According to Eq. (51), ω F = ω D or τ = τ D gives c v = 2. This corresponds to the qualitative change ofthe excitation spectrum in the liquid, the loss of trans-verse modes. Therefore, we expect to find an interestingbehavior of c v around c v = 2 and its crossover to a newregime. This is indeed the case as discussed in the nextsection.Due to the qualitative change of particle dynamics,the FL separates the states with different macroscopicproperties, consistent with experimental data [107]. Thisincludes diffusion constant, viscosity, thermal conductiv-ity, speed of sound and other properties [111, 113, 114].For example, the fast sound discussed earlier disappearsabove the FL due to the loss of shear resistance at allavailable frequencies. Depending on the temperature andpressure path on the phase diagram, the crossover of aparticular property may not take place on the FL directlybut close to it.We note a different proposal to define a line above thecritical point, the Widom line. At the critical point, ther-modynamic functions have divergent maxima. Above thecritical point, these maxima broaden and persist in thelimited range of pressure and temperature. This enablesone to define lines of maxima of different properties such7as heat capacity, thermal expansion, compressibility andso on. Close to the critical point, system properties canbe expressed in terms of the correlation length, the max-ima of which is the Widom line [115].The physical significance of the Widom line was origi-nally attributed to the effect of persisting critical fluctu-ations on system’s dynamical properties [115]. Followingthe detection of PSD above the critical point [74, 116],the Widom line was proposed to separate two supercrit-ical states where PSD does and does not operate [75](see [111] for the discussion of extrapolating the line tohigh pressure and temperature where no maxima exist).The states with and without PSD were called “liquid-like” and “gas-like” because they resemble the presenceand absence of PSD in subcritical liquids and gases. Thediscussion of the effect of the Widom line on thermody-namic, dynamical and transport properties followed (see,e.g., [117, 118]).Persisting critical anomalies and fluctuations related tothe Widom line certainly affect system properties closeto the critical point. At the same time, the physical ori-gin of the Widom line and the FL is different, as evidentfrom the above discussion. A detailed discussion of thispoint is outside the scope of this review. Here, we includetwo brief remarks: (a) the FL is not physically relatedto the critical point and critical fluctuations and existsin systems where the boiling line and the critical pointare absent such as the soft-sphere system [113]; and (b)the persisting maxima of thermodynamic functions andthe Widom line decay around (1.5-2) T c and strongly de-pend on the property (e.g. heat capacity, compressibilityand so on) and on the path on the phase diagram (i.e.the location of the Widom line depends on whether theproperty is calculated along isobars, isotherms and soon) [119–122]. This is in contrast to the FL which ex-ists at arbitrarily high temperature and pressure and isproperty- and path-independent. Heat capacity above the Frenkel line
A confirmation of the above theoretical proposal thatthe specific heat undergoes a crossover around c v = 2comes from molecular dynamics simulations in the su-percritical state [111, 123]. c v of the model Lennard-Jones liquid is shown in Figure 11. We first observe afairly sharp decrease of c v from about 3 to 2, similar tothe previously discussed behavior in Figure 9. This isfollowed by the flattening and slower decrease at highertemperature. The crossover takes place at around c v = 2as predicted.Understanding the slower decrease of c v above the FLinvolves the discussion of how the remaining longitu-dinal mode evolves with temperature (recall that twotransverse modes disappear at the FL). When the FLis crossed from below, particles lose the oscillatory mo- c r o s s o v e r a tt h e F r e n k e l l i n e Tc v FIG. 11: Colour online. c v ( k B = 1) as a function of tempera-ture from the molecular dynamics simulation of the Lennard-Jones (LJ) liquid using the data from Ref. [111]. Temperatureis in LJ units. Density is ρ = 1 in LJ units. The region ofdynamical crossover at c v = 2 is highlighted in red and by thearrow. tion around their quasi-equilibrium positions, and startundergoing purely diffusive jumps with distances compa-rable with the interatomic distance a . Further increaseof particle energy at higher temperature increases themean free path of particles L , the average distance whichthe particles travel before colliding. L sets the minimalwavelength of the remaining longitudinal mode, λ L : in-deed, oscillation wavelength can only be larger than L .Therefore, the propagating longitudinal mode above theFL has the wavelengths satisfying λ > L (63)We observe that the oscillations of the longitudinalmode in (63) disappear with temperature starting withthe highest frequency (smallest wavelength) above theFL, in interesting contrast to the evolution of transversemodes in (11) where transverse modes disappear startingwith the smallest frequency. The difference of tempera-ture evolution of collective modes below and above theFL is responsible for the crossover of c v at the FL dis-cussed below.The energy of the above longitudinal mode, E l , can becalculated using Eq. (48) as E l = ω L (cid:90) E ( ω, T ) g ( ω ) dω (64)where ω L = πλ L c = πL c is the minimal frequency.8Taking g ( ω ) = Nω ω as before and E ( ω, T ) = T inthe classical case gives E l = N T (cid:16) ω L ω D (cid:17) , or N T (cid:0) aL (cid:1) .The total energy of the system is the sum of the kineticenergy, N T , and potential energy. Using the equiparti-tion theorem, the potential energy can be written as E l .This gives the total energy of the non-rigid gas-like fluidabove the FL as E = 32 N T + 12
N T (cid:16) aL (cid:17) (65)Just above the FL, L ≈ a . According to Eq. (65),this gives c v = 2, the result that also follows from theEquation (51) describing the rigid liquid. When L (cid:29) a at high temperature, Eq. (65) gives c v = as expected.The crossover of c v seen in Figure 11 is therefore at-tributed to two different mechanisms governing the de-crease of c v . Below the FL, c v decreases from the solidvalue of 3 to 2 due to the progressive disappearance oftwo transverse modes with frequency ω > ω F . Above theFL, c v decreases from 2 to the ideal-gas value of dueto the disappearance of the remaining longitudinal modestarting with the shortest wavelength governed by L . Re-maining long-wavelength longitudinal oscillations, sound,make only small contribution to the system energy andheat capacity.The softening of the phonon frequencies with temper-ature can be accounted for in the same way as in thecase of subcritical fluids above (see Eqs. (52, 53)), giving[123]: E = 32 N T + 12
N T (cid:18) αT (cid:19) (cid:16) aL (cid:17) (66)The actual decrease of c v between c v = 2 and c v = can be calculated if temperature dependence of L isknown. This dependence can be taken from the inde-pendent measurement of the gas-like viscosity of the su-percritical fluid: η = 13 ρ ¯ uL (67)where ¯ u is the average velocity defined by temperature.Taking η from the experiment, calculating L using (67)and using it in Eq. (65) or Eq. (66) enables us to cal-culate E and c v . This gives good agreement with theexperimental c v for several supercritical systems, includ-ing noble and molecular fluids [123]. In these systems, c v slowly decreases with temperature as is seen in Figure 11in the high-temperature range.We note that in our discussion of liquid thermodynam-ics throughout this paper, we assumed that the mode en-ergy is T (in the classical case). This applies to harmonic c r o s s o v e r a t t h e F r e n k e l l i n e T i n c r e a s e T i n c r e a s e l m i n / a l m a x / a . . . l / a c v FIG. 12: c v as a function of the characteristic wavelengths λ max (maximal transverse wavelength in the system) and λ min (minimal longitudinal wavelength in the system) illustratingthat most important changes of thermodynamics of the dis-ordered system take place when both wavelengths becomecomparable to the fundamental length a . waves. In weakly-anharmonic cases, the anharmonicitycan be accounted for in the Gr¨uneisen approximation (seeEq. 54 and related discussion). If the anharmonicity isstrong, the mode energy can substantially differ from T .This can include the case of very high temperature orinherently anharmonic systems such as the hard-spheressystem as an extreme example where heat capacity isequal to the ideal-gas value. HEAT CAPACITY OF LIQUIDS AND SYSTEM’SFUNDAMENTAL LENGTH
The behavior of liquid c v in its entire range from thesolid value, c v = 3, to the ideal-gas value, c v = , can beunified and generalized in terms of wavelengths.Lets consider c v in the rigid liquid state, Eq. (51) andin the non-rigid gas-like fluid, (65). We do not consideranharmonic effects related to phonon softening: thesegive small corrections ( αT (cid:28)
1) to the energy in Eqs.(54), (66). An interesting insight comes from combiningEqs. (51) and (65) and interpreting both of them in termsof wavelengths [46] (see Figure 12).The minimal frequency of transverse modes that a liq-uid supports, ω F , corresponds to the maximal transversewavelength, λ max , λ max = a ω D ω F = a ττ D , where a is theinteratomic separation, a ≈ − c v remains close to its solid-state value of 3 in al-most entire range of available wavelengths of transversemodes until ω F starts to approach ω D , including in the9viscous regime discussed below, or when λ max starts toapproach a . When λ max = a , c v becomes c v = 2 accord-ing to Eq. (51) and undergoes a crossover to anotherregime given by Eq. (65). In this regime, the minimalwavelength of the longitudinal mode supported by thesystem is λ min = L . According to Eq. (65), c v remainsclose to the ideal gas value of in almost entire rangeof the wavelengths of the longitudinal mode until λ min approaches a . When λ max = a , c v becomes c v = 2, andmatches its low-temperature value at the crossover asschematically shown in Figure 12.Consistent with the above discussion, Figure 12 showsthat c v remains constant at either 3 or over many or-ders of magnitude of λa , including the regime of viscousliquids and glasses discussed below, except when λa be-comes close to 1 by order of magnitude.Figure 12 emphasizes a transparent physical point:modes with the smallest wavelengths comparable to in-teratomic separations a contribute most to the energyand c v in the disordered systems (as they do in crystals)because they are most numerous. Consequently, condi-tions λ max ≈ a for two transverse modes and λ min ≈ a forone longitudinal mode, corresponding to the disappear-ance of modes with wavelengths comparable to a , givethe largest changes of c v as is seen in Figure 12.The last result is tantamount to the following generalassertion: the most important changes in thermodynam-ics of the disordered system are governed by its funda-mental length a only. Because this length is not affectedby disorder, this assertion holds equally in ordered anddisordered systems.Interestingly, the above assertion does not follow fromthe hydrodynamic approach to liquids. The hydrody-namic approach works well at large wavelengths, but maynot correctly describe effects at length scales compara-ble to a . Yet, as we have seen, this scale which playsan important role in governing system’s thermodynamicproperties. EVOLUTION OF COLLECTIVE MODES INLIQUIDS: SUMMARY
We can now summarize the above discussion of howcollective modes change in liquids with temperature.This is illustrated in Figure 13.Figure 13 illustrates that at low temperature, liquidshave the same set of collective modes as in solids: onelongitudinal mode and two transverse modes. In the vis-cous regime at low enough temperature where τ (cid:29) τ D or ω F (cid:28) ω D , the liquid energy is almost entirely given bythe vibrational energy due to these modes, as discussed inthe next section. On temperature increase, the numberof transverse modes propagating above the frequency ω F decreases. At the FL where particles lose the oscillatorycomponent of motion and start moving diffusively as in a gas, the two transverse modes disappear. This pictureis consistent with the results of molecular dynamics sim-ulations where transverse modes are directly calculatedfrom the transverse current correlation functions [124].Above the FL, the collective mode is the remaininglongitudinal mode with the wavelength larger than L ,and its energy progressively decreases with temperatureuntil it becomes close to the ideal gas.The evolution of collective modes and related changesof liquid energy and heat capacity are intimately relatedto the change of microscopic dynamics of particles andthe relative weights of diffusive and oscillatory compo-nents. We will return to this point below when we dis-cuss the mixed state of liquid dynamics as contrasted topure dynamical states of solids and gases. VISCOUS LIQUIDS
In this section, we discuss how energy and heat capac-ity of viscous liquids can be understood on the basis ofcollective modes. “Viscous” or “highly-viscous” liquidsare loosely defined as liquids where τ (cid:29) τ D (68)More generally, viscous liquids are discussed as systemsthat avoid crystallization and enter the glass transforma-tion range. When τ exceeds the experimental time scaleof 10 − s and particle jumps stop operating duringthe observation time (in the field of glass transition, par-ticle jumps are often referred to as “alpha-relaxation”),the system forms glass. This defines glass transition tem-perature as τ ( T g ) = 10 − s [34].Properties of viscous liquids have been widely dis-cussed due to the interest in the problem of liquid-glasstransition, the problem which consists of several unusualeffects and includes persisting controversies (see, e.g.,[34, 35, 41, 42, 51–55]). Understanding viscous liquidsabove T g is thought to facilitate explaining effects in-volved in the actual liquid-glass transition at T g and pos-sibly effects below T g [34, 35, 41, 51, 53–55].We find that in some respects, the glass transitionproblem is more controversial that it needs to be. This ispartly because the controversies emerged before good-quality experimental data became available. For ex-ample, the crossover from the VFT to the Arrhenius(or nearly Arrhenius) behavior at low temperature [57–59, 61, 62] removes the basis for discussing possible di-vergence and associated ideal glass transition at the VFTtemperature T , as discussed above.0 [Frenkel line] [ (cid:31) = long only][ (cid:31) = long and short] [Longitudinal waves][Longitudinal waves][ T ] [Transverse waves][No transverse waves][ (cid:31) = short only] FIG. 13: Colour online. Evolution of transverse and longitudinal waves in disordered matter, from viscous liquids and glassesat low temperature to gases at high. The variation of colour from deep blue at the bottom to light red at the top correspondsto temperature increase. The Figure shows that the transverse waves (left) start disappearing with temperature starting withlong wavelength modes and completely disappear at the Frenkel line. The longitudinal waves (right) do not change up to theFrenkel line but start disappearing above the line starting with the shortest wavelength, with only long-wavelength longitudinalmodes propagating at high temperature.
Energy and heat capacity
Perhaps unexpectedly, understanding basic thermody-namic properties of viscous liquids such as energy andheat capacity is easier than of low-viscous liquids. It doesnot involve expanding the energy into the oscillatory anddiffusive parts as in Eq. (44) or integrating over the oper-ating phonon states as in Eqs. (48-50). The main resultscan be obtained on the basis of one parameter only, τ D τ (or ω F ω D ) using a simple yet rigorous statistical-mechanicalargument [125].The jump probability for a particle is the ratio betweenthe time spent diffusing and oscillating. The jump eventlasts on the order of Debye vibration period τ D ≈ . τ is the time between two consecutive particlejumps, and therefore is the time that the particle spendsoscillating. Therefore, the jump probability is τ D τ . Instatistical equilibrium, this probability is equal to the ratio of diffusing atoms, N dif , and the total number ofatoms, N . Then, at any given moment of time: N dif N = τ D τ (69)If E dif is the energy associated with diffusing particles, E dif ∝ N dif . Together with E tot ∝ N , Eq. (69) gives E dif E tot = τ D τ (70)Eq. (70) implies that under condition (68), the contri-bution of E dif to the total energy at any moment of timeis negligible.We note that Eq. (70) corresponds to the instanta-neous value of E dif which, from the physical point ofview, is given by the smallest time scale of the system,1 τ D . During time τ D , the system is not in equilibrium.The equilibrium state is reached when the observationtime exceeds system relaxation time, τ . After time τ ,all particles in the system undergo jumps. Therefore, weneed to calculate E dif that is averaged over time τ .Let us divide time τ into m time periods of duration τ D each, so that m = ττ D . Then, E dif , averaged over time τ , E avdif , is E avdif = E + E + ... + E m dif m (71)where E i dif are instantaneous values of E dif featured inEq. (70). E avdif E tot is E avdif E tot = E + E + ... + E m dif E tot · m (72)Each of the terms E i dif E tot in Eq. (72) is equal to τ D τ ,according to Eq. (70). There are m terms in the sum inEq. (72). Therefore, E avdif E tot = τ D τ (73)We therefore find that under the condition (68), theratio of the average energy of diffusion motion to thetotal energy is negligibly small, as in the instantaneouscase. Consequently, the energy of the liquid under thecondition (68) is, to a very good approximation, givenby the remaining vibrational part. Similarly, the liquidconstant-volume specific heat, c v, l = N d E l d T is entirelyvibrational in the regime (68): E l = E vibl c v, l = c vib v, l (74)The vibrational energy and specific heat of liquids inthe regime (68) is readily found. When regime (68) isoperative, E vibl to a very good approximation is E vibl =3 N T (here and below, k B = 1). Indeed, a solid supportsone longitudinal mode and two transverse waves in therange 0 < ω < τ D . The ability of liquids to support shearmodes with frequency ω > τ , combined with τ (cid:29) τ D inEq. (68), implies that a viscous liquid supports mostof the shear modes present in a solid. Furthermore andimportantly, it is only the high-frequency shear modesthat make a significant contribution to the liquid vibra-tional energy, because the vibrational density of states isapproximately proportional to ω . Hence in the regime(68), E vibl = 3 N T to a very good approximation, as in asolid.We now now consider Eqs. (74) in harmonic and an-harmonic cases. In the harmonic case, Eqs. (74) give the energy and specific heat of a liquid as 3
N T and 3,respectively, i.e. the same as in a harmonic solid: E hl = E hs = 3 N Tc h v, l = c h v, s = 3 (75)where s corresponds to the solid and h to the harmoniccase.In the anharmonic case, Eqs. (74) are modified by theintrinsic anharmonicity related to softening of phononfrequencies, and become Eqs. (55) and (56) as discussedabove.Three pieces of evidence support the above picture.First, experimental specific heat of liquid metals at lowtemperature is close to 3, consistent with the above pre-dictions [21, 22]. As experimental techniques advancedand gave access to high pressure and temperature, spe-cific heats of many noble, molecular and network liquidswere measured in a wide range of parameters including inthe supercritical region [107]. Similarly to liquid metals,the experimental c v of these liquids was found to be closeto 3 at low temperature where Eq. (1) applies (see Ref.[105] for a compilation of the NIST and other data of c v for over 20 liquids of different types).Second, condition τ (cid:29) τ D becomes particularly pro-nounced in viscous liquids approaching liquid-glass tran-sition where τ D τ becomes as small as τ D τ ≈ − . Ex-periments have shown that in the highly viscous regimejust above T g , C p measured at high frequency and rep-resenting the vibrational part of heat capacity coincideswith the total low-frequency heat capacity usually mea-sured [126, 127], consistent with Eq. (74). In the glasstransformation range close to T g , the two heat capacitiesstart to differ due to non-equilibrium effects and freezingof configurational entropy, and coincide again below T g in the solid glass.Third, representing c v by its vibrational term in thehighly viscous regime above T g gives the experimentallyobserved change of heat capacity in viscous liquids above T g as compared to glasses below T g . This is discussed inthe next section.We recall that the only condition used to make theabove assertions is Eq. (68). For practical purposes, thiscondition is satisfied for τ > ∼ τ D . Perhaps not widelyrecognized, the condition τ ≈ τ D holds even for low-viscous liquids such as liquid metals (Hg, Na, Rb andso on) and noble liquids such as Ar near their meltingpoints, let alone for more viscous liquids such as room-temperature olive or motor oil, honey and so on.Notably, the condition τ > ∼ τ D corresponds to almostthe entire range of τ at which liquids exist. This factwas not fully appreciated in earlier theoretical work onliquids. Indeed, on lowering the temperature, τ increasesfrom its smallest limiting value of τ = τ D ≈ . τ ≈ s where, by definition, the liquid forms glass2at the glass transition temperature T g . Here, τ changesby 16 orders of magnitude. Consequently, the condition τ D τ (cid:28)
1, Eq. (68), or τ > ∼ τ D , applies in the range10 − − s, spanning 15 orders of magnitude of τ .This constitutes almost entire range of τ where liquidsexist as such. Entropy
Although Eq. (73), combined with Eq. (68), impliesthat the energy and c v of a liquid are entirely vibrationalas in a solid, this does not apply to entropy: the diffu-sional component to entropy is substantial, and can notbe neglected [125].Indeed, if Z vib and Z dif are the contributions to thepartition sum from vibrations and diffusion, respectively,the total partition sum of the liquid is Z = Z vib · Z dif .Then, the liquid energy is E = T T (ln( Z vib · Z dif )) = T T ln Z vib + T T ln Z dif = E vib + E dif (here and be-low, the derivatives are taken at constant volume). Next, E avdif E tot (cid:28) E dif E vib (cid:28)
1, where,for brevity, we dropped the subscript referring to theaverage. Therefore, the smallness of diffusional energy, E dif E vib (cid:28)
1, gives dd T ln Z difdd T ln Z vib (cid:28) S = dd T ( T ln( Z vib · Z dif )), is: S = T dd T ln Z vib + ln Z vib + T dd T ln Z dif + ln Z dif (77)The condition (76) implies that the third term in Eq.(77) is much smaller than the first one, and can be ne-glected, giving S = T dd T ln Z vib + ln Z vib + ln Z dif (78)Eq. (78) implies that the smallness of E dif , expressedby Eq. (76), does not lead to the disappearance of allentropy terms that depend on diffusion because the termln Z dif remains. This term is responsible for the excessentropy of liquid over the solid. On the other hand, thesmallness of E dif does lead to the disappearance of termsdepending on Z dif in the specific heat. Indeed, c v, l = T d S d T (here, S refers to entropy per atom or molecule),and from Eq. (78), we find: c v, l = T dd T (cid:18) T dd T ln Z vib (cid:19) + T dd T ln Z vib + T dd T ln Z dif (79) Using Eq. (76) once again, we observe that the thirdterm in Eq. (79) is small compared to the second term,and can be neglected, giving c v, l = T dd T (cid:18) T dd T ln Z vib (cid:19) + T dd T ln Z vib (80)As a result, c v does not depend on Z dif , and is givenby the vibrational term that depends on Z vib only. Asexpected, Eq. (80) is consistent with c v in Eq. (75).Physically, the inequality of liquid and solid entropies, S l (cid:54) = S s , is related to the fact that the entropy mea-sures the total phase space available to the system, whichis larger in the liquid due to the diffusional componentpresent in Eq. (78). However, the diffusional component,ln Z dif , although large, is slowly varying with tempera-ture according to Eq. (76), resulting in a small contribu-tion to c v (see Eqs. (79) and (80)). On the other hand,the energy corresponds to the instantaneous state of thesystem (or averaged over τ ), and is not related to explor-ing the phase space. Consequently, E l = E vib , yieldingEq. (76) and the smallness of diffusional contribution to c v despite S l (cid:54) = S s .We note that the common thermodynamic descriptionof entropy does not involve time: it is assumed that theobservation time is long enough for the total phase spaceto be explored. In a viscous liquid with large τ , thisexploration is due to particle jumps, and is complete atlong times t (cid:29) τ only, at which point the system becomesergodic.If extrapolated below T g , configurational entropy ofviscous liquids reaches zero at a finite temperature, con-stituting an apparent problem known as the widely dis-cussed Kauzmann paradox. More recently, issues in-volved in separating configurational and vibrational en-tropy and interpreting experimental data became appar-ent, affecting the way the Kauzmann paradox is viewedand extent of the problem (see, e.g., Refs. [34], [128] andreferences therein). LIQUID-GLASS TRANSITION
In the previous section, we have ascertained that in theviscous regime τ (cid:29) τ D , liquid energy and heat capacityare essentially given by the vibrational terms. What hap-pens to heat capacity when temperature drops below theglass transition temperature T g and we are dealing withthe solid glass, the non-equilibrium system where τ ex-ceeds observation time?Figure 14 gives a typical example of the change of theconstant-pressure specific heat, c p , in the glass transfor-mation range around T g . If c l p and c g p correspond to thespecific heat above and below T g on both sides of theglass transformation range, c l p c g p = 1 . − . c p at T g is considered as the3
220 240 260 280 300 3201.01.21.41.6
Temperature (K)
Heat capacity (Jg-1K-1)
FIG. 14: Heat capacity of Poly(a-methyl styrene) measuredin calorimetry experiments [128]. Glass transformation rangeoperates in the interval of about 260 −
270 K. “thermodynamic” signature of the liquid-glass transition,and serves to define T g in the calorimetry experiments. T g measured as the temperature of the change of c p coin-cides with the temperature at which τ reaches 10 − s and exceeds the observation time.Most researchers do not consider the change of c p asa signature of the phase transition. This is supportedby the numerous data testifying that the structure of theviscous liquid above T g and the structure of glass arenearly identical. What causes the change of c p at T g ?Recall that liquid response includes viscous responserelated to diffusive jumps and solid-like response. Whenthe viscous response stops at T g during the experimentaltime scale (from the definition of T g ) and only the elasticresponse remains, system’s bulk modulus B and thermalexpansion coefficient α change. This results in different c p above and below T g [103].Lets consider that pressure P is applied to a liquid. Ac-cording to the Maxwell-Frenkel viscoelastic picture, thechange of liquid volume, v , is v = v el + v r , where v el and v r are associated with solid-like elastic deformationand viscous relaxation process. Lets now define T g asthe temperature at which τ exceeds the observation time t . This implies that particle jumps are not operative at T g during the time of observation. Therefore, v at T g isgiven by purely elastic response as in elastic solid. Then,we write P = B l v el + v r V P = B g v g V (81)where V and V are initial volumes of the liquid and the glass, v g is the elastic deformation of the glass and B l and B g are bulk moduli of the liquid and glass, respectively.Let ∆ T be a small temperature interval that separatesthe liquid from the glass such that τ in the liquid, τ l ,is τ l = τ ( T g + ∆ T ) and ∆ TT g (cid:28)
1. Then, V ≈ V .Similarly, the difference between the elastic response ofthe liquid and the glass can be ignored for small ∆ T ,giving v el ≈ v g . Combining the two expressions in (81),we find: B l = B g (cid:15) + 1 (82)where (cid:15) = v r v el is the ratio of relaxational and elasticresponse to pressure.The coefficients of thermal expansion of the liquid andthe glass, α l and α g , can be related in a similar way. Letsconsider liquid relaxation in response to the increase oftemperature by ∆ T . We write α l = 1 V l0 v el + v r ∆ Tα g = 1 V g0 v g ∆ T (83)where v el and v r are volume changes due to solid-likeelastic and relaxational response as in Eq. (81) but nowin response to temperature variation and v g is elasticresponse of the glass. Combining the two expressions for α l and α g and assuming V = V and v el = v g as before,we find α l = ( (cid:15) + 1) α g (84)where (cid:15) = v r v el is the ratio of relaxational and elasticresponse to temperature.Eqs. (82,84) describe the relationships between B and α in the liquid and the glass due to the presence of par-ticle jumps in the liquid above T g and their absence inthe glass at T g , insofar as T g is the temperature at which t < τ . Consistent with experimental observations, theseequations predict that liquids above T g have larger α andsmaller B as compared to below T g .We are now ready to calculate c p below and above T g . In the previous section, we have seen that in thehighly viscous regime, c v is given by the vibrationalcomponent of motion only (see Eq. (80)). Hence, weuse c v = 3(1 + αT ) from Eq. (56) which accounts forphonon softening due to inherent anharmonicity. Writ-ing constant-pressure specific heat c p as c p = c v + nT α B ,where n is the number density, we find c p above and be-low T g as c l p = 3 (1 + α l T ) + nT α B l , T > T g c g p = 3 (1 + α g T ) + nT α B g , T < T g (85)4 -3 -2 -1 Tg (K) q (K/s)
FIG. 15: Increase of T g in Pd Ni P Si glass with thequench rate q [131]. Eq. (85) predicts that temperature dependence of c p should follow that of α , in agreement with simultaneousmeasurements of c p and α showing that both quantitiesclosely follow each other across T g [130].From Eq. (85), c l p c g p can be calculated using experimen-tal B and α above and below T g . This gives good agree-ment with the experimentally observed c l p c g p [103].We have related the change of c p at T g to the change ofsystem’s thermal and elastic properties when the liquidfalls out of equilibrium at T g . It is important to notethat T g is not a fixed temperature. T g decreases with theobservation time, or increases with the quench rate q (see,e.g., [37, 131]). This is a generic effect involved in manyglass transition phenomena where a typical relaxationtime exceeds the experimental time scale.In Figure 15, we show an example of how T g , defined asthe temperature of the jump of heat capacity, increaseswith the quench rate q in an appreciably large tempera-ture range.This effect can be explained as follows. Recall thatthe jump of c p at T g takes place when the observationtime t crosses liquid relaxation time τ . This implies thatbecause q = ∆ Tt , τ at which the jump of heat capacitytakes place is τ ( T g ) = ∆ Tq , where ∆ T is the tempera-ture interval of glass transformation range. Combiningthis with τ ( T g ) = τ D exp( U/T g ) (here U is approximatelyconstant because τ is nearly Arrhenius around T g as dis-cussed in the previous section “Continuity of solid and liquid states...”) gives T g = U ln ∆ Tτ − ln q (86)According to Eq. (86), T g increases with the loga-rithm of the quench rate q . In particular, this increase ispredicted to be faster than linear with ln q . This is con-sistent with experiments [37, 131] and the data in Figure15. We note that Eq. (86) predicts no divergence of T g because the maximal physically possible quench rate isset by the minimal internal time, Debye vibration period τ D , so that ∆ Tτ D in Eq. (86) is always larger than q .Can the “glass transition line” be identified on thephase diagram separating the combined oscillatory anddiffusive particle motion above the line from the purelyoscillatory motion observed below T g during the exper-imental time scale? This would serve as the oppositeto the Frenkel line which separates the combined oscilla-tory and diffusive particle motion below the line fromthe purely diffusive particle motion above the line athigh temperature. As we have seen above, T g dependson the observation time (or frequency), and so no well-defined glass transition line exists on the phase diagrambecause the low-temperature state is a non-equilibriumliquid. The Frenkel line, on the other hand, separatestwo equilibrium states of matter.To summarize this section, we have seen that severalimportant experimental results of the glass transition, in-cluding the heat capacity jump and dependence of T g onthe quench rate can be understood in the picture viewingthe glass as the viscous liquid that falls out of equilibriumat T g .There are several other interesting non-equilibrium ef-fects involved in liquid physics. These serve as goodexamples and case studies that inspire thinking aboutmore general issues, including the foundations of statis-tical mechanics and their modification and extension tonon-equilibrium conditions (see, e.g., [132]). PHASE TRANSITIONS IN LIQUIDS
We have ascertained several solid-like properties of liq-uids in the above discussion. The important basic prop-erty that immediately follows from this picture is thatduring time shorter than τ , the local structure of the liq-uid does not change. This implies the presence of well-defined short- and medium-range order and transverse-like excitations as in solids and gives the possibilityfor the structure to undergo a phase transition. Phasetransitions in liquids have been indeed found, althoughthey were discovered fairly recently and their explorationstarted much later than of phase transitions in solids.First-order transitions in liquids demarcate differentlocal structures and thermodynamic properties. The5transitions are common in multi-component systems andliquid crystals where composition and molecular orienta-tion serve as order parameters. On the other hand, thepossibility of the first-order transition in simple isotropicliquids was not known until about 2-3 decades ago, ex-cept for earlier theoretical works (see, e.g, Ref. [133]).In recent years, phase transformations have been found[47, 134–150] in several different types of liquids, includ-ing in elementary liquids (e.g., P [134, 137, 138, 149],Se [134, 136, 139, 140], S [134, 136], Bi [134, 136], Te[134, 136]), oxide liquids (e.g., H O [134, 150], Y O -Al O [137], GeO [141], B O [146], P O [148]), halo-genides (e.g., AlCl [142], ZnCl [142], AgI [143]), andchalcogenides (e.g., AsS [47], As S [147], GeSe [144]).Pressure-induced transformations are accompanied bystructural changes in both short-range and intermediate-range order as well as changes of all physical properties.Moreover, multiple pressure-induced phase transitionsmay take place in one system: for example, AsS under-goes the transformation between the molecular and cova-lent liquid, followed by the transformation to the metallicphase [47]. The transformations take place in the narrowpressure range and with large changes of structure andmajor properties such as viscosity.Transformations in simple liquids can be both sharpand smeared. The analysis suggests that sharp transi-tions take place in liquids whose parent crystals undergophase transitions with large changes of the short-rangeorder structure and bonding type [151].One of the first examples of sharp liquid-liquid transi-tions is the semiconductor-metal transformation in liquidSe [139]. The transition is accompanied by the changeof the short-range order structure, volume and enthalpyjumps as well as by very large jump of conductivity. Nearthe melting curve, the transition occurs at 700 C and 4GPa. At very high temperatures this transition becomessmooth and finally almost disappears.Another clear example of the sharp liquid-liquid tran-sition in a simple isotropic system is the transition in liq-uid phosphorus [138, 149]. In Figure 16 we show sharpchanges of the structure factor taking place in a nar-row range of pressure and temperature. An abrupt andreversible structural transformation takes place betweenthe low-pressure molecular liquid and the high-pressurepolymeric liquid. This is shown in the phase diagram inFigure 16b. As for Se, the line of liquid-liquid transfor-mation is terminated at very high temperature only andabove 2200 ◦ C.The key to understanding these transitions lies in liq-uid dynamical properties. Indeed, if, as was often thecase in the field, we consider a liquid as a structure-less dense gas, no sharp phase transformations are possi-ble. On the other hand, the oscillatory-diffusive pictureof liquid dynamics based on τ offers a different insight.In the regime τ (cid:29) τ D , particles perform many oscilla-tions around fixed positions before jumping to the nearby P = 0 . 6 6 G P a , T = 1 5 5 0 ° C S(q)
P = 0 . 6 8 G P a , T = 1 6 2 0 ° C ( a ) Temperature ( ºC ) P r e s s u r e ( G P a ) q ( Å - 1 )( b ) p o l y m e r i c l i q u i dm o l e c u l a r l i q u i d o r t h o r h o m b i c c r y s t a l FIG. 16: (a) Sharp changes in the structure factor in liquidphosphorus in a narrow range of pressure and temperature.(b) Phase diagram showing the transition line between molec-ular and polymeric phosphorus. The data are from Ref. [149]. quasi-equilibrium sites. Therefore, a well-defined short-and medium-range order exists during time τ . In thiscase, liquids not far above the melting point can supportpressure-induced sharp or smeared structural changes,similarly to their solid analogues.Interestingly, the critical point of the liquid-gas tran-sition in phosphorus is around 695 ◦ C and 8.2 MPa. Thismeans that the transition between the molecular andpolymeric fluids takes place in the supercritical state.This may have come as a surprising finding in view of theperceived similarity of the supercritical state in terms ofphysical properties. Yet, as discussed above, well-definedshort- and medium-range order exist in liquids above thecritical point as long as the system is in the “rigid”-liquidstate below the FL where τ (cid:29) τ D . We can thereforepredict that liquid-liquid transitions in the supercriticalstate operate in the rigid-liquid below the FL but not inthe non-rigid gas-like fluid state above the line.Not surprisingly and similar to solids, liquid-liquidphase transitions are accompanied by the change of spec-6trum of collective modes. Recently, the evidence forthis has started to come from experiments and model-ing [152].Understanding liquid structure and its response topressure and temperature will continue to benefit fromthe development of experimental techniques and in-situexperiments in particular (see, e.g., Refs. [153, 154] forreview). QUANTUM LIQUIDS: SOLID-LIKE ANDGAS-LIKE APPROACHES
A quantum liquid is a liquid at temperature low enoughwhere the effects related to particle statistics, Bose-Einstein or Fermi-Dirac, become operable. Quantumliquids is a large area of research (for review, see, e.g.[2, 3, 155–158]), largely stimulated by superfluidity inliquid helium. Here, we point to gas-like and solid-likeapproaches to quantum liquids and to similarities anddifferences of these approaches to those used in classicalliquids discussed earlier.The solid-like approach to the thermodynamics ofquantum liquids is due to Landau. Emphasizing stronginteractions in the liquid and rejecting earlier proposalswhich did not, Landau asserted that the energy of a low-temperature quantum liquid, such as liquid helium atroom pressure, is the energy of the longitudinal phononmode [2].In this consideration, the quantum nature of the liq-uid simplifies the understanding of its thermodynamics:Landau argued that any weakly perturbed state of thequantum system is a set of elementary excitations, orquasi-particles. In the low-temperature quantum liquid,the quasi-particles are phonons and are the lowest energystates in the system. This gives the solid-like heat capac-ity of a quantum liquid equal to that in the quantum solidbut with one longitudinal mode only [2]: c v = 2 π n
15 (¯ hu ) T (87)where n is the number density and u is the speed ofsound.Eq. (87) is in agreement with the experimental heatcapacity of liquid helium at room pressure.Interestingly, Landau assumed that only one longitudi-nal mode contributes to the energy of a low-temperaturequantum liquid and did not consider high-frequencytransverse modes predicted earlier by Frenkel. This hasbeen consistent with the absence of direct experimentalevidence of transverse modes in liquid helium at roompressure. However, it is interesting to ask whether oneshould generally consider transverse modes in a hypo-thetical low-temperature liquid. As we have seen ear-lier (see section “Phonon excitations at low temperature” above), transverse modes do not contribute to the liquidenergy in the limit of zero temperature. Hence Landau’sassumption turned out to be correct.In addition to explaining the experimental heat ca-pacity, the solid-like phonon picture of liquid heliumexplained superfluidity. Superfluidity emerges due tothe impossibility to excite phonons in the liquid movingslower than the critical velocity. In the original Landautheory, the critical velocity is the speed of sound. Con-siderably lower critical velocity found experimentally waslater attributed to other effects such as energy-absorbingvortices.The above low-temperature picture is discussed in thelinear dispersion regime, (cid:15) = cp . At higher tempera-tures, higher phonon branches become excited, includ-ing the roton part of the spectrum. Interestingly, theroton part, originally thought to be specific to helium,later discussed in the context of the Bose-Einstein con-densate (BEC) [155] and thought to be unusual in morerecent discussions [158], is seen in many classical high-temperature liquids (see, e.g., [76], [70] and Figure 7).In addition to the solid-like approach to liquid he-lium mentioned above, the hydrodynamic approach hasbeen widely used to discuss hydrodynamic effects (nat-urally) such as density waves (first sound) and temper-ature or entropy waves (second sound) and their veloc-ities [3, 156]. Interestingly and similar to the classicalliquids, two regimes of wave propagation and two soundsare distinguished depending on ω . Waves with ωτ < ωτ q > τ q is the lifetime of the quasi-particle excitation [156], and is analogous to the solid-likeelastic modes in classical liquids discussed above.Interesting problems related to gas-like versus solid-approach emerge when the question of BEC in liquid he-lium is considered. As in the previous discussion, we canidentify two approaches: gas-like and solid-like. The gas-like approach is due to Bogoliubov, and starts with theHamiltonian describing weakly perturbed states of theBose gas: H = (cid:88) p p m a +p a p + 12 (cid:88) U p (cid:48) p (cid:48) p p a + p (cid:48) a + p (cid:48) a p a p (88)where the first and second terms represent kinetic andpotential energy, a + p , a p are creation and annihilationoperators and U p (cid:48) p (cid:48) p p is the matrix element of the pairinteraction potential U ( r ).Without the second term, the ground state of the sys-tem is the BEC gas state. For weak interactions, theenergy levels of the system can be calculated in the per-turbation theory. As a result, the diagonalised Hamilto-7nian reads [2]: H = E + (cid:88) (cid:15) ( p ) b +p b p (cid:15) ( p ) = (cid:115) u p + (cid:18) p m (cid:19) u = (cid:115) π ¯ h nam (89)where n is concentration, a = m π ¯ h U and U is the vol-ume integral of the pair interaction potential.According to Eq. (89), the presence of interactionsmodifies the energy spectrum of the Bose gas and resultsin the emergence of the low-energy collective mode withthe propagation speed u . At small momenta, (cid:15) = up .This result is analogous to the gas-like approach toclassical liquids where the weak interactions result in thelow-frequency sound. What happens when interactionsare strong as in liquid helium and when the perturbationtheory does not apply? Here, we face the same problemof strong interactions as in the classical case.Landau rejected the possibility of BEC in a strongly-interacting system: in his view, the low-energy statesof the strongly-interacting system are collective modesrather than single-particle states as in gases, the picturesimilar to quantum solids where phonons are the lowestenergy states and where BEC is irrelevant. In later devel-opments, BEC was generalized for the case of strongly-interacting system on the basis of macroscopic occupa-tion of some one-particle state. It was estimated that inlow-temperature liquid helium, about 10% of atoms arein the BEC state while the rest is in the normal state(in this picture, the interactions “deplete” BEC) [156].The BEC component is then related to the superfluidcomponent, and its weight changes with temperature.It is probably fair to say that compared to well-studiedeffects of BEC in gases, operation of BEC in liquids is notunderstood in a consistent and detailed picture. Pinesand Nozieres remark [156] that a quantitative micro-scopic theory of liquid helium is yet to emerge. Leggettcomments on the challenge of obtaining direct experi-mental evidence of BEC in liquid helium as compared togases [157].We now recall our starting picture of liquids where par-ticle motion includes two components: oscillatory anddiffusive. Can quantum liquids be understood on thebasis of these two types of motion only, similarly to clas-sical liquids? An interesting insight has come from path-integral simulations [159]: the emergence of macroscopicexchanges of diffusing atoms contributes to the λ -peakin the heat capacity, confirming the earlier Feynman re-sult [160], and is related to momentum condensation andsuperfluidity.This picture enables one to adopt the solid-like ap-proach to quantum liquids (instead of the commonly dis- cussed gas-like approach): we approach the system fromthe solid state where strong interactions and resultingcollective modes are considered as a starting point, andintroduce diffusive particle jumps as in the classical case.From the thermodynamic point of view, these jumps onlymodify the phonon spectrum in classical liquids. In quan-tum liquids, they additionally contribute to the exchangeenergy because particle jumps enable the effect of quan-tum exchange [161].Can the exchange energy be related to exchange fre-quency ω F , as is the case for liquid energy in Eqs. (51),(54) or (60)? This would amount to a Frenkel reductiondiscussed earlier but applied to the exchange energy. Wethink interesting insights may follow. We feel that gen-erally developing closer ties between the areas and toolsof classical and quantum liquids should result in new un-derstanding. MIXED AND PURE DYNAMICAL STATES:LIQUIDS, SOLIDS, GASES
The emphasis of our review has been on understand-ing experimental and modeling data and on providingrelationships between different physical properties. Inaddition to this rather practical approach, we can revisita more general question alluded to in the Introduction:how are we to view and classify liquids in terms of theirproximity to gases or solids? Throughout the history ofliquid research, different ways of addressing this questionwere discussed [1, 2, 4–8, 10, 12, 15, 16].On the basis of discussion in this paper, our answeris that liquids do not need such a classification, or anyother compartmentalizing for that matter. With their in-teresting and unique properties, liquids belong to a stateof their own. Throughout this review, we have seen thatmost important properties of liquids and supercritical flu-ids can be consistently understood in the picture wherewe are compelled to view them as distinct systems in thenotably mixed dynamical state. Particles undergo bothoscillatory motions and diffusive jumps, and the relativeweight of the two components of motion changes withtemperature. As discussed in the section “Viscous liq-uids” above, this relative weight is quantified by the ratio τ D τ , which we now define as parameter R : R = τ D τ = ω F ω D (90)We have seen that R enters the energy of both low-viscous liquids (see Eqs. (51),(54),(60)) and highly-viscous liquids including in the glass transformationrange (see Eqs. (69),(70),(73)) and is implicitly presentthroughout our discussion.In liquids, R varies between 0 and 1, and defines theliquid’s proximity to the solid or gas state. This enables8us to delineate solids and gases as two limiting cases interms of dynamics and thermodynamics.In solids, particle motion is purely oscillatory, corre-sponding to R = 0. Indeed, τ → ∞ in ideal crystals orbecomes astronomically large in familiar glasses such asSiO at room temperature [103].In gases, particle motion is purely diffusive. This cor-responds to R = 1, as is the case in the supercritical stateabove the FL where the oscillatory component of particlemotion is lost and where τ ≈ τ D .We note that R = 1 at the FL above the critical pointor in subcritical liquids constitutes a microscopic andphysically transparent criterion of the difference betweenliquids and gases [112]. Indeed, existing common criteriainclude distinctions such as that gas fills available vol-ume but liquid does not, or that gas does not possessa cohesive state but liquid does. These criteria are ei-ther not microscopic, are tied to a particular pressurerange or can not be implemented in practice [112]. Onthe other hand, asserting that the gas state is charac-terized by purely diffusive dynamics of particles whereasthe liquid state includes both diffusive and oscillatorycomponents of particle motion gives a microscopic andphysically transparent criterion.We therefore find that R = 0 and R = 1 give solidsand gases as two limiting cases of dynamical properties.In this sense, gases and solids are pure states of matterin terms of their dynamics. It is for this reason that theyhave been well understood theoretically. Liquids, on theother hand, are a mixed state in terms of their dynamics,the state that combines solid-like and gas-like motions. Itis the mixed state which has been the ultimate problemfor the theory of liquids.On the basis of R -parameter, we see that liquids canonly be viewed as solid-like or gas-like when R is eitherclose to 0 or 1. In all other cases, liquids are thermody-namically close to neither state. This becomes apparentfrom looking at the experimental thermodynamic datasuch as in Figure 9. This highlights our earlier pointabout the distinct mixed dynamical state of liquids andassociated rich physics.Once the last assertion is appreciated, theorists be-come better informed about what approach to liquids ismore appropriate. The best starting point for liquid the-ory is to make no assumptions regarding the proximityof liquids to gases or solids and seek no extrapolationsof the hydrodynamic regime to the solid-like regime andvice versa. Instead, the best starting point is to con-sider the microscopic picture of liquid dynamics and itsmixed character from the outset, and recognize that therelative weights of diffusive and oscillatory componentschange with temperature. Depending on the property inquestion, we can encounter several possibilities.If we are concerned with long-time and low-energy ob-servables only ( t > τ or ωτ < ωτ >
1. In this case,we can focus on the solid-like regime of liquid dynamicsfrom the outset and treat it separately and independentlyfrom the hydrodynamic regime. In this approach, we donot need to extrapolate the hydrodynamic descriptioninto the solid-like regime as is done in generalized hydro-dynamics and where extrapolation schemes may be anissue.Each regime, hydrodynamic or solid-like, can be an-alyzed separately. There are also mixed cases where,for example, we observe solid-like high-frequency modes( ωτ >
1) at long times ( t > τ ) because we are inter-ested in their propagation length. Here, we can start witheither hydrodynamic or elasticity equations and modifythem appropriately. This gives the same results as wehave seen above.
CONCLUSIONS AND OUTLOOK
Our important conclusion regarding the theoreticalview of liquids has already been made in the previoussection. In this review, we discussed how this viewevolved and how different ideas proposed at very differ-ent times were developing. With the recent evidence forhigh-frequency solid-like modes in liquids, it has now be-come possible to use the solid-like approach to liquids anddiscuss their most important thermodynamic propertiessuch as energy and heat capacity. We have reviewed howthis can be done for liquids in different regimes: low-viscous subcritical liquids, high-temperature supercriti-cal gas-like fluids, viscous liquids in the glass transfor-mation range and systems at the liquid-glass transition.In each case, we have noted limitations and caveats ofthis approach throughout this review.As alluded to in the Introduction, liquids have beenviewed as inherently complicated systems lacking usefultheoretical concepts such as a small parameter. New un-derstanding of liquids, including the increasing amount ofhigh-energy experimental data and its quantitative agree-ment with predicted thermodynamic properties, changethis traditional perspective. We are beginning to un-derstand liquid thermodynamics on the basis of high-frequency collective modes. Contrary to the pessimisticand vague picture often drawn about them, liquids areemerging as exciting and unique systems amenable totheoretical understanding in a consistent picture.Several points can be mentioned that may advance liq-9uid research. The evidence for high-frequency collectivemodes and transverse modes in particular has started toemerge fairly recently. It will be interesting to widenthe number of systems with mapped solid-like disper-sion curves and go beyond simple liquids. It will also beinteresting to extend the experiments to high tempera-ture and pressure including the supercritical state andto follow the evolution of collective modes as predictedtheoretically. This can be directly compared to the con-comitant variation of thermodynamic properties such asheat capacity.We have not reviewed molecular dynamics (MD) sim-ulations although in places we discussed modeling aimedat backing up experiments and theory. MD simulationsof liquids are as old as the method itself: indeed, the needfor MD simulations was originally rationalized by the dif-ficulty to construct liquid theory [162], with simulationsplaying the role of testing the theory. With the firstsimulation of liquids performed in 1957, the generateddata exceeds what is feasible to review. For more recentexamples, an interested reader can consult liquid text-books and review papers cited throughout this review.A common issue faced by computer simulations is thesame as in experiments: understanding and interpretingthe data. With reliable interatomic potentials existingtoday, it is not hard to calculate c v shown in Figure 9,but understanding the results requires a physical model.As far as liquid heat capacity is concerned, it is fair to saythat MD simulations have not resulted in understandingliquid c v such as shown in Figure 9. Once liquid ther-modynamics is better understood, MD simulations willprovide interesting microscopic insights and potentiallyuncover novel effects. These can include the operationof collective modes in the solid-like elastic regime andtheir evolution at conditions not currently sampled byexperiments including in the supercritical state.We feel that bringing concepts and tools from classi-cal and quantum liquids closer may result in new un-derstanding, particularly in the area of thermodynamicsand operation of BEC in real strongly-interacting liquids.Exploring the mixed state of liquid dynamics and the sep-aration of solid-like oscillatory and gas-like diffusive par-ticle dynamics in quantum liquids may bring unexpectednew insights.We are grateful to S. Hosokawa and A. Mokshin fordiscussions and providing data and to EPSRC for sup-port. [1] Frenkel J 1947 Kinetic Theory of Liquids (Oxford Uni-versity Press)[2] Landau L D and Lifshitz E M 1969
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