On the universality of AdS_2 diffusion bounds and the breakdown of linearized hydrodynamics
OOn the universality of AdS diffusion boundsand the breakdown of linearized hydrodynamics Ning Wu , Matteo Baggioli , ∗ and Wei-Jia Li † Institute of Theoretical Physics, School of Physics,Dalian University of Technology, Dalian 116024, China.Wilczek Quantum Center, School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China andShanghai Research Center for Quantum Sciences, Shanghai 201315.
The chase of universal bounds on diffusivities in strongly coupled systems and holographic modelshas a long track record. The identification of a universal velocity scale, independent of the presenceof well-defined quasiparticle excitations, is one of the major challenges of this program. A recentanalysis, valid for emergent IR fixed points exhibiting local quantum criticality, and dual to IR AdS geometries, suggests to identify such a velocity using the time and length scales at which hydrody-namics breaks down – the equilibration velocity. The latter relates to the radius of convergence ofthe hydrodynamic expansion and it is extracted from a collision between a hydrodynamic diffusivemode and a non-hydrodynamic mode associated to the IR AdS region. In this short note, weconfirm this picture for holographic systems displaying the spontaneous breaking of translationalinvariance. Differently from the case of energy and charge diffusion, here, the collision determiningthe breakdown of the hydrodynamics expansion appears for real values of the wave-vector. More-over, we find that, at zero temperature, the lower bound set by quantum chaos and the upperone defined by causality and hydrodynamics exactly coincide, determining uniquely the diffusionconstant. Finally, we comment on the meaning and universality of this newly proposed prescription. INTRODUCTION
The more specific we are, the more universalsomething can become. Jaqueline Woodson
The search for universal features in the transportproperties of many-body quantum systems, stronglycoupled materials and holographic models has a longhistory. In this context, universality is intended asinsensitivity to the specific microscopic details of thesystem and it therefore resonates nicely with the conceptof hydrodynamics [1]. Hydrodynamics is an effective ∗ [email protected] † [email protected] description controlling the long time and large scalesdynamics, where all the short-lived operators carryingthe microscopic information get washed out. In thissense, the universal properties remaining are carriedby the long-lived quantities and they are consequentlyrelated to the so-called hydrodynamic modes and thecorresponding conservation equations. Importantly,using this broad terminology, hydrodynamics can beapplied to any physical systems and is not restricted tothe description of fluids [2].In this ballpark, a milestone result has been the iden-tification of a universal lower bound on the ratio of shearviscosity η to entropy density s , which supposedly holds a r X i v : . [ h e p - t h ] F e b for any system in nature. The resulting inequality ηs ≥ (cid:126) π k B (1)takes the name of Kovtun-Son-Starinets (KSS) bound [3]and it has been derived using a dual gravitational descrip-tion in terms of the black hole horizon dynamics and thegravitons absorption rate therein [4, 5]. Notice how thisbound immediately connects with hydrodynamics sincethe η/s ratio coincides exactly with the transverse mo-mentum dimensionless diffusion constant in a neutral rel-ativistic fluid [6]. Indeed, for neutral relativistic systems,the KSS bound can be re-written as: D shear ≥ c π τ pl (2)where D shear is the diffusion constant of the shearmode, c the restored speed of light and τ pl ≡ (cid:126) /k B T the so-called Planckian time [7, 8]. This last quantityplays an important role and it has been involved inseveral discussions and experimental observations aboutuniversality and transport [3, 9–19]Despite the great success of the KSS bound even whenconfronted with realistic experimental data [20–24], itsoon became clear that the inequality in Eq.(1) couldbe violated by breaking explicitly and/or spontaneouslyspacetime symmetries, such as translations and rota-tions . The references reporting on these violations areindeed several [26–32]. From the physical point of view,these cases were accompanied by the observation thatthe ratio η/s plays a very special role only in relativisticneutral fluids, while it is not connected with any specifictransport properties otherwise. A clear example is thatof a non-relativistic system in which the KSS bound canbe violated just by increasing the number of differentspecies [33]. In view of these facts, the universal char-acter of the KSS bound has been recently discredited [34].In a parallel line of investigation [35–37], the superior(in the sense of more general and universal) role ofthe diffusion constants (compared to the η/s ratio forexample) has been outlined in the context of realisticliquids and simple bounds on momentum and energydiffusion have been derived in terms of few fundamentalphysical constants.Inspired by the equivalent formulation of the KSSbound in terms of the shear diffusion constant expressedin Eq.(2), a more general universal bound was later pro-posed in [38]. The idea is that any diffusive process in Curiously, this is slightly imprecise for the case of rotations. In-deed, contrary to the explicit breaking scenario, the spontaneousbreaking of rotations does not imply a violation of the KSS bound[25] nature is bounded from below by a certain combinationof two unknown velocity and time scales as: D ≥ v τ ? . (3)This last expression recovers immediately the KSSbound by setting D = D shear , v ? = c and τ ? = τ pl .The inequality in Eq.(3) applies to physical diffusionconstants, it is very general and it can be consistentlydefined for any system possessing a diffusive hydrody-namic process which corresponds to the time evolutionof a certain conserved quantity. Nevertheless, it appearsquite void and not practical unless one specifies in detailwhich are the scales appearing in the r.h.s. of Eq.(3).Additionally, one would call such an expression universalonly if the same velocity and time scales bounded allthe diffusive processes in the system. On the contrary,a statement like Eq.(3) would become quite poor if, forany diffusion constant D i , different scales in the r.h.s.had to be used. Finally, following this logic, one wouldexpect the scales in the r.h.s. of Eq.(3) to be infrared(IR) quantities, independent of the ultraviolet (UV)microscopic physics, and therefore universal.A first, and partially successful, attempt to make thebound in Eq.(3) more concrete has originated from theidea of identifying the scales in the r.h.s. using physicalobservables from quantum chaos. In particular, Refs.[39, 40] have proposed to identify: v ? = v B , τ ? = τ L , (4)where v B is the butterfly velocity and τ L the Liapunovtime. Both these quantities can be directly extractedusing the out-of-time-order correlator (OTOC) [41]. Insummary, the final proposal coming from [39, 40] wasthat any diffusion constant D i has to be bounded frombelow as follows: D i ≥ i v B τ L , (5)where i is an O (1) number which depends on the spe-cific diffusive process as well as IR fixed point considered.Despite the considerable success of this proposal [18, 42–60], it was soon realized that in the case of charge dif-fusion D i ≡ D Q = σ/χ ρρ (with σ the electric con-ductivity and χ ρρ the charge susceptibility) the boundin Eq.(5) could be violated [61, 62]. One more time,this is not surprising from a physical point of view. Infact, the quantum chaos data ( v B and λ L ) are extractedholographically from the gravitational sector of fluctua-tions and in general are totally agnostic about the chargesector to which the charge diffusion constant attains.When the diffusive process considered is that of energy, D i ≡ D (cid:15) = κ/c v (with κ the thermal conductivity and c v the specific heat), the bound in Eq.(5) is much morerobust and hard to break. Nevertheless, there are at leasttwo known cases [63, 64] where this happens.Taking a similar perspective, one could ask whether the Figure 1. Given an arbitrary diffusive process and its relateddiffusion constant D , does D obey any universal lower/upperbounds and in terms of which physical quantities? This is thequestion we address in this note. diffusion constants are also bounded from above or theycan grow indefinitely (see cartoon in Fig.1). It turnsout that causality, and in particular the requirement ofavoiding superluminal propagation, imposes a strong up-per bound on diffusion which takes the form [19]: D i ≤ v τ eq , (6)where v lightcone is the velocity, setting the causal lightconein the theory, and τ eq the equilibration time at whichthe system thermalizes, and after which hydrodynamicsstarts to apply. The equilibration time can be universallydefined using the imaginary part of the first damped, andtherefore non-hydrodynamic, mode as: τ − eq ≡ ω eq = | Im ω | , (7)where ω is the frequency of the lowest of those modes.Contrary to the equilibration time, the definition ofthe lightcone velocity related to the causal structure isfar from trivial in systems which do not enjoy relativisticinvariance and/or systems with emergent IR lightconestructures. In relativistic systems, the lightcone ve-locity is obviously set by the speed of light c . Onesimple example is given by Israel-Stewart relativistichydrodynamics [65]. There, the lightcone speed isimmediately identified with the speed of light c andthe equilibration time with the IR relaxation time τ π which is pheomenologically introduced in the framework.Indeed, within the Israel-Stewart formalism, the absenceof superluminality can be re-written exactly as an upperbound on the shear diffusion constant D shear < c τ π [19, 34].A first check of the upper bound in Eq.(6) wasperformed in Ref.[34] using different velocity scales. Amore formal derivation, based on technical mathematicalproperties of the hydrodynamic perturbative expansion,was discussed in [66]. Interestingly, if one considers the momentum diffusion constant, instead of the η/s ratio, all the known violations related to the breaking ofspacetime symmetries disappear [34].Given the important role of hydrodynamics, recently,Ref.[67] proposed to connect the lightcone velocity withthe equilibration velocity. In particular, Ref.[67] pro-posed a new bound which takes the following form: D i ≤ v i,eq τ eq . (8)Here, v i,eq is the equilibration velocity for the i th diffusivemode, defined as: v eq ≡ ω eq k eq . (9)where, to avoid clutter, the i index has been dropped.This idea uses the recent definition of the radius of con-vergence of hydrodynamics presented in [68–70] and dis-cussed further in [71–73]. In particular, the pair ( ω eq , k eq )corresponds to the location of the first (closest to the ori-gin) critical point of the hydrodynamic perturbative se-ries. This point coincides with the collision (in general inthe complex plane) between a first (in this case diffusive)hydrodynamic mode and a nearby non-hydrodynamicmode (or a tower of them), and it determines the ra-dius of convergence of the hydrodynamic series. Moreprecisely, we utilize the following definitions: k eq ≡ | k ∗ | , ω eq ≡ | ω ∗ | , (10)where ( ω ∗ , k ∗ ) is the first critical point – the positionof the collision for complex frequency and momentum, k ∗ , ω ∗ ∈ C . In simple words, such a point determines thescale at which considering only conserved quantities isnot enough anymore and the hydrodynamics descriptionmust be improved.Few comments are in order. (I) The definition of theequilibration velocity is specific to the diffusive processconsidered. In this sense, it is quite a stretch to considerthe bound in Eq.(8) as universal. Notice for example thecrucial difference with the butterfly velocity proposal inEq.(5), in which the velocity scale on the r.h.s. is thesame for all the diffusion constants considered. (II) Itis not clear how the bound in Eq.(8) connects with thatin Eq.(6). In particular, Eq.(8) does not follow from therequirement of causality and furthermore v eq does notdefine nor the lightcone velocity nor the causal struc-ture of any propagating process. (III) The equal sign inEq.(8) follows trivially from assuming that the diffusivedispersion relation: ω = − i D k (11)is valid until the critical point ( ω ∗ , k ∗ ).In particular simple algebra gives ω eq = D k eq → D ω eq v eq → D = v eq τ eq . (12)That said, the observation of [67] is interesting and itboils down to understand the following questions: • Are there situations where the corrections to thehydrodynamic dispersion relation can be neglecteduntil the critical point determining the radius ofconvergence of linearized hydrodynamics? • Which conditions ensure the existence of such ascenarios and what is their meaning?In this note, we consider the proposal of Ref.[67] in ho-mogeneous holographic models with long-range order,i.e. with spontaneously broken translational invariance.These systems display an AdS IR near-horizon geom-etry and a peculiar new diffusive mode labelled crystaldiffusion [74–76]. In these holographic models, the lowerbound Eq.(5) for crystal diffusion has already been ver-ified in [34]. Our task now is to determine whether theinequivalence in Eq.(8) applies also to the same modeand which are the scales involved. Moreover, we analyzethe connections and interplay between the lower boundon diffusion dictated by quantum chaos and this new up-per bound determined by the breakdown of the hydrody-namic perturbative expansion. Finally, we provide somecomments and thoughts for the future.
THE HOLOGRAPHIC MODEL
We consider a large class of holographic axion models[77] introduced and discussed in [78–81] and defined asfollows: S = (cid:90) d x √− g (cid:20) R − m V ( X ) (cid:21) , (13)where X ≡ g µν ∂ µ φ I ∂ ν φ I and we have set the AdSradius to be unit. We choose an isotropic profile for theaxion fields given by φ I = x I , (14)which represents a trivial solution of the equations of mo-tion because of the global shift symmetry φ I → φ I + b I ofthe action (13). The background geometry in Eddington-Filkenstein coordinates is written as: ds = 1 u (cid:2) − f ( u ) dt − dt du + dx + dy (cid:3) , (15)where u ∈ [0 , u h ] is the radial holographic direction goingfrom the boundary u = 0 to the horizon, f ( u h ) = 0.Finally, we have: f ( u ) = u (cid:90) u h u dv (cid:20) v − m V ( v ) v (cid:21) , (16)and consequently the temperature T is defined as T = − f (cid:48) ( u h )4 π = 6 − m V (cid:0) u h (cid:1) π u h , (17) ●●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● crystal diffusion k eq ω eq
20 40 60 80 100 120 kT - Im ( ω ) π T ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● kT - Im ( ω ) π T Figure 2. The longitudinal spectrum of quasinormal modesfor N = 3 and m/T = 1000 (top panel), m/T = 500 (bottompanel). For simplicity, the attenuation constant of the longi-tudinal sound modes has been manually removed. The grayregion emphasizes the location of the collision between thecrystal diffusion mode and the first non-hydrodynamic AdS mode, from which the parameters ω eq , k eq are extracted asshown by the dashed lines. Similar pictures can be obtainedfor different values N and of m/T . while the entropy density is given by s = 2 π/u h .In the rest of the note, we will focus on the monomialform: V ( X ) = X N . (18)We refer to the previous literature [82–89] for more de-tails concerning these models and their properties.In these holographic models, the UV expansion of the φ I bulk fields reads: φ I ( t, x, u ) = φ I (0) ( t, x ) + φ I (1) ( t, x ) u − N + . . . (19)Assuming standard quantization, one could verifiy thatfor N < / φ I = x I plays therole of an external source, while for N > / O I , dual tothe axion fields. Following this argument, the breakingof translations is explicit for N < / N > /
2. In this note, we will only consider the case
N > / RESULTS
The longitudinal spectrum of systems with sponta-neously broken translations features a peculiar modewith diffusive dispersion relation which is usually labelled“crystal diffusion”. Using the correct hydrodynamic de-scription [90], the diffusion constant of such a mode reads: D = ξ ( B + G − P ) χ ππ s (cid:48) T v L . (20)The various parameters appearing in the equationabove are: the Goldstones diffusion constant ξ , thebulk modulus B , the shear modulus G , the momen-tum susceptibility χ ππ , the temperature derivativeof the entropy s (cid:48) and finally the so called crystalpressure P . The hydrodynamic formula Eq.(20) hasbeen successfully matched to the holographic results[88, 89] after some initial, and then resolved, tension [87].The dispersion relation of the crystal diffusion modeis shown in Fig.2 for very low values of temperature.The characteristic quadratic scaling and the correspond-ing slope are consistent with the previous theoreticalcomputations. In Fig.2, the lowest non-hydrodynamicmodes ω = − i ω n are shown as well. As explained in theprevious literature [91, 92] and discussed in [67], thesemodes have a quite flat dispersion relation at low tem-peratures and they appear equally separated. In partic-ular, this tower of non-hydrodynamic modes belongs tothe IR AdS spectrum and it is given by: ω n = 2 π T ( δ + n ) for T → , (21)where n is the index labelling the modes and δ a constantdepending on the specific operator considered .Given the presence of these modes, we can immediatelyidentify the value of the lowest ( n = 0) frequency inEq.(21) with the equilibration timescale: ω eq = τ − eq = ω = 2 π δ T . (22)We plot the value of the normalized equilibration fre-quency in function of m/T at low temperatures in Fig.3.The curves approach nicely a common asymptotic T = 0 For bulk fields with canonically normalized kinetic terms, thisconstant δ corresponds to the conformal dimension of the dualoperator in the AdS IR fixed point evaluated at zero momen-tum, δ = ∆( k = 0)[67]. In our case, given the non-canonicalaction, a more detailed analysis is needed. It is currently inprogress and it will be reported somewhere else. N = = = δ
100 500 1000 5000 1 × × × mT ω eq π T Figure 3. The equilibration frequency ω eq in function of thedimensionless temperature for various N = 3 , ,
5. All thecurves approach asymptotically a constant value given by δ =3 / value which is given by δ = 3 /
2. This number does notcoincide with the conformal dimension of a standardmassless scalar operator in the AdS IR fixed point [91].Importantly, this value is completely independent of thechoice of the potential and in particular the value of thepower N .Let us now move to discuss the interplay between thediffusive hydrodynamic mode and the first non-hydromode. We have zoomed in the area in which the crystaldiffusion mode and the first non-hydrodynamic mode ap-proach each other, which is indicated with a gray shadedregion in Fig.2. The results are more clearly shown inFig.4. From there, it is evident that the two modes dis-play a double-collision dynamics. In particular, the twomodes first collide with each other on the imaginary axes(Re ω = 0), then, they move off axis acquiring a finalreal part and finally they collide again on the imaginaryaxes and they move apart. We have observed the samedynamics independently of the value of N > / T /m . Because of this feature, one can take thereal value of momentum at which the first collision hap-pens (see dashed red line in the top panel of Fig.4) as thelengthscale determining the breakdown of the diffusivedispersion relation within the linearized hydrodynamicsframework. More precisely, we can take: k eq ≡ | k ∗ | , (23)which, together with the critical frequency ω ∗ , is easilyreadable from the dispersion relation of the modes inthe longitudinal spectrum, as shown in the two panelsof Fig.4. Notice the fundamental differences with theenergy diffusion mode and the charge diffusion modeanalyzed in Ref.[67]. There, no collision was present forreal values of the momentum and a more complicatedanalysis in the complex momentum space had to beperformed. ●●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●●●● ●●●●● ●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●● ●●● ●●● ●●● ●●●●● ●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● kT - - ( ω ) π T k * / T crystal diffusionmode kT - Im ( ω ) π T ●●●●●● ●●● ● ●● ●● ●● ●● ● ●● ●●● ●●● ● ●●● ●● ●● ●● ● ●● ● ●●● ●●● ●●●● ●● ●●●● ●● ●●●● ●●● ●●● ●●●● ●● ●● ●● ●●●● ●● ●●● ●●● ●●● ●● ●●● ●● ●● ●● ●●● ●●● ●● ●● ●●●● ●●● ●●● ●●●● ●●●● ●●●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ● ●● ●● ● ●●● ●● ●● ●●● ●● ●●● ●● ●●● ●● ● ●●● ● ●●●●●●●●●● the collisionpoint ω * / π T - - - ( ω ) π T - Im ( ω ) π T Figure 4.
Top:
A zoom of the area around the double colli-sion corresponding to the gray region in Fig.2. Here we take m/T = 500 and N = 3. The inset shows the correspondingreal part. Bottom:
The two collisions on the imaginary axesand the identification of the critical frequency ω ∗ for the samedata of the top panel. In summary, we can extract both parameters, ω eq , k eq ,simply by looking at the dispersion relations of the twolowest modes as shown in Figures 2 and 4. At this point,we can also straightforwardly define the equilibration ve-locity as: v eq ≡ ω eq k eq . (24)We show the behaviour of the non-normalized equilibra-tion velocity in function of temperature in the top panelof Fig.5. Interestingly, we observed a very clear T / scaling close to zero temperature. Given that ω eq ∼ T atlow temperature (Eq.(21)), we can derive that in such aregime the radius of convergence of hydrodynamics goesas: k eq ∼ T / . (25)This is consistent with the idea that the convergenceproperties of hydrodynamics become worse and worse ■■■■■■■■■■■■■ ■■■■■■■■■■■■■ ■■■■■■■■■■■■■ ■ N = ■ N = ■ N = / T
500 1000 5000 1 × × × mT v eq ■■■■■■■■■■■■■ ■■■■■■■■■■■■■ ■■■■■■■■■■■■■ ■ N = ■ N = ■ N = δ
100 500 1000 5000 1 × × × mT v eq v B Figure 5.
Top:
The behaviour of the equilibration velocityin function of the dimensionless temperature m/T for variousvalues of N = 3 , ,
5. The dashed line shows the T / scaling. Bottom:
The comparison with the butterfly velocity. Atlow temperatures, the ratio of the two velocities approach aconstant value, √ δ . upon lowering the temperature [73]. In other words,hydrodynamics breaks down at larger distances going to-wards zero temperature. Eq.(25) is also consistent withprevious results for charged holographic backgroundsand realistic liquids [71–73].At this stage, we want to compare the behaviour of theequilibration velocity with that of the butterfly velocity,which has a fundamental role in the diffusion boundsdiscussed in the introduction. The butterfly velocity canbe obtained from horizon data and in our background isgiven by: v B = π Tu h . (26)In the limit of m/T → ∞ , the radius of the horizon goesto a constant and therefore we obtain the expected scal-ing v B ∼ T / . Interestingly, the ratio between the twovelocity scales approaches a constant, given by √ δ , in thelow temperature limit. We notice that for canonicallynormalized operators with unitary conformal dimensionin the AdS IR fixed point, the two velocities would ex-actly coincide at low temperature. More in general, wefind that: v eq > v B . (27)If one considers the butterfly velocity as an emergentlightcone speed of some (non-relativistic) quantumchaotic system, the relation just obtained looks quitedangerous since it would imply a superluminal propa-gation with speed v eq outside of the causal lightcone.Nevertheless, the equilibration velocity v eq does notcorrespond to any propagating modes. In other words, inthese cases, there is absolutely no excitation propagatingat such speed. In any case, this relation looks certainlyinteresting and it deserves further understanding. Italso implies that the thermalization speed, definedas in Eq.(24), is faster than the speed of informationscrambling. It would be important to understand howuniversal this hierarchy is and which are the physicalconsequences.After having identified all the scales entering in thebound of Eq.(8), we can finally test its validity for thecrystal diffusion mode. In Fig.6, we plot the dimension-less ratio D/v eq τ eq in function of the dimensionless in-verse temperature m/T at small temperatures and forvarious potentials, N = 3 , ,
5. In all the cases, we noticethat this ratio approaches unity at T → D = v eq τ eq for T → . (28)Moreover, we obtain that in general D ≤ v eq τ eq (29)and that therefore the bound Eq.(8) proposed in [67] in-deed holds. This constitutes an explicit confirmation thateven for the crystal diffusion mode, the diffusion constantis bounded from above by the equilibration scales v eq and τ eq . CONCLUSIONS
In summary, in this short note, we have confirmedthe validity of the diffusion bound defined in termsof the hydrodynamics breakdown data proposed inRef.[67] for the crystal diffusion mode present in allholographic systems with spontaneously broken transla-tions. Importantly, in our case, the collision determiningthe equilibration scales happens for real values of themomentum k and therefore it is easily extracted fromthe dispersion relation of the lowest quasinormal modes. ■■■■■■■■■■■■■ ■■■■■■■■■■■■■ ■■■■■■■■■■■■■ ■ N = ■ N = ■ N =
100 500 1000 5000 1 × × × mT Dv eq2 τ eq Figure 6. The dimensionless ratio D/ ( v eq τ eq ) in functionof the inverse dimensionless temperature for various powers N = 3 , ,
5. All the curves approach the unit value at lowtemperature. Moreover, all curves are consistent with theinequality D ≤ v eq τ eq . Moreover, we do find that, for arbitrary values of thetemperature, the diffusion constant is confined in anrange determined by: v B τ L ≤ D ≤ v eq τ eq , (30)which is shown in Fig.7. Importantly, approaching thezero temperature limit this allowed region shrinks andat exactly zero temperature the two limits collapse oneach other. This means that the universal bounds deter-mine uniquely the value of the diffusion constant at zerotemperature: (cid:0) v B τ L (cid:1) | T =0 = D | T =0 = (cid:0) v eq τ eq (cid:1) | T =0 . (31)This is a very interesting and new outcome whose uni-versal character must be investigated further. Final comments
We conclude with several comments and ideas for thefuture. • One interesting point is the connection between thediffusivity bound D ≤ v eq τ eq and the higher ordercorrections to the diffusive dispersion relation. Aswe have shown in the main text, if the diffusivebehaviour ω = − iDk persists until the collisionpoint, the equality trivially holds. What happensif that is not the case? In general, the dispersionrelation is given by a perturbative expansion in mo-mentum of the type: ω = − i (cid:0) Dk + a k + a k + . . . (cid:1) , (32)where for simplicity we have ignored any possiblereal part and considered a purely imaginary mode. Figure 7. A cartoon of the allowed region for the diffusionconstant D . The upper edge comes from the scales related tothe breakdown of linearized hydrodynamics. While, the lowerone is dictated by the data of quantum chaos [34]. Both curvesmerge in the limit of T → Let us consider that, upon reaching the criticalpoint ( ω ∗ , k ∗ ), the first higher order correction can-not be neglected. Then, we have: ω eq = Dk eq + a k eq (33)which, after some manipulations, gives: D = v eq τ eq − a v eq τ eq . (34)The extension to higher order terms is trivial andfor simplicity not shown here. By looking atEq.(34), one can infer that the validity of the up-per diffusion bound depends crucially on the signsof the higher order corrections. In particular, thehigher order terms in the diffusive dispersion re-lations introduce corrections of order O (1 /v neq τ neq ),controlled by the various new coefficients a n . Isthere any physical requirement (e.g. causality) thatfixes the sign of these coefficients and therefore thevalidity of the upper bound? That seems indeed thecase. The univalence property of the hydrodynam-ics expansion put stringent bounds on all the higherorder coefficients [66]. For example, it constraintsthe first of them, labelled a above, to be positive.In other words, it is very tempting to claim that thevalidity of the upper bound proposed in [67] can beformally derived using mathematical properties ofthe hydrodynamic series [66]. A simple scenariowhere this mechanism appears is the telegrapherequation [93]: ω + iω/τ = v k . (35)In this case, the purely diffusive dispersion law getscorrected before the poles collision as: ω = − i D k − i v τ k + . . . (36) and the corresponding higher order coefficient reads a = v τ . Stability, and more precisely the re-quirement of having τ > a > • It would be interesting to understand better thezero temperature relation in Eq.(31). To the bestof our knowledge, the collapse of the two boundsat zero temperature has not been observed nor dis-cussed before. How universal is this feature? Whatcan we learn from it? Is it possible to maintain afinite range of allowed values at zero temperatureor the two bounds always collapse? • In all this discussion, the value of the constant δ plays a fundamental role. In particular, the con-crete value √ δ controls the ratio between the equi-libration velocity and the butterfly velocity at lowtemperature. It would be interesting to understandif any physical requirement (e.g. unitarity) con-straints δ > • A priori, it is not clear what is the relation be-tween the equilibration velocity v eq and the causalstructure of the system. In particular, such a ve-locity in general does not correspond to any propa-gating mode. Interestingly, taking the telegrapherequation (35), one can derive that the equilibrationvelocity v eq coincides exactly with the sound speedof the emergent propagating mode at large momen-tum. In this simplified scenario, it does correspond-ing to a propagating mode at short distance. • In the main text, we have derived that in the lowtemperature limit, τ − eq = 2 π δ T . This relaxationtime has the same temperature dependence of thePlanckian time and the Liapunov time but witha different numerical prefactor. In particular wehave: ( τ pl , τ L , τ eq ) T = (cid:18) , π , π δ (cid:19) . (37)Moreover, the hierarchy of these timescales dependscrucially on the value of δ . Also, the situationmight be substantially different away from maxi-mal chaos [94] where the Maldacena bound [13] isnot saturated and τ L > π T . Which is the orderof these timescales and how can it be changed? • In [67], the equilibration velocity has been definedusing the critical point which determines the break-down of the hydrodynamics expansion and in par-ticular of the diffusive dispersion relation. In prin-ciple, there are other points in the complex planewhich assume a particular role – the pole-skippingpoints [95–98]. Given a certain Green function,those are the points at which the zeros and the polescross, rendering the Green function indeterminate.Following [67], one could define a pole-skipping ve-locity: v O skip ≡ | ω skip || k skip | , (38)where ( ω skip , k skip ) indicates the location of thefirst pole skipping point in the complex plane andthe index O the operator whose Green functionis considered. Notice that if one considers theenergy-energy correlator, one finds v (cid:15)skip = v B and ω skip = τ − L [95]. In this sense, for the energy cor-relator, we have the following identification: v skip ω skip = v B τ L (39)and therefore one can immediately write down alower bound of the type: D (cid:15) ≥ v skip ω skip . (40)The question whether a more general bound: D O ≥ v O ,skip ω O ,skip (41)exists for an arbitrary operator O is valuable andit can be easily investigated with the existing tech-niques. Preliminary indications [99] seem to sug-gest our hypothesis. • Finally, it would be interesting to consider IRfixed point with dangerously irrelevant deforma-tions [42]. There, the equilibration time is expectedto be parametrically longer, τ eq (cid:29) T − , and the fullpicture could change substantially. Hyperscaling-Lifshitz IR geometries are also a straightforwardgeneralization of this program.The emerging global picture suggests intriguing andpossibly fundamental connections between transport,quantum chaos, hydrodynamics and pole skipping whichare left to be revealed.We plan to come back to some of these questions inthe near future. Acknowledgments
We thank Sebastian Grieninger for providing the nu-merical codes used in previous works and for useful com-ments. We thank Saso Grozdanov, Keun-Young Kim,Yongjun Ahn and Hyun-Sik Jeong for reading a pre-liminary version of the manuscript and providing usefulcomments and suggestions. We thank Keun-Young Kim,Yongjun Ahn, Hyun-Sik Jeong and Ya-Wen Sun for shar-ing with us unpublished results. N.W. and W.J.L. aresupported by NSFC No.11905024 and No.DUT19LK20.M.B. acknowledges the support of the Shanghai Mu-nicipal Science and Technology Major Project (GrantNo.2019SHZDZX01). [1] L. Landau and E. Lifshitz,
Fluid Mechanics: Volume 6 ,v. 6 (Elsevier Science, 2013).[2] P. C. Martin, O. Parodi, and P. S. Pershan, Phys. Rev.A , 2401 (1972).[3] G. Policastro, D. T. Son, and A. O. Starinets, Phys.Rev. Lett. , 081601 (2001), arXiv:hep-th/0104066[hep-th].[4] S. S. Gubser, I. R. Klebanov, and A. A. Tseytlin, Nucl.Phys. B , 217 (1997), arXiv:hep-th/9703040.[5] I. R. Klebanov, Nucl. Phys. B , 231 (1997),arXiv:hep-th/9702076.[6] P. Kovtun, J. Phys. A , 473001 (2012),arXiv:1205.5040 [hep-th].[7] J. Zaanen, Nature , 512 (2004).[8] J. Zaanen, SciPost Phys. , 61 (2019).[9] J. A. N. Bruin, H. Sakai, R. S. Perry, and A. P. Macken-zie, Science , 804 (2013).[10] K. Behnia and A. Kapitulnik, Journal of Physics: Con-densed Matter , 405702 (2019).[11] S. Sachdev, Phys. Rev. X , 041025 (2015).[12] A. A. Patel and S. Sachdev, Phys. Rev. Lett. ,066601 (2019). [13] J. Maldacena, S. H. Shenker, and D. Stanford, JHEP , 106 (2016), arXiv:1503.01409 [hep-th].[14] C. H. Mousatov and S. A. Hartnoll, Nature Physics(2020), 10.1038/s41567-020-0828-6.[15] Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies-Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, andP. Jarillo-Herrero, Phys. Rev. Lett. , 076801 (2020).[16] V. Martelli, J. L. Jim´enez, M. Continentino, E. Baggio-Saitovitch, and K. Behnia, Phys. Rev. Lett. ,125901 (2018).[17] J. Zhang, E. D. Kountz, K. Behnia, andA. Kapitulnik, Proceedings of the NationalAcademy of Sciences , 216601 (2019),arXiv:1809.07769 [cond-mat.str-el].[19] T. Hartman, S. A. Hartnoll, and R. Mahajan, Phys.Rev. Lett. , 141601 (2017), arXiv:1706.00019 [hep-th].[20] T. Schafer and D. Teaney, Rept. Prog. Phys. , 126001(2009), arXiv:0904.3107 [hep-ph]. [21] S. Cremonini, Mod. Phys. Lett. B25 , 1867 (2011),arXiv:1108.0677 [hep-th].[22] M. Luzum and P. Romatschke, Phys. Rev. C , 034915(2008), [Erratum: Phys.Rev.C 79, 039903 (2009)],arXiv:0804.4015 [nucl-th].[23] J. L. Nagle, I. G. Bearden, and W. A. Zajc, New J.Phys. , 075004 (2011), arXiv:1102.0680 [nucl-th].[24] C. Shen, U. Heinz, P. Huovinen, and H. Song, Phys.Rev. C , 044903 (2011), arXiv:1105.3226 [nucl-th].[25] J. Erdmenger, P. Kerner, and H. Zeller, Physics LettersB , 301 (2011).[26] L. Alberte, M. Baggioli, and O. Pujolas, JHEP , 074(2016), arXiv:1601.03384 [hep-th].[27] S. A. Hartnoll, D. M. Ramirez, and J. E. Santos, JHEP , 170 (2016), arXiv:1601.02757 [hep-th].[28] P. Burikham and N. Poovuttikul, Phys. Rev. D94 ,106001 (2016), arXiv:1601.04624 [hep-th].[29] A. Rebhan and D. Steineder, Phys. Rev. Lett. ,021601 (2012), arXiv:1110.6825 [hep-th].[30] X.-H. Ge, S.-K. Jian, Y.-L. Wang, Z.-Y. Xian, andH. Yao, (2018), arXiv:1810.00669 [hep-th].[31] M. P. Gochan, H. Li, and K. S. Bedell,(2018), 10.1088/2399-6528/ab292b, arXiv:1801.08627[cond-mat.str-el].[32] J. P. Figueroa and K. Pallikaris, JHEP , 090 (2020),arXiv:2006.00967 [hep-th].[33] T. D. Cohen, Phys. Rev. Lett. , 021602 (2007).[34] M. Baggioli and W.-J. Li, SciPost Phys. , 007 (2020),arXiv:2005.06482 [hep-th].[35] K. Trachenko and V. V. Brazhkin, Science Advances (2020).[36] K. Trachenko, V. Brazhkin, and M. Baggioli, (2020),arXiv:2003.13506 [hep-th].[37] K. Trachenko, M. Baggioli, K. Behnia, andV. V. Brazhkin, Phys. Rev. B , 014311 (2021),arXiv:2009.01628 [cond-mat.stat-mech].[38] S. A. Hartnoll, Nature Phys. , 54 (2015),arXiv:1405.3651 [cond-mat.str-el].[39] M. Blake, Phys. Rev. Lett. , 091601 (2016),arXiv:1603.08510 [hep-th].[40] M. Blake, Phys. Rev. D94 , 086014 (2016),arXiv:1604.01754 [hep-th].[41] A. I. Larkin and Y. N. Ovchinnikov, Soviet Journal ofExperimental and Theoretical Physics , 1200 (1969).[42] R. A. Davison, S. A. Gentle, and B. Gouteraux, Phys.Rev. Lett. , 141601 (2019), arXiv:1808.05659 [hep-th].[43] Y. Gu, A. Lucas, and X.-L. Qi, JHEP , 120 (2017),arXiv:1708.00871 [hep-th].[44] Y. Ling and Z.-Y. Xian, JHEP , 003 (2017),arXiv:1707.02843 [hep-th].[45] Y. Gu, A. Lucas, and X.-L. Qi, SciPost Phys. , 018(2017), arXiv:1702.08462 [hep-th].[46] M. Blake and A. Donos, JHEP , 013 (2017),arXiv:1611.09380 [hep-th].[47] M. Blake, R. A. Davison, and S. Sachdev, Phys. Rev. D96 , 106008 (2017), arXiv:1705.07896 [hep-th].[48] S.-F. Wu, B. Wang, X.-H. Ge, and Y. Tian, Phys. Rev.
D97 , 106018 (2018), arXiv:1702.08803 [hep-th].[49] W. Li, S. Lin, and J. Mei, Phys. Rev.
D100 , 046012(2019), arXiv:1905.07684 [hep-th].[50] X.-H. Ge, S.-J. Sin, Y. Tian, S.-F. Wu, and S.-Y. Wu,JHEP , 068 (2018), arXiv:1712.00705 [hep-th]. [51] W.-J. Li, P. Liu, and J.-P. Wu, JHEP , 115 (2018),arXiv:1710.07896 [hep-th].[52] H.-S. Jeong, Y. Ahn, D. Ahn, C. Niu, W.-J. Li, and K.-Y. Kim, JHEP , 140 (2018), arXiv:1708.08822 [hep-th].[53] M. Baggioli and W.-J. Li, JHEP , 055 (2017),arXiv:1705.01766 [hep-th].[54] K.-Y. Kim and C. Niu, JHEP , 030 (2017),arXiv:1704.00947 [hep-th].[55] I. L. Aleiner, L. Faoro, and L. B. Ioffe, An-nals Phys. , 378 (2016), arXiv:1609.01251 [cond-mat.stat-mech].[56] A. A. Patel, D. Chowdhury, S. Sachdev, and B. Swingle,Phys. Rev. X , 031047 (2017), arXiv:1703.07353 [cond-mat.str-el].[57] A. A. Patel and S. Sachdev, Proc. Nat. Acad. Sci. ,1844 (2017), arXiv:1611.00003 [cond-mat.str-el].[58] A. Bohrdt, C. Mendl, M. Endres, and M. Knap, NewJ. Phys. , 063001 (2017), arXiv:1612.02434 [cond-mat.quant-gas].[59] Y. Werman, S. A. Kivelson, and E. Berg, arXive-prints , arXiv:1705.07895 (2017), arXiv:1705.07895[cond-mat.str-el].[60] X. Chen, R. M. Nandkishore, and A. Lucas, Phys. Rev.B , 064307 (2020), arXiv:1912.02190 [cond-mat.stat-mech].[61] A. Lucas and J. Steinberg, JHEP , 143 (2016),arXiv:1608.03286 [hep-th].[62] M. Baggioli, B. Gouteraux, E. Kiritsis, and W.-J. Li,JHEP , 170 (2017), arXiv:1612.05500 [hep-th].[63] A. Lucas and J. Steinberg, JHEP , 143 (2016),arXiv:1608.03286 [hep-th].[64] H.-K. Wu and J. Sau, “A classical model for sub-planckian thermal diffusivity in complex crystals,”(2021), arXiv:2101.05353 [cond-mat.mtrl-sci].[65] W. Israel and J. Stewart, Annals of Physics , 341(1979).[66] S. Grozdanov, (2020), arXiv:2008.00888 [hep-th].[67] D. Arean, R. A. Davison, B. Gout´eraux, and K. Suzuki,(2020), arXiv:2011.12301 [hep-th].[68] S. c. v. Grozdanov, P. K. Kovtun, A. O. Starinets, andP. Tadi´c, Phys. Rev. Lett. , 251601 (2019).[69] S. Grozdanov, P. K. Kovtun, A. O. Starinets, andP. Tadi´c, Journal of High Energy Physics , 97(2019).[70] B. Withers, JHEP , 059 (2018), arXiv:1803.08058[hep-th].[71] N. Abbasi and S. Tahery, (2020), arXiv:2007.10024[hep-th].[72] A. Jansen and C. Pantelidou, (2020), arXiv:2007.14418[hep-th].[73] M. Baggioli, (2020), arXiv:2010.05916 [hep-th].[74] A. Donos, D. Martin, C. Pantelidou, and V. Ziogas,JHEP , 218 (2019), arXiv:1905.00398 [hep-th].[75] M. Baggioli, (2020), arXiv:2001.06228 [hep-th].[76] M. Baggioli and M. Landry, (2020), arXiv:2008.05339[hep-th].[77] M. Baggioli, K.-Y. Kim, L. Li, and W.-J. Li, (2021),arXiv:2101.01892 [hep-th].[78] M. Baggioli and O. Pujolas, Phys. Rev. Lett. ,251602 (2015), arXiv:1411.1003 [hep-th].[79] L. Alberte, M. Baggioli, A. Khmelnitsky, and O. Pujo-las, JHEP , 114 (2016), arXiv:1510.09089 [hep-th]. [80] M. Baggioli, Ph.D. thesis, Barcelona U. (2016),arXiv:1610.02681 [hep-th].[81] M. Baggioli, Applied Holography , Ph.D. thesis, Madrid,IFT (2019), arXiv:1908.02667 [hep-th].[82] L. Alberte, M. Ammon, M. Baggioli, A. Jimenez, andO. Pujolas, JHEP , 129 (2018), arXiv:1708.08477[hep-th].[83] L. Alberte, M. Ammon, M. Baggioli, A. Jimenez-Alba,and O. Pujolas, (2017), arXiv:1711.03100 [hep-th].[84] M. Baggioli and K. Trachenko, JHEP , 093 (2019),arXiv:1807.10530 [hep-th].[85] T. Andrade, M. Baggioli, and O. Pujolas, Phys. Rev. D100 , 106014 (2019), arXiv:1903.02859 [hep-th].[86] M. Ammon, M. Baggioli, and A. Jimenez-Alba, JHEP , 124 (2019), arXiv:1904.05785 [hep-th].[87] M. Ammon, M. Baggioli, S. Gray, and S. Grieninger,JHEP , 064 (2019), arXiv:1905.09164 [hep-th].[88] M. Baggioli and S. Grieninger, JHEP , 235 (2019),arXiv:1905.09488 [hep-th].[89] M. Ammon, M. Baggioli, S. Gray, S. Grieninger, andA. Jain, (2020), arXiv:2001.05737 [hep-th].[90] J. Armas and A. Jain, (2019), arXiv:1908.01175 [hep-th]. [91] T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, Phys.Rev. D , 125002 (2011), arXiv:0907.2694 [hep-th].[92] M. Edalati, J. I. Jottar, and R. G. Leigh, JHEP ,075 (2010), arXiv:1001.0779 [hep-th].[93] M. Baggioli, M. Vasin, V. Brazhkin, andK. Trachenko, Physics Reports (2020),https://doi.org/10.1016/j.physrep.2020.04.002.[94] C. Choi, M. Mezei, and G. S´arosi, (2020),arXiv:2010.08558 [hep-th].[95] S. Grozdanov, K. Schalm, and V. Scopelliti, Phys. Rev.Lett. , 231601 (2018), arXiv:1710.00921 [hep-th].[96] M. Blake, R. A. Davison, S. Grozdanov, and H. Liu,JHEP , 035 (2018), arXiv:1809.01169 [hep-th].[97] M. Blake, H. Lee, and H. Liu, JHEP , 127 (2018),arXiv:1801.00010 [hep-th].[98] Y. Ahn, V. Jahnke, H.-S. Jeong, K.-Y. Kim, K.-S. Lee,and M. Nishida, (2020), arXiv:2010.16166 [hep-th].[99] “Private communication with Keun-Young Kim,Yongjun Ahn, Hyun-Sik Jeong and Ya-Wen Sun,”.[100] S. L. Grieninger, Non-equilibrium dynamics in Holog-raphy , Ph.D. thesis, Jena U. (2020), arXiv:2012.10109[hep-th].
EQUATIONS FOR THE PERTURBATIONS
We align the momentum k along the y direction. The perturbations in the longitudinal sector are given by { h x, s = 1 / h xx + h yy ) , h x, a = 1 / h xx − h yy ) , δφ y , h tt , h ty } , (42)2We use a radial gauge. The final set of equations for the perturbations reads uf (cid:48) δφ (cid:48) y ˙ V + 2 u f δφ (cid:48) y ¨ V + u f δφ (cid:48)(cid:48) y ˙ V − f δφ (cid:48) y ˙ V − k u δφ y ˙ V − k u δφ y ¨ V + i k u h x, a ˙ V − i k u h x, s ¨ V + 2 i u ω δφ y ¨ V + u h (cid:48) ty ˙ V + 2 i u ω δφ (cid:48) y ˙ V − i ω δφ y ˙ V − h ty (cid:16) ˙ V − u ¨ V (cid:17) = 0 (43) u ( f (cid:16) uf (cid:48) h (cid:48) x, s − f h (cid:48) x, s − i k m u δφ y ¨ V − u h (cid:48)(cid:48) tt + 4 h (cid:48) tt (cid:17) + k h ty ( i u f (cid:48) − i f + 2 u ω ) + h x, s (cid:16) m u f ¨ V + ω ( i u f (cid:48) − i f + 2 u ω ) (cid:17) )+ h tt (cid:16) u (cid:16) − uf (cid:48)(cid:48) + 4 f (cid:48) + 2 m u ˙ V (cid:17) − f + k u − m V − i u ω + 6 (cid:17) = 0 (44)2 h ty (cid:16) u (cid:16) f (cid:48) + m u ˙ V (cid:17) − f − m V + 3 (cid:17) − u (cid:16) u f h (cid:48)(cid:48) ty − f h (cid:48) ty + i k u h (cid:48) tt + k u ω h x, s + k u ω h x, a − i m u ω δφ y ˙ V + i u ω h (cid:48) ty (cid:17) + 2 i k u h tt = 0 (45) h x, s (cid:16) u (cid:16) f (cid:48) + m u ˙ V (cid:17) − f + k u − m V + 4 i u ω + 6 (cid:17) − u f (cid:48) h (cid:48) x, s − u f (cid:48) h (cid:48) x, a + 2 u h x, a f (cid:48) − u f h (cid:48)(cid:48) x, s − u f h (cid:48)(cid:48) x, a + 4 u f h (cid:48) x, s + 2 u f h (cid:48) x, a − f h x, a + k u h x, a + 2 i k u h ty + 2 m u h x, a ˙ V − m h x, a V − i u ω h (cid:48) x, s − i u ω h (cid:48) x, a − u h (cid:48) tt + 6 h tt ( u ) + 2 i u ω h x, a + 6 h x, a = 0 (46) h x, s (cid:16) u (cid:16) f (cid:48) + m u ˙ V (cid:17) − f + k u − m V + 4 i u ω + 6 (cid:17) − u f (cid:48) h (cid:48) x, s + u f (cid:48) h (cid:48) x, a − u h x, a f (cid:48) − u f h (cid:48)(cid:48) x, s + u f h (cid:48)(cid:48) x, a + 4 u f h (cid:48) x, s − u f h (cid:48) x, a + 6 f h x, a + k u h x, a − i k m u δφ y ˙ V − i k u h (cid:48) ty + 6 i k u h ty − m u h x, a ˙ V + 2 m h x, a V − i u ω h (cid:48) x, s + 2 i u ω h (cid:48) x, a − u h (cid:48) tt + 6 h tt − i u ω h x, a − h x, a = 0 (47) − h tt + u ( u f (cid:48) h (cid:48) x, s − f h (cid:48) x, s − i k m u δφ y ¨ V + i k u h (cid:48) ty − i k h ty + 2 m u h x, s ¨ V − u h (cid:48)(cid:48) tt + 4 h (cid:48) tt + 2 i u ω h (cid:48) x, s − i ω h x, s ) = 0 (48) k u (cid:0) h (cid:48) x, s + h (cid:48) x, a (cid:1) − i u (cid:16) m δφ (cid:48) y ˙ V + h (cid:48)(cid:48) ty (cid:17) + 2 i h (cid:48) ty = 0 (49) h (cid:48)(cid:48) x, s = 0 , (50)where the following notations ˙ V ≡ dV ( X ) /dX, ¨ V ≡ d V ( X ) /dX2