Non-Hydrodynamic Initial Conditions are Not Soon Forgotten
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Non-Hydrodynamic Initial Conditions are Not Soon Forgotten
T.R. Kirkpatrick , D. Belitz , and J.R. Dorfman Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA Department of Physics and Institute for Fundamental Science, University of Oregon, Eugene, OR 97403, USA Materials Science Institute, University of Oregon, Eugene, OR 97403, USA (Dated: February 23, 2021)Solutions to hydrodynamic equations, which are used for a vast variety of physical problems, areassumed to be specified by boundary conditions and initial conditions on the hydrodynamic variablesonly. Initial values of other variables are assumed to be irrelevant for a hydrodynamic description.This assumption is not correct because of the existence of long-time-tail effects that are ubiquitousin systems governed by hydrodynamic equations. We illustrate this breakdown of a hydrodynamicdescription by means of the simple example of diffusion in a disordered electron system.
Hydrodynamic descriptions of matter have a long his-tory, with classical fluids the most obvious example [1].Others include plasmas [2, 3], superfluids, spin transportin magnets, and excitations in liquid crystals and in solids[4, 5]. More recently, hydrodynamic descriptions havebeen used to study active matter [6], general relativity[7], supersymmetric field theories and quantum gravity[8], quark-gluon plasmas [9], and the unusual propertiesof Weyl and Dirac metals and semi-metals [10].The basic assumption of any hydrodynamic theory isthat the behavior of a macroscopic system on lengthand time scales large compared to the microscopic onescan be described in terms of a small number of ‘hydro-dynamic’ variables, while all other degrees of freedomare effectively integrated out. One of the simplest hy-drodynamic processes is diffusion, with applications inPhysics, Chemistry, Biology, and Engineering [11–13].It is phenomenologically described by Fick’s law, whichstates that the current density j associated with a con-served density n is proportional to the gradient of n , j ( x , t ) = − D ∇ n ( x , t ) , with D the diffusion coefficient.Together with the continuity equation for n , this leads tothe diffusion equation ∂ t n ( x , t ) − D ∇ n ( x , t ) = 0 . (1)This is expected to be valid for times t ≫ t and wavenumbers q ≪ q , with t and /q the microscopic timeand length scales that depend on the nature of the diffu-sive process. In this description there is only one hydro-dynamic variable, viz., the density. With n the constantequilibrium density, Eq. (1) describes the relaxation of adensity perturbation δn ( x , t ) = n ( x , t ) − n . It is solvedby a spatial Fourier transform and a temporal Laplacetransform [4, 14]. The result is the well known diffusionpole in the complex-frequency ( z ) plane, δn ( q , z ) = δn ( q , t = 0) i/ [ z + iDq s z ] (2a)with q the wave vector, and s z = sgn Im ( z ) . In thetime domain, this corresponds to exponential decay witha relaxation time that diverges as /q for small q , δn ( q , t ) = δn ( q , t = 0) e − Dq t . (2b) According to Eq. (2b), the initial value δn ( q , t = 0) com-pletely determines the relaxation for t ≫ t and q ≪ q .As we will show, the relaxation behavior described bythe diffusion equation is not complete. First, kinetictheory shows that δn ( q , t ) also depends on initial con-ditions in higher angular-momentum channels; e.g., aninitial current. Within diffusion theory, these initial con-ditions are ‘non-hydrodynamic’ since they involve quan-tities other than the density. As we will show, solvingthe standard Boltzmann equation for the scattering ofelectrons by static point-like impurities yields δn ( q , t ≫ t ) = [ δn ( q , t = 0) + δn ⊥ ( q , t = 0)] e − Dq t . (3a)Here the diffusion coefficient is D = v F τ / , with v F theFermi velocity and τ the elastic mean-free time. In thiscase, the microscopic time and wave number are t = τ and q = 1 /v F τ , respectively. δn ⊥ ( q , t = 0) is a set ofnon-hydrodynamic initial conditions that are orthogonalin angular-momentum ( l ) space to the density, with thecurrent density j ( l = 1 ) the leading contribution, δn ⊥ ( q , t = 0) ≈ − iτ q · j ( q , t = 0) . (3b)The density relaxation is still given by the solution of thediffusion equation, Eq. (2b), if we replace δn ( t = 0) by δn taken at an adjusted initial time t s defined by δn ( q , t s ) ≡ δn ( q , t = 0) e − Dq t s = δn ( q , t = 0) + δn ⊥ ( q , t = 0) . (3c)The ‘slip time’ t s is the temporal analog of the ‘sliplength’ used to describe the flow of a fluid near a sur-face, i.e., the distance between the actual surface andthe fictitious one where hydrodynamic boundary con-ditions can be used [15, 16]. If the initial conditionsare such that δn and δn ⊥ are related by Fick’s law, δn ⊥ ( q , t = 0) ≈ Dq τ δn ( q , t = 0) , then | t s | ≈ τ , butgenerically this will not be the case.Second, the exponential decay predicted by Eq. (3a) isnot the true asymptotic long-time behavior: Correlationeffects lead to a non-analytic frequency dependence ofthe diffusion coefficient. As a function of time, this corre-sponds to a power-law decay of δn . Such non-exponential‘long-time tails’ (LTTs) are not described by the Boltz-mann equation [17, 18], yet are present in all systemsdescribed by hydrodynamics [6, 8, 15].In this Letter we show that the combination of LTTsand non-hydrodynamic initial conditions yields a long-time behavior of the density relaxation that cannot bereconciled with the diffusion equation by adjusting theinitial condition as in Eq. (3c), even with a frequency-dependent diffusivity that yields the correct LTT.As an example, we consider the density relax-ation of noninteracting conduction electrons in a three-dimensional (3-d) system with weak quenched disorder.The latter leads to LTTs known as weak-localization(WL) effects [19, 20], but in 3-d systems it does not leadto Anderson localization; the system remains metallic.Let ǫ F and k F be the Fermi energy and wave number,respectively, and m the effective mass. The density ofstates per spin at the Fermi surface is N F = k F m/ π ,and the mean-free path is ℓ = v F τ . For weak disorder( k F ℓ ≫ , or ǫ F τ ≫ ), and and t ≫ /Dq , we find δn ( q , t ) = δn ( q , t = 0) (cid:20) √ π k F ℓ q/k F ( Dq t ) / + O ( 1 t / ) (cid:21) + δn ⊥ ( q , t = 0) 98 √ π k F ℓ q/k F ( Dq t ) / . (4a)The different LTTs in the two terms make it impossibleto account for the non-hydrodynamic initial conditionsrepresented by δn ⊥ ( t = 0) by means of a slip time as inEqs. (3). A diffusion pole, Eq. (2a), generalized to allowfor the LTT, thus fails to describe the density relaxationat long times. This remains true for intermediate times τ ≪ t ≪ /Dq , where the leading time dependence(ignoring an O (1 /k F ℓ ) contribution to the constant partof the δn ⊥ term) is δn ( q , t ) = δn ( q , t = 0) (cid:2) O ( Dq t ) (cid:3) + δn ⊥ ( q , t = 0) (cid:20) − √ π k F ℓ q/k F ( Dq t ) / (cid:21) . (4b)Here the difference between the hydrodynamic and non-hydrodynamic terms is even more striking than inEq. (4a): The former is a constant, as described by thediffusion equation, whereas the latter has a /t / long-time tail. Again, the concept of a slip time breaks down.Equations (4), and the related comments, representour main result which we will now derive from kinetictheory. The dependence on non-hydrodynamic initialconditions arises already at the level of the Boltzmannequation, while the WL LTTs require a more sophisti-cated treatment of collision processes. Kinetic equation for the single-particle distribu-tion
Consider the single-electron distribution function f ( p , x , t ) as a function of the electron momentum p , real-space position x , and time t . In the absence of externalforces, the kinetic equation that governs f reads ( ∂ t + p · ∇ x /m ) f ( p , x , t ) = ( ∂f /∂t ) coll . (5) This is completely general: The total time derivative of f on the left-hand side equals the collision integral, i.e., thetemporal change of f due to collision, on the right-handside. The equilibrium distribution function is given bythe Fermi-Dirac distribution f ( p ) = 1 / ( e ( ǫ p − µ ) /T ) + 1) ,with ǫ p = p / m the single-electron energy, µ the chem-ical potential, and T the temperature. We parameterizethe deviation from equilibrium, δf = f − f , by δf ( p , x , t ) = w ( ǫ p ) φ ( p , x , t ) , (6)where w ( ǫ p ) = − ∂f /∂ǫ p is a weight function. For laterreference, the number density n and the number currentdensity j are n ( x , t ) = 1 V X p f ( p , x , t ) , j ( x , t ) = 1 V X p p f ( p , x , t ) . (7) Boltzmann equation
The Boltzmann collision integralfor elastic scattering by impurities is [21, 22] (cid:18) ∂f∂t (cid:19)
Bcoll = − N F V X p ′ δ ( ǫ p − ǫ p ′ ) 1 τ W ( ˆ p , ˆ p ′ ) × [ δf ( p , x , t ) − δf ( p ′ , x , t )] . (8a)The delta-function reflects the elastic nature of the colli-sions, and W is a form factor. For point-like scatterers,the latter is a constant equal to one and we obtain (cid:18) ∂f∂t (cid:19) Bcoll = − τ (cid:2) δf ( p , x , t ) − δ ¯ f ( ǫ p , x , t ) (cid:3) , (8b)where δ ¯ f ( ǫ p , x , t ) = P p ′ δ ( ǫ p − ǫ p ′ ) δf ( ǫ p , x , t ) /V N F is δf averaged over the ǫ p -energy shell. Integrating overthe momentum yields P p δ ¯ f ( ǫ p , x , t ) = P p δf ( p , x , t ) ,so the number density, Eq. (7), is conserved. After aFourier-Laplace transform as in Eq. (2a), the linearizedBoltzmann equation reads (cid:18) − iz + im p · q + 1 τ (cid:19) φ ( p , q , z ) = φ ( p , q , t = 0)+ 1 τ ¯ φ ( ǫ p , q , z ) , (9)with φ from Eq. (6) and ¯ φ defined in analogy to δ ¯ f . Nowconsider δn = n − n , the deviation of the particle-numberdensity from its equilibrium value n . From Eq. (9) wefind δn ( q , z ) = J ( q , z ) + iτ V X p w ( ǫ p ) ¯ φ ( ǫ p , q , z ) z − p · q /m + i/τ , (10a)where J ( q , z ) = iV X p δf ( p , q , t = 0) z − p · q /m + i/τ . (10b)For T ≪ ǫ F we can replace p · q /m in Eq. (10a) by v F ˆ p · q ,which neglects corrections of O ( T ) to the q -dependenceof δn . The resulting closed equation for δn yields δn ( q , z ) = J ( q , z ) / (1 − iJ ( q , z ) /τ ) , (11a)where J ( q , z ) = 14 π Z d Ω p z − v F ˆ p · q + i/τ , (11b)with Ω p the solid angle associated with p . An expan-sion in the limit of small wave numbers ( qℓ ≪ ) andfrequencies ( zτ ≪ ) yields J ( q , z ) = iδn ( q , t = 0) z + i/τ + i q · j ( q , t = 0)( z + i/τ ) + O ( q ) , (12a) − iτ J ( q , z ) = − iτ ( z + iD q ) + O ( z , q ) , (12b)where D = v F τ / is the Boltzmann diffusion coefficient.Inserting these results into Eq. (11a), and performing aLaplace back transform, yields Eqs. (3a, 3b).Equations (10b) and (12a) show that the solution de-pends, in addition to the initial density perturbation, onthe initial current density, as well as higher modes. Atthe level of the Boltzmann equation, after a few mean-free times the various initial-condition terms all multiplythe same time dependence and therefore can be incorpo-rated into a ‘slip time’ as in Eq. (3c). This is no longerthe case if one uses a more sophisticated collision integralthat accounts for LTTs. Weak-localization effects
The LTTs associated with WLeffects arise from two-particle correlations that are notincluded in the Boltzmann collision integral and lead toa frequency-dependent diffusivity that is nonanalytic atzero frequency [23, 24]. Reference 25 showed that theseeffects can be incorporated into the kinetic equation forthe single-particle distribution by adding an additionalterm to the collision integral. This additional term canstill be written in the form of Eq. (8a) [26], with two mod-ifications: First, the scattering rate /τ must be replacedby a time-dependent memory function α (to be specifiedbelow) that encodes correlations. The WL contributionto the collision integral then has the form (cid:18) ∂f∂t (cid:19) WLcoll = − Z t dt ′ α ( t − t ′ ) 1 N F V X p ′ δ ( ǫ p − ǫ p ′ ) × W WL ( ˆ p , ˆ p ′ ) [ δf ( p , x , t ′ ) − δf ( p ′ , x , t ′ )] . (13)Second, the form factor is strongly angle dependent,which reflects the fact that the WL effects are due tobackscattering events [27]: W WL ( ˆ p , ˆ p ′ ) = 14 π δ ( ˆ p + ˆ p ′ ) − , (14a) The additional contribution to the collision integral doesnot change the total scattering rate [26]: Z d Ω p ′ W WL ( ˆ p , ˆ p ′ ) = 0 . (14b)The WL contribution to the collision integral becomes (cid:18) ∂f∂t (cid:19) WLcoll = Z t dt ′ α ( t − t ′ ) (cid:2) δf ( − p , x , t ′ ) − δ ¯ f ( ǫ p , x , t ′ ) (cid:3) . (15a)The function α ( t ) and its Laplace transform are [28, 29] α ( t ) = 1 πN F τ V X k e − D k t ( t > . (15b) α ( z ) = iπN F τ V X k z + iD k s z (15c) = s z τ π k F ℓ h c − π ζ / + O ( ζ ) i , (15d)where ζ = z s z /iD k F , c is the non-universal zero-frequency contribution [28], and Eq. (15d) is valid for d = 3 . Note the nonanalytic frequency dependence thatresults from integrating over the diffusion pole. Only thiscontribution to α matters for the arguments that follow.The kinetic equation that generalizes Eq. (9) now reads (cid:18) − iz + im p · q + 1 τ (cid:19) φ ( p , q , z ) − α ( z ) φ ( − p , q , z ) = φ ( p , q , t = 0) + (cid:18) τ − α ( z ) (cid:19) ¯ φ ( ǫ p , q , z ) . (16)By letting p → − p we obtain two coupled equations for φ ( p ) and φ ( − p ) that can be solved exactly. The resultfor the density relaxation generalizes Eq. (11a) to δn ( q , z ) = ˜ J ( q , z ) + iα ( z ) K ( q , z )1 − [ i/τ − iα ( z )][ ˜ J ( q , z ) + iα ( z ) K ( q , z )] . (17a)In terms of a denominator N ( p , q , z ) = z − p · q /m + i/τ + ( α ( z )) z + p · q /m + i/τ (17b)we find ˜ J ( q , z ) = iV X p δf ( p , q , t = 0) /N ( p , q , z ) (17c) ˜ J ( q , z ) = 14 π Z d Ω p /N ( k F ˆ p , q , z ) (17d) K ( q , z ) = iV X p δf ( − p , q , t = 0) N ( p , q , z )( z + p · q /m + i/τ ) , (17e) K ( q , z ) = Z d Ω p π N ( k F ˆ p , q , z )( z + v F ˆ p · q + i/τ ) . (17f)Expanding the numerator in Eq. (17a) to O ( q ) and thedenominator to O ( q ) , and neglecting corrections to thediffusion pole that are analytic in z , we find δn ( q , z ) = iδn ( q , t = 0) z + iD ( z ) q + iδn ⊥ ( q , t = 0)[ z + iD ( z ) q ][1 + s z α ( z )] (18a)with δn ⊥ from Eq. (3b), and the renormalized diffusivity D ( z ) = D s z / [1 + s z α ( z ) τ ] . (18b)The leading nonanalytic z -dependence of δn followsfrom using Eq. (15d) in Eqs. (18). In the asymptotic low-frequency regime z ≪ D q , the latter is z / in the firstterm on the right-hand side of Eq. (18a), and z / in thesecond one. For z ≫ D q , the first term has no nonana-lytic z -dependence, while the second term goes as z − / .The corresponding behavior in the time domain can beobtained by using the concept of generalized functionsand their Fourier transforms [30, 31]. All of the relevantnonanalyticities can be expressed in terms of H ν ( z ) = − i [ z ν − ( − z ) ν ] . (19a)The Laplace back transform for non-integer ν is [30] H ν ( t ) = − π sin( πν ) sin( πν/
2) Γ(1 + ν ) / | t | ν , (19b)with Γ the Gamma function. H − ( t ) is a constant. Ap-plying these results to Eqs. (18) we obtain Eqs. (4).We conclude with two general remarks.(1) A basic tenet of Statistical Mechanics is that a de-scription that drastically reduces the number of degreesof freedom is necessary, desirable, and physically sensi-ble. In equilibrium one goes from a description in termsof N = O (10 ) variables to one in terms of two thermo-dynamic variables, say, density n and energy density ǫ .In a nonequlibrium fluid one also includes the fluid ve-locity u . We have shown, using kinetic theory, that thesedescriptions are necessarily incomplete. One cannot sim-ply integrate out the other variables and still correctlydescribe transport or other nonequilibrium effects.(2) There is no simple remedy for this incompleteness.A description in terms of the single-particle distribution f , as used in this Letter, yields a solution in terms ofthe initial condition f ( p , x , t = 0) . This is not completeeither, since in order to obtain a closed description interms of f alone one has to effectively integrate out higherorder distribution functions, such as the pair correlationfunction G (2) = f (2) − f f , with f (2) the two-particledistribution function. A complete description in termsof f therefore depends on G (2) ( t = 0) , which in generalis nonzero and also multiplies a LTT contribution [15].The conclusion is that, due to LTT effects, no reduceddescription can be truly complete.To summarize: Long-time-tail effects have been inves-tigated in a host of physical systems including classicalfluids, superfluids, granular matter, active matter, elec-trons in solids, quark-gluon plasmas, and quantum grav-ity. The main message of this Letter is that a purely hy-drodynamic description that is routinely used in all these fields is incomplete because non-hydrodynamic initial-condition effects cannot be ignored in general.We thank Alex Levchenko for a discussion. [1] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Perg-amon, Oxford, 1987).[2] E. M. Lifshitz and L. P. Pitaevskii,
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