Comment on "Collective modes and gapped momentum states in liquid Ga: Experiment, theory, and simulation"
aa r X i v : . [ c ond - m a t . d i s - nn ] A ug Comment on ”Collective modes and gapped momentum states inliquid Ga: Experiment, theory, and simulation”
Taras Bryk , , Ihor Mryglod , Giancarlo Ruocco , Institute for Condensed Matter Physics,National Academy of Sciences of Ukraine,UA-79011 Lviv, Ukraine Institute of Applied Mathematics and Fundamental Sciences,Lviv National Polytechnic University, UA-79013 Lviv, Ukraine Center for Life Nano Science @Sapienza, Istituto Italiano di Tecnologia,295 Viale Regina Elena, I-00161, Roma, Italy and Dipartimento di Fisica, Universita’ di Roma ”La Sapienza”, I-00185, Roma, Italy (Dated: August 13, 2020)
Abstract
We show that the presented in [1] theoretical expressions for longitudinal current spectral func-tion C L ( k, ω ) and dispersion of collective excitations are not correct. Indeed, they are not com-patible with the continuum limit and C L ( k, ω →
0) contradicts the continuity equation. . INTRODUCTION In a recent paper [1] the authors formulated their ”overarching goal of this research pro-gramme ... to reach the stage where, despite the complexity of their theoretical description,liquids emerge as systems amenable to theoretical understanding at the level comparable togases and solids”. Looking at Figs.4 and 5 of [1] one can really make sure that the authorsof [1] reached their ambitious goal in perfect agreement between the proposed theory andcomputer simulations. In this paper the authors proposed theoretical expressions for thelongitudinal current spectral function C L ( k, ω ), with k and ω being wave number and fre-quency, and for the dispersion of longitudinal collective excitations ω Lc ( k ). Their expressionsfor C L ( k, ω ) (Eq.18) and ω Lc ( k ) (Eq.20), as one can judge from their Figs.4 and 5, recoverwith high precision the molecular dynamics (MD) data in a wide range of wave numbersand temperatures. The C L ( k, ω ) in their theoretical scheme was obtained from a simplecontinued fraction shown in their Eq.11. Although the standard approach for description ofcollective dynamics in liquids is to represent the Laplace-transformed density-density timecorrelation function as a continued fraction[2, 3], in [1] the authors derived the continuedfraction for the longitudinal currrent-current correlations. Applying different closures for thechain of memory functions like in [4–6] one can obtain formal solution for C L ( k, ω ) withina precision of several its frequency moments.However, such an approach of Ref.[1] is not really consistent with the hydrodynamics[4, 7],which is a collection of local conservation laws. Any liquid system on the spatial scales muchlarger than the mean interatomic distance must behave similarly from the point of view ofslow collective modes derived by fluctuations of conserved quantities. In [1] the proposedtheoretical approch is developed from a single conserved dynamic variable, longitudinal com-ponent of total momentum J L ( k, t ), which is the slowest dynamic variable in the presentedapproach. It is well known from the textbooks [4, 7] as well as from other multivariableapproaches [8–10] which dynamic variables are responsible for description of the viscoelas-tic transition in dispersion of collective excitations [11, 12]. The theoretical approach [1]does not contain coupling of longitudinal current fluctuations with the fluctuations of otherconserved quantities, namely density n ( k, t ) and energy e ( k, t ) ones. The energy (or heat)density fluctuations reflect specific for liquids fluctuations of local temperature[13], andlong-wavelength heat relaxation processes are responsible for the central Rayleigh peak of2he dynamic structure factor S ( k, ω ) for one-component liquids at sufficiently small wavenumbers k . Outside the hydrodynamic regime the short-wavelength density fluctuations n ( k, t ) reflect the processes connected with structural relaxation and instead of heat relax-ation form the leading contribution to the central peak of S ( k, ω ) [12, 14, 15]. The presenceof heat and density relaxation, therefore, are essential ingredients for a correct descriptionof the spectra, including the propagating density fluctuations regions, which are the maintarget of Ref.[1].The poor theoretical approach presented in Ref.[1], missing the coupling with the mostimportant for liquids slow processes, is an oversimplified theory. It is, therefore, difficultto understand why it is able to reproduce to a very good degree of accuracy the moleculardynamics (MD) data for C L ( k, ω ) in some region of wave numbers as it is shown in their Fig.4[1]. Moreover, we were motivated to understand why their expressions were able to recoverthe adiabatic speed of sound in the long-wavelength region of their Fig.5. Our question was:is it possible within the proposed fit-free theoretical scheme to obtain in the long-wavelengthlimit the propagating modes with adiabatic speed of sound c s ? The multivariable approachesbased on the set of dynamic variables { J L ( k, t ) , ˙ J L ( k, t ) , .... } usually can produce in the long-wavelength limit the propagating modes only in elastic regime with propagation speed beingthe high-frequency one c ∞ slightly renormalized due to the coupling to faster kinetic modes.No viscoelastic effects like positive sound dispersion can be expected in this theory.Motivated by the surprisingly good agreement shown in their Fig.4 we will check theexpressions (Eqs.17-20) of [1] and behavior of their ”relaxation parameters” ∆ i ( k ) in the k → i ( k ) andtheir Eqs.17-20. In the next Section we provide details of our MD simulations and calcu-lations of corresponding correlators. Then we will present our resuts and discuss them incomparison with the Eqs.17-20 of [1]. The last Section contains conclusion of this study. II. DETAILS OF MD SIMULATIONS
We performed molecular dynamics simulations for supercritical Ne at T=295 K anddensity 1600 kg/m using its Lennard-Jones potentials the same as in our previous study[16]. A model system of 4000 particles was simulated in microcanonical ensemble with perfect3 < d J | d J > , < d J | J > (r ed . un . ) k [A (cid:16)° -1 ]
FIG. 1: Check of the properties of time derivatives of longitudinal current for static correlators h ˙ J L ( − k ) ˙ J L ( k ) i ≡ −h ¨ J L ( − k ) J L ( k ) i (a) and h ¨ J L ( − k ) ¨ J L ( k ) i ≡ −h ... J L ( − k ) ˙ J L ( k ) i (b) for supercriticalNe at T=295 K and density 1600 kg/m . energy conservation over the whole production run of 300 000 time steps. The time step was0.5 fs. Our main task was in sampling the space-Fourier components of all hydrodynamicvariables, i.e. of density n ( k, t ), mass-current J ( k, t ) and energy e ( k, t ), as well as of theirtime derivatives, in particular, of the mass-current up to the third order ... J ( k, t ). We sampledall the possible wave vectors corresponding to the same absolute value, and used all themin spherical average of the corresponding correlators. The smallest wave number sampled inthis MD study was 0.143598˚A − .In order to check reliability of the sampled time derivatives of the longitudinal mass-current and of our calculated static correlators we made use of the exact relations, whichfollow from a property of time derivatives of time correlations functions [4] h ˙ J L ( − k ) ˙ J L ( k ) i ≡ −h ¨ J L ( − k ) J L ( k ) ih ¨ J L ( − k ) ¨ J L ( k ) i ≡ −h ... J L ( − k ) ˙ J L ( k ) i . One can see in Fig.1 that perfect equivalence (difference less than 0.2% for any k -point)is the evidence of correct direct sampling of J L ( k, t ), ˙ J L ( k, t ) ¨ J L ( k, t ) and ... J L ( k, t ) in MDsimulations. These dynamic variables are needed for calculations of quantities ∆ i ( k ) , i =1 , , C L ( k, ω ) and ω Lc ( k ) in [1]. Throughout this paper we will usereduced units of energy k B T = 1, mass m = 1 and time τ σ = 1 . ps II. RESULTS AND DISCUSSION
As we mentioned above the perfect agreement between the proposed in [1] fit-free theoryand MD results for C L ( k, ω ) in their Fig.4 looks too good to be true. Indeed, a simplestcheck of their Eq.18 in the ω → C L ( k, ω = 0) C L ( k, ω = 0) = 1 π ∆ ( k )∆ ( k )∆ ( k ) / B ( k ) ≡ π ∆ ( k )∆ ( k )∆ ( k ) / , (1)while any viscoelastic theory must result in C L ( k, ω = 0) ≡ C L ( k, ω → ∝ ω behavior from their fit-free theory (their Eq.18).We calculated from their Eqs.16-17 the ”relaxation parameters” ∆ i ( k ) , i = 1 , , ( k ) + ∆ ( k ) = h ¨ J ( − k ) ¨ J ( k ) ih ˙ J ( − k ) ˙ J ( k ) i , where the right hand side tends to a constant in long-wavelength limit and is simply theratio of k -dependences shown in Fig.1(a,b), and∆ ( k ) = [ h ... J ( − k ) ... J ( k ) i − h ¨ J L ( − k ) ¨ J L ( k ) i h ˙ J L ( − k ) ˙ J L ( k ) i ] / [ h ¨ J L ( − k ) ¨ J L ( k ) i − h ˙ J L ( − k ) ˙ J L ( k ) i h J L ( − k ) J L ( k ) i ] . In Fig.2 we show the k -dependence of the ”relaxation parameters”[1] and one can see theparameters ∆ ( k ) and ∆ ( k ) tending in the long-wavelength limit to non-zero values while∆ ( k ) ≡ h ˙ J L ( − k ) ˙ J L ( k ) ih J L ( − k ) J L ( k ) i behaves in k → ∝ c ∞ k with c ∞ being the high-frequency speed of sound.Now we can estimate how large is the deviation of C L ( k, ω = 0) from the correct zerovalue. Since the ∆ ( k ) goes to zero in the long-wavelength limit and ∆ ( k →
0) and∆ ( k →
0) tend to finite non-zero values, the resulting C L ( k, ω = 0) taken from Eq.18of [1] should diverge for k →
0. Indeed, in Fig.3 one can observe the strong increase of C L ( k, ω = 0) ∝ k − in [1], that means wrong theoretical result comparing with the exactrelation C L ( k, ω →
0) = 0.Now we will analyze the expression for dispersion of collective excitations [1]. Sinceonly the ”relaxation parameter” ∆ ( k ) tends to zero as k in the long-wavelength limit,5
10 100 1000 10000 100000 0 0.5 1 1.5 2 2.5 3 ∆ i τ σ k [A (cid:16)° -1 ] ∆ ∆ ∆ FIG. 2: Dependence of the ”relaxation parameters” ∆ i , i = 1 , , . C L ( k , ω = ) (r ed . un . ) k [A (cid:16)° -1 ]Eq.18 of [1] FIG. 3: Dependence of the zero-frequency value C L ( k, ω = 0) (Eq.18 of [1]) on wave numbers forsupercritical Ne at T=295 K and density 1600 kg/m . and higher ”relaxation parameters” ∆ , ( k ) tend to constants in that limit, one can easilyestimate, that their Eq.(20) for ω Lc ( k ) tends to a constant for k → ω Lc ( k →
0) = ∆ (0) q (0) − ∆ (0)] , while the correct dispersion law had to recover in that limit the hydrodynamic dispersionlaw ω ( k →
0) = c s k . In Fig.4 we show the dispersion of collective acoustic modes estimatedfrom the peak positions of MD-derived C L ( k, ω ) (plus symbols with error bars) and compareit with the dispersion of ”bare” (non-damped) high-frequency modes which in the long-wavelength limit have linear dispersion with the high-frequency (elastic) speed of sound6 ∞ ω ∞ ( k →
0) = [ h ˙ J L ( − k ) ˙ J L ( k ) ih J L ( − k ) J L ( k ) i ] / | k → → c ∞ k . (2)The coupling to the faster dynamic modes (connected with higher time derivatives of thelongitudinal current) can only slightly renormalize down the theoretical dispersion law, how-ever it will never result in the hydrodynamic speed of sound c s and positive sound dispersion[11]. Within the proposed in [1] theoretical approach is impossible to obtain the propagat-ing modes with adiabatic speed of sound, because in order to obtain it one has to includecoupling with density and energy (or heat) density fluctuations into the theoretical scheme.And, as it was expected from the wrong behavior of C L ( k, ω ) discussed above, the proposedexpression for dispersion of longitudinal collective excitations is wrong too. In Fig.4 onlyfor two lowest k -values we obtained the positive expression under the square root in theirEq.(20). For higher wave numbers the expression under the square root became negative,i.e. no propagating modes for those wave numbers. It is not clear how in Fig.5 of [1] theauthors were able to reproduce perfectly the MD data by using their Eq.20 and even reachthe adiabatic speed of sound in the long-wavelength region, that is impossible to do in theirtheoretical approach. Even conceptually their theoretical approach, which does not containcoupling to fluctuations of conserved quantities, density n ( k, t ) and energy density e ( k, t ),and Eq.20 cannot result in the long-wavelength limit in the linear dispersion with the adi-abatic speed of sound. In their run for the ”overarching goal of this research programme”the authors forgot about the existing methodologies of calculations and theories of collectiveexcitations in liquids, which correctly satisfy exact relations and a large number of sumrules.Another point we want to discuss here is the claimed ”gapped momentum states”[1].It sounds strange that the authors are trying to represent the well known in the literatureshear waves with a propagation gap as some special finding and rename them as the ”gappedmomentum states”. The title of [1] stating ”Collective modes and gapped momentum states...” clearly discriminates between the ”collective modes” and ”gapped momentum states”that is not correct because there is no difference between ordinary collective shear wavesand ”gapped momentum states”. Moreover, it has been known for long time that othercollective propagating processes in liquids have very similar behavior of their dispersion,like heat waves [10, 12, 17] or optic-like modes in binary liquids with demixing tendencies[12, 18]. We would like to remind the readers that by 2017 the same group assured the7 F r equen cy [ p s - ] k [A (cid:16)° -1 ] MD ∆ Eq.20 of [1]
FIG. 4: Peak positions of the longitudinal current spectral function C L ( k, ω ), obtained fromMD simulation (plus symbols with error bars). The dispersion of the nondamped high-frequencyacoustic-like modes with long-wavelength asymptote (2) is shown by line-connected cross symbols.Eq.20 of [1] (line-connected star symbols) contains positive expression unders square root only fortwo lowest k -points, for larger k -values no real ω Lc ( k ) exist. community in Frenkel-like dispersion of the transverse excitations in liquids[19, 20], i.e.when the transverse excitations in liquids exist only above the so-called Frenkel frequencycut-off, that contradicted the existed theories of transverse exsitations[4, 22, 23] and MD data(see our discussion in [16]), which evidenced on existing the long-wavelength propagationgap for shear waves. In 2017 the same authors revealed that the dispersion of shear wavesindeed starts from zero frequency outside the propagation gap and published a paper [21]in which claimed that the propagation gap originates from the Frenkel jumps and is definedby the single-particle Frenkel time (quoting [21]: ” τ is understood to be the full period ofthe particles jump motion equal to twice Frenkels τ ”). That claim again contradicted theexisted theory of transverse excitations in liquids[4, 22, 23], in which the collective shearstress relaxation with Maxwell relaxation time is responsible for the propagation gap andwe showed several times that there is huge difference between the collective and single-particle relaxation processes in their effect on transverse dynamics[24, 25]. Now, in [1] thesame group started to rename the ordinary shear waves of liquid dynamics as ”gappedmomentum states”. 8 V. CONCLUSION
The proposed in [1] theoretical scheme for description of longitudinal collective excitationsin simple liquids is not consistent with hydrodynamics, because only one hydrodynamicvariable, the longitudinal current, was used in that scheme, that rised questions whether theobtained in [1] expressions for longitudinal current spectral function C L ( k, ω ) and for thedispersion of collective excitations are correct. We performed molecular dynamics simulatinson a simple supercritical Ne at 295K and density 1600 kg/m with a purpose of numericalcheck of these expressions.We showed that the proposed in [1] expression for C L ( k, ω ) does not have correct low-frequency limit C L ( k, ω →
0) and even diverges in the long-wavelength limit, that is wrong,while according to the continuity equation it must be C L ( k, ω = 0) ≡
0. Why their Fig.4shows perfect agreement of their theoretical C L ( k, ω ) with MD data we cannot explain.Within the proposed in [1] theoretical scheme it is impossible to recover the hydrodynamicdispersion law in k → c s we cannot explain. We wouldsuggest the authors of [1] to show their similar checks for the correlators h ¨ J L ( − k ) J L ( k ) i and h ... J L ( − k ) ˙ J L ( k ) i as we presented in Fig.1, as well as to reveal the k -dependence of their∆ i ( k ). This defintely will allow to find out why the low-frequency limit of C L ( k, ω ), theirEq.18, and the long-wavelength limit of ω Lc ( k ), their Eq.20, do not correspond to the datain their Figs.4 and 5, respectively. [1] R.M. Khusnutdinoff, C. Cockrell, O.A. Dicks, A.C.S. Jensen, M.D. Le, L. Wang, M.T. Dove,A.V. Mokshin, V.V. Brazhkin, K. Trachenko, Phys. Rev. B , 214312 (2020).[2] J.R.D. Copley, S.W. Lovesey, Rep. Prog. Phys. , 461 (1975).[3] T. Scopigno, G. Ruocco and F. Sette, Rev.Mod.Phys., , 881 (2005).
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