Hydrodynamic tails and a fluctuation bound on the bulk viscosity
HHydrodynamic tails and a fluctuation bound on the bulk viscosity
Mauricio Martinez and Thomas Sch¨afer
Department of Physics, North Carolina State University, Raleigh, NC 27695
Abstract
We study the small frequency behavior of the bulk viscosity spectral function using stochasticfluid dynamics. We obtain a number of model independent results, including the long-time tailof the bulk stress correlation function, and the leading non-analyticity of the spectral function atsmall frequency. We also establish a lower bound on the bulk viscosity which is weakly dependenton assumptions regarding the range of applicability of fluid dynamics. The bound on the bulkviscosity ζ scales as ζ min ∼ ( P − E ) (cid:80) i D − i , where D i are the diffusion constants for energyand momentum, and P − E , where P is the pressure and E is the energy density, is a measureof scale breaking. Applied to the cold Fermi gas near unitarity, | λ/a s | ∼ > λ is the thermalde Broglie wave length and a s is the s -wave scattering length, this bound implies that the ratio ofbulk viscosity to entropy density satisfies ζ/s ∼ > . h/k B . Here, ¯ h is Planck’s constant and k B isBoltzmann’s constant. a r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec . INTRODUCTION Hydrodynamic tails reflect the fact that fluid dynamics is an effective theory, in which theclassical equations of motions are the lowest order approximation to a more complete theoryinvolving averages over fluctuations of the fundamental variables. The classical equations ofmotion in fluid dynamics describe the evolution of conserved quantities such as mass, energy,and momentum. These equations depend on the form of the associated currents [1]. In fluiddynamics the currents are expanded in gradients of hydrodynamic variables, and the corre-sponding expansion coefficients are known as transport coefficients. Transport coefficientscontrol dissipative effects and fluctuation-dissipation relations imply that dissipative termsmust be accompanied by stochastic forces. The presence of stochastic terms manifests itselfin the form of long time, non-analytic, tails in correlation functions [2–4].Long time tails have been observed in computer simulations of fluids [7, 8], but theyare more difficult to detect experimentally. In the present work we will study the corre-lation function of the bulk stress, with an emphasis on dilute quantum fluids, such as thedilute Fermi gas near unitarity. Bulk stresses are interesting because the bulk viscosity canbe strongly enhanced near a phase transition [9], and quantum fluids provide attractiveapplications because hydrodynamic fluctuations are enhanced in systems in which the mi-croscopic transport coefficients are small. The existing literature contains only very limitedinformation on the bulk stress correlation function. The only calculation of the bulk tail ina non-relativistic theory away from the critical point that we have been able to find appearsto be wrong [4]. There are a number of studies of hydrodynamic tails near the liquid-gasendpoint and the superfluid transition [5], and there is a calculation of the bulk tail in arelativistic non-conformal fluid at zero mean charge density in [6].In this work we compute the long time tail of the bulk stress correlation function in anon-relativistic fluid. We apply the result to the dilute Fermi gas near unitarity, and derive anovel bound on the bulk viscosity of a non-conformal fluid. This bound only depends on theshear viscosity and thermal conductivity of the fluid, combined with a measure of conformalsymmetry breaking in the equation of state. The bound is similar to lower bounds on theshear viscosity in relativistic and non-relativistic fluids that have been derived in [10–12].Finally, we discuss constraints on the bulk viscosity spectral function of a non-relativisticfluid. 2
I. KUBO FORMULA
In this section we will determine the relation between the bulk viscosity and the lowfrequency behavior of the retarded correlation function of the stress tensor. This relation,known as the Kubo formula, can be determined by matching the linear response relationfor the stress induced by an external strain to the low frequency behavior of the responsepredicted by fluid dynamics. The Kubo formula for the shear and bulk viscosity of a non-relativistic fluid has been rederived many times [1, 13], but there are a number of subtletiesthat we would like to emphasize. We will make use of a formalism developed in [14–17], whichis based on studying the response of the fluid to a non-trivial background metric g ij ( t, (cid:126)x ).Correlation functions of the stress tensor are determined using linear response theory, andthe constraints of Galilean symmetry can be incorporated by requiring the equations of fluiddynamics to satisfy diffeomorphism invariance.The retarded correlation function of the stress tensor Π ij is defined by G ijklR ( ω, k ) = − i (cid:90) dt (cid:90) d x e iωt − i k · x Θ( t ) (cid:104) [Π ij ( t, x ) , Π kl (0 , )] (cid:105) . (1)The retarded correlator determines the stress induced by a small strain g ij ( t, x ) = δ ij + h ij ( t, x ) δ Π ij ( ω, k ) = − G ijklR ( ω, k ) h kl ( ω, k ) . (2)In fluid dynamics the stress tensor is expanded in terms of gradients of the thermodynamicvariables. We write Π ij = Π ij + Π ij + . . . , whereΠ ij = ρv i v j + P g ij (3)is the ideal fluid part, and Π iij with i (cid:54) = 0 are viscous corrections. Here, ρ is the mass densityof the fluid, v i is the velocity, and P is the pressure. At first order in the gradient expansionΠ ij = − ησ ij − ζg ij (cid:104) σ (cid:105) with σ ij = ∇ i v j + ∇ j v i + ˙ g ij − g ij (cid:104) σ (cid:105) , (4) (cid:104) σ (cid:105) = ∇ · v + ˙ g g , (5)where σ ij is the shear stress tensor, η is the shear viscosity, ζ is the bulk viscosity, g isthe determinant of the metric, and ∇ i is the covariant derivative associated with g ij . Theterms involving time derivatives of the metric are fixed by diffeomorphism invariance [15].3oughly, we can think of these terms as arising from the non-relativistic reduction of agenerally covariant stress tensor, σ ij ∼ ∇ i u j ∼ u Γ ij ∼ ˙ g ij , where u , u i are the temporaland spatial components of the four-velocity, and Γ αµν is the Christoffel symbol.We will consider a harmonic perturbation of the form h ij ( t, x ) = δ ij he − iωt . At the levelof ideal fluid dynamics this perturbation induces two terms in the stress tensor. The first, δ Π ij = P h ij , arises from the direct coupling of Π ij to the background metric. The secondterm follows from the equations of ideal fluid dynamics in a non-trivial background. Thecontinuity equation implies δρ = iω hρ , where ρ is the unperturbed mass density. Thisleads to a shift in the pressure δP = ( ∂P ) / ( ∂ρ ) s δρ .At first order in gradients the response is carried by the coupling to the backgroundmetric in equ. (4). As expected, the response to a bulk strain h ij ∼ δ ij is independent of theshear viscosity. At order O ( ω ) we get19 G iijjR ( ω, ) = − (cid:32) P − (cid:32) ∂P∂ρ (cid:33) s ρ (cid:33) − iωζ , (6)where repeated indices are summed over. The Kubo relation is ζ = − lim ω → ω Im G iijjR ( ω, ) . (7)In the following we will derive a slightly more convenient version of this Kubo relation. Bulkviscosity is a measure of scale breaking, and we would like to find a version of the Kuborelation in which this property is manifest. In the local rest frame of the fluid the trace ofthe stress tensor is proportional to the pressure. In [17] we showed that in equilibrium scalebreaking can be characterized by the quantity ∆ Tr P = P − E . (8)Here, we use E to denote the energy density in the rest frame of the fluid. In ideal fluiddynamics the total energy density is given by E = E + ρ v . We can now make use of thefact that the energy density of the fluid is conserved ∂ E ∂t + ∇ · (cid:15) = 0 , (9) We use the subscript Tr to distinguish the trace anomaly ∆ Tr P from the quantity ∆ P , which is afluctuation in the pressure. (cid:15) is the energy current. This relation implies that for ω (cid:54) = 0 the retarded Greenfunction G (cid:15)iiR ( ω, k ) of the energy density and the trace of the stress tensor must vanish as k →
0. A more formal proof of this statement using Ward identities was given in [18], seealso [19, 20]. We conclude that we can use any linear combination of the form O = Π ii + c E to define the Kubo relation for the bulk viscosity. Here, we will use c = − . This choicehas the nice property that the Kubo relation ζ = − lim ω → ω Im G OO R ( ω, ) , O = 13 (cid:16) Π ii − E (cid:17) (10)involves an operator which is manifestly sensitive to the trace anomaly in the hydrodynamiclimit, O = ∆ Tr P = P − E . III. HYDRODYNAMIC FLUCTUATIONS
There are many possible strategies for evaluating the retarded correlation function of O = ∆ Tr P . An example is the microscopic calculation in [21], where we compute the bulkviscosity in a dilute Fermi gas based on a perturbative calculation of quasi-particle properties.In this work we will employ a different strategy and compute the retarded correlation usinga macroscopic theory of the long distance properties of the fluid. This theory it stochasticfluid dynamics [13]. As we will show this theory provides a universal prediction of the leadingnon-analyticity in G OO R ( ω, ) as ω →
0. It also provides a lower bound on ζ , but this boundis sensitive to microscopic physics.In order to explore the role of hydrodynamic fluctuations we will expand ∆ P to secondorder in hydrodynamic variables. Higher order terms can be computed, but they providecorrections that are subleading in ω/ω br . Here ω br is the breakdown scale of hydrodynamics,which we will define more carefully below. The probability of a fluctuation of the hydrody-namic variable is proportional to exp(∆ S ), where ∆ S is the change in entropy of the fluid[22]. We can write S = (cid:90) d x s ( ρ, E ) , (11)so that ∆ S = (cid:90) d x (cid:32) ∂s∂ρ (cid:33) E ∆ ρ + (cid:32) ∂s∂ E (cid:33) ρ ∆ E + 12 (cid:32) ∂ s∂ρ (cid:33) E (∆ ρ ) + ∂ s∂ρ∂ E ∆ ρ ∆ E + 12 (cid:32) ∂ s∂ ( E ) (cid:33) ρ (∆ E ) + . . . , (12)5e can use the conservation laws for the mass density ρ and the energy density E to showthat the linear terms vanish. The quadratic terms can be simplified by using a set ofthermodynamic variables that diagonalizes the quadratic form. A suitable set of variablesif provided by ( ρ, T ) [13, 23]. The entropy functional that governs fluctuations in ρ, T and v is ∆ S = − T (cid:90) d x (cid:40) ρ (cid:32) ∂P∂ρ (cid:33) T (∆ ρ ) + c V T (∆ T ) + ρ v + . . . (cid:41) , (13)where ( T , ρ ) denote the mean values of the temperature and density, and (∆ T, ∆ ρ, v ) arelocal fluctuations. We can expand O = ∆ Tr P to second order in (∆ T, ∆ ρ ), O = O + a ρ ∆ ρ + a T ∆ T + a ρρ (∆ ρ ) + a ρT ∆ ρ ∆ T + a T T (∆ T ) + . . . . (14)The hydrodynamic tails are determined by the second order terms. The correspondingcoefficients can be expressed in terms of thermodynamic quantities. We find a ρρ = 12 ∂∂ρ (cid:34) c T − (cid:32) hm − T ακ T ρ (cid:33)(cid:35) T , (15) a ρT = ∂c T ∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ − ∂c V ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T , (16) a T T = 12 T (cid:32) − ρ ∂∂ρ (cid:33) T − ∂∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ c V . (17)Here, c T is the isothermal speed of sound, h is the enthalpy per particle, α is the thermalexpansion coefficient, κ T is the bulk modulus, and c V is the specific heat at constant volume.We define these quantities in the appendix. The coefficients a αβ with α, β = ( ρ, T ) aresensitive to conformal symmetry breaking, and vanish in the ideal gas limit. A numericalestimate of a αβ therefore requires a non-trivial equation of state. As an example we considera dilute Fermi gas governed by an s -wave interacting with scattering length a s . In the hightemperature limit the trace anomaly is given by [21]∆ Tr P = 2 π m a s ρ T , (18)where we employ units ¯ h = k B = 1. In the limit a s → ∞ the dilute Fermi gas is scaleinvariant and the trace anomaly vanishes. Using equ. (18) we find( a ρρ , a ρT , a T T ) = 2 π m T a s (cid:16) T , − ρT, ρ (cid:17) . (19)6 V. HYDRODYNAMIC TAILS: FORMALISM
In order to study hydrodynamic tails we consider the correlation function of ∆ Tr P ex-panded to second order in (∆ ρ, ∆ T ). In statistical field theory it is convenient to start fromthe symmetrized correlation function G OO S ( ω, k ) = (cid:90) d x (cid:90) dt e i ( ωt − k · x ) (cid:28) {O ( t, x ) , O (0 , } (cid:29) . (20)This function is related to the retarded correlator by the fluctuation-dissipation theorem.For ω → G S ( ω, k ) (cid:39) − Tω Im G R ( ω, k ) . (21)At second order in (∆ ρ, ∆ T ) and at the level of the Gaussian entropy functional the sym-metrized correlation function factorizes into a set of two-point functions G OO S ( ω,
0) = (cid:90) dω (cid:48) π (cid:90) d k (2 π ) (cid:104) a ρρ ∆ ρρS ( ω (cid:48) , k )∆ ρρS ( ω − ω (cid:48) , k ) (22)+ a ρT ∆ ρρS ( ω (cid:48) , k )∆ T TS ( ω − ω (cid:48) , k ) + 2 a T T ∆ T TS ( ω (cid:48) , k )∆ T TS ( ω − ω (cid:48) , k ) (cid:105) . where ∆ ρρS is the symmetrized density correlation function∆ ρρS ( ω, k ) = (cid:90) d x (cid:90) dt e i ( ωt − k · x ) (cid:28) { ρ ( t, x ) , ρ (0 , } (cid:29) , (23)and ∆ T TS is the temperature correlation function. Note that by working with (∆ T, ∆ ρ ) weavoid off-diagonal correlation functions such as ∆ ρTS . Also note that in hydrodynamics thesymmetrized functions ∆ S reduces to the statistical correlation function.The Kubo relation involves the retarded, not the symmetrized, correlation function. Wecan reconstruct the retarded function using the fluctuation-dissipation relation (21). Con-sider the first term in equ. (22). At low frequency the contribution to G R can be written as[10, 11] G OO R ( ω, (cid:12)(cid:12)(cid:12) ρρ = 2 a ρρ (cid:90) dω (cid:48) π (cid:90) d k (2 π ) (cid:104) ∆ ρρR ( ω (cid:48) , k )∆ ρρS ( ω − ω (cid:48) , k )+ ∆ ρρS ( ω (cid:48) , k )∆ ρρR ( ω − ω (cid:48) , k ) (cid:105) . (24)This is an example of a more general relation that one can prove using hydrodynamiceffective actions, which shows that the retarded correlation functions can be derived usinga perturbative expansion based on a combination of retarded and symmetrized propagators[5, 6, 24–29]. 7he two-point functions of the temperature and density in first order dissipative hydrody-namics are well known [1]. The temperature correlation function is dominated by a diffusiveheat wave. The symmetric and retarded correlation functions are∆ T TS ( ω, k ) = 2 T c P D T k ω + ( D T k ) , (25)∆ T TR ( ω, k ) = Tc P − D T k − iω + D T k , (26)where c P is the specific heat at constant pressure, D T = κ/c P is the thermal diffusionconstant, and κ is the thermal conductivity. The two-point function of the density is morecomplicated, because the density couples to both propagating sound modes and diffusiveheat modes. The symmetric correlation function is [1]∆ ρρS ( ω, k ) = 2 ρT (cid:40) Γ k ( ω − c s k ) + (Γ ωk ) + ∆ c P c s D T k ω + ( D T k ) − ∆ c P c s ( ω − c s k ) D T k ( ω − c s k ) + (Γ ωk ) (cid:41) , (27)where k = k , c s is the speed of sound, and ∆ c P = ( c P − c V ) /c V . We have also defined thesound attenuation constantΓ = 43 ηρ + ζρ + κ (cid:18) c V − c P (cid:19) = 43 ηρ (cid:34) ζη + 34 ∆ c P Pr (cid:35) , (28)where Pr = ( c P η ) / ( ρκ ) is the Prandtl number, the ratio of the momentum and thermal diffu-sion constants. At high temperature ∆ c P = 2 / Pr = 2 / c P / Pr → ρρR,S ( ω, k ) = ∆ sd R,S ( ω, k ) + ∆ ht R,S ( ω, k ) + ∆ m R,S ( ω, k ) . (29)In the long wavelength limit the sound contribution can be written as∆ sd S ( ω, k ) = ρT Γ k ωc s (cid:40) ω − c s k ) + ( Γ2 k ) − ω + c s k ) + ( Γ2 k ) (cid:41) (30)∆ sd R ( ω, k ) = ρ Γ k c s (cid:40) ω − c s k + i Γ2 k − ω + c s k + i Γ2 k (cid:41) (31)8 ∆ TT ∆ TT a TT a TT (cid:2) ∆ ρρ ∆ ρρ a ρρ a ρρ (cid:3) ∆ TT ∆ ρρ a ρT a ρT a) b) c)FIG. 1: Diagrammatic representation of the leading contribution of thermal fluctuations to thebulk stress correlation function. The dashed line corresponds to the operator O = P − E . Solidlines denote the diffusive temperature correlator, and wavy lines denote the density correlationfunction, determined by the sound pole and the diffusive heat mode. and the diffusive heat mode is∆ ht S ( ω, k ) = 2 ρT ∆ c P c s D T k ω + ( D T k ) , (32)∆ ht R ( ω, k ) = ρ ∆ c P c s − D T k − iω + D T k . (33)Finally, there is a term that is sensitive to both sound and diffusive modes∆ m S ( ω, k ) = − ρT ∆ c P c s kD T c s (cid:40) ω − c s k ( ω − c s k ) + ( Γ2 k ) − ω + c s k ( ω + c s k ) + ( Γ2 k ) (cid:41) , (34)∆ m R ( ω, k ) = ρ ∆ c P c s iωkD T c s (cid:40) ω − c s k + i Γ2 k − ω + c s k + i Γ2 k (cid:41) . (35) V. HYDRODYNAMIC TAILS: ONE-LOOP DIAGRAMS
In this section we will compute the leading infrared behavior of the three one-loop dia-grams shown in Fig. 1. The two-point function of the density has three distinct contributions,see equ. (29), and as a result there are ten one-loop diagrams total. As we will see, onlyfour of them contribute to the low frequency behavior of G R ( ω, ).1. The simplest diagram involves diffusive fluctuations of the temperature only. Weconsider equ. (24) with ( ρρ ) → ( T T ) and use the retarded and symmetrized functions givenin equ. (25,26). We perform the frequency integral by closing the contour in the complex ω plane. We find G OO R ( ω, ) (cid:12)(cid:12)(cid:12) ht T T = − a T T T c P (cid:90) d k (2 π ) k k − iω D T , (36)9here T T refers to the presence of two temperature correlation functions, and ht indicatesthat these modes are dominated by a diffusive heat mode. The integral in equ. (36) isultraviolet divergent. We will regularize the integral using a momentum cutoff Λ. Wewill see that there are two types of terms. Hydrodynamic tails are non-analytic in ω andindependent of the cutoff. Fluctuation terms are sensitive to the cutoff and contribute to G R in the same way as transport coefficients. This implies that the cutoff dependence canbe absorbed into the bare transport parameters. However, we will see that this procedureimplies bounds on the transport coefficients.After introducing a cutoff we can compute the integral in equ. (36) by expanding in ω .The leading terms are G OO R ( ω, ) (cid:12)(cid:12)(cid:12) ht T T = − a T T T c P L ( ω, Λ , D T ) , (37)where we have defined L ( ω, Λ , D T ) = 12 π (cid:40) Λ iω Λ2 D T − π √ i ) (cid:18) ω D T (cid:19) / + . . . (cid:41) . (38)Note that the small parameter in the low frequency expansion is (cid:15) ≡ ω/ ( D T Λ ). We observethat the Λ term can be viewed as a contribution to the compressibility term in equ. (6),and the iω Λ term is a contribution to the bulk viscosity. This term is sensitive to scalebreaking via the coefficient a T T , and it scales inversely with the thermal conductivity. Thelast term is a hydrodynamic tail. The imaginary part can be viewed as a √ ω contribution tothe frequency dependent bulk viscosity ζ ( ω ), and real part is a 1 / √ ω contribution the bulkviscosity relaxation time. This term signals the breakdown of second order deterministicfluid dynamics in the low frequency limit.2. A similar diffusive heat contribution appears in the two point function of the density.Comparing equ. (25,26) to equ. (32,33) we observe that this contribution is equal to theprevious term up to an overall factor. We get G OO R ( ω, ) (cid:12)(cid:12)(cid:12) ht ρρ = − a ρρ T ρ (∆ c P ) c s L ( ω, Λ , D T ) (39)In the case of a dilute gas equ. (39) and equ. (37) are comparable in magnitude, but ingeneral the two contributions can be different.3. Another diffusive heat contribution is contained in the mixed ∆ ρρ ∆ T T term, shown asthe third diagram in Fig. 1. We get G OO R ( ω, ) (cid:12)(cid:12)(cid:12) ht ρT = − a ρT ρT ∆ c P c P c s L ( ω, Λ , D T ) . (40)10. The two point function of the density also contains a sound contribution. This term isquite different, because sound is a propagating mode, and sound attenuation is controlled byΓ, which is not only sensitive to κ but also to the shear viscosity η and a possible microscopiccontribution to ζ . We determine this term using the two point functions in equ. (30, 31).We observe that there are two types of contributions, characterized by the relative sign ofthe real part of the pole position, ω (cid:48)± = ± c s k + O ( ω, k ). We first consider diagrams wherethe poles are on opposite sides of the real axis. We get G OO R ( ω, ) (cid:12)(cid:12)(cid:12) sd ρρ = − a ρρ T ρ c s L ( ω, Λ , Γ) , (41)where the index sd indicates the contribution from the sound mode. The diagram wherethe two poles are on the same side gives G OO R ( ω, ) (cid:12)(cid:12)(cid:12) sd ρρ = − a ρρ T ρ c s (cid:90) d k (2 π ) k ( ω − c s k + i Γ k )( c s k − i Γ k ) . (42)This integral is UV divergent, but it is less IR sensitive then equ. (36). In particular, thelow frequency behavior is governed by c s k (cid:29) Γ k . As a result, the contribution to the iω term in G OO R ( ω, ) is suppressed by a factor (ΓΛ /c s ) relative to equ. (37).5. The remaining diagrams fall into two categories. The first class involves mixed di-agrams in which a diffusive heat mode is coupled to a propagating sound mode. Thesediagrams are suppressed because if one of the propagators is put on shell the other propa-gator is far off shell, and the diagram is not infrared sensitive. The other diagrams involvethe mixed sound-heat propagator in equ. (34,). The on-shell residue of this propagator issuppressed. We finally collect the contributions from equ. (37-43). We get G OO R ( ω, ) = − A T L ( ω, Λ , D T ) − A Γ L ( ω, Λ , Γ) , (43)where we have defined A T = 2 a T T T c P + 2 a ρρ ρ T (∆ c P ) c s + a ρT ρT ∆ c P c P c s , A Γ = a ρρ ρ Tc s . (44) VI. PHENOMENOLOGICAL ESTIMATESA. Hydrodynamic tail
In the previous section we showed that the ω / term in the retarded correlation functionis uniquely determined in terms of the equation of state and the transport parameters. This11erm has several physical effects: It determines the long time tail of the correlation function,it governs the small frequency limit of the bulk viscosity spectral function, and it determinesthe ω → C ζ ( t ) = (cid:90) dω π G OO S ( ω, ) e − iωt . (45)For t → ∞ we obtain a t − / tail C ζ ( t ) = T π / (cid:32) A T (2 D T ) / + A Γ Γ / (cid:33) t / , (46)This contribution is computed most easily by starting from the momentum integral inequ. (36), and then perform the frequency integral before the momentum integral. Thehydrodynamic tail in the bulk stress correlator was first computed by Pomeau and R´esibois[4], but their result does not appear to be correct. In particular, the expression for C ζ ( t )given in [4] does not vanish for a scale invariant fluid. In our work C ζ ( t ) ∼ a αβ ∼ (∆ Tr P ) automatically vanishes for a scale invariant fluid.The contribution of critical fluctuations to the tail in the bulk stress correlation functionwas computed by Onuki [5], both in model H (liquid-gas endpoint) and model F (superfluidtransition). In principle the model F result for T > T c is directly applicable to the Fermigas near unitarity. Model F contains two hydrodynamic variables, a linear combination ofthe energy density E and the density ρ , as well as the the superfluid density. Above T c only the energy density like variable contributes. In this regime there are two differencescompared to our analysis: 1) We keep both both E and ρ ; 2) The model F analysis uses amore complicated functional form of the thermal conductivity κ ( k , t ) with t = ( T − T c ) /T c ,which reduces to a simple constant for t (cid:29)
1. This implies that the model F tail should besimilar to our tail for large t . This is difficult to verify, because the coupling between theenergy density-like variable to the bulk stress does not manifestly respect scale invariance.The bulk tail in a relativistic non-conformal fluid was computed by Kovtun and Yaffe [6].These authors assume that the mean density of the fluid vanishes, so that we cannot directlycompare to the non-relativistic limit. B. Spectral function
A second quantity of interest is the spectral function ζ ( ω ) = − ω Im G OO R ( ω, ) . (47)12he existence of a hydrodynamic tail implies that ζ ( ω ) = ζ (0) − (cid:32) A T (2 D T ) / + A Γ Γ / (cid:33) √ ω √ π . (48)This result can be combined with other model independent information about the spectralfunction. The high frequency tail of the bulk viscosity was determined using the operatorproduct expansion [31] ζ ( ω ) = C π √ mω
11 + a s mω , (49)where C is the contact density [32, 33]. The contact density is directly related to the traceanomaly near unitarity ∆ Tr P = C πma s . (50)In the high temperature limit C can be computed using the virial expansion [34]. Nearunitarity we find C = 4 πn λ (cid:40) √ (cid:32) λa s (cid:33) + . . . (cid:41) , (51)which implies ζ ( ω ) ∼ λ − (cid:32) zλa s (cid:33) (cid:18) Tω (cid:19) / , (52)where λ = [(2 π ) / ( mT )] / is the thermal de Broglie wave length and z = nλ is the fugacityof the gas. Finally, there is a sum rule for the bulk viscosity spectral function [35–37]1 π (cid:90) dω ζ ( ω ) = 172 πma ∂ C ∂a − s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s/n . (53)In the next section we will combine these constraints with the fluctuation bound to providea simple model of the viscosity spectral function. C. Fluctuation bound
The cutoff dependent term in the bulk viscosity ζ Λ = 118 π (cid:18) A T Λ2 D T + A Γ ΛΓ (cid:19) , (54)has to combine with the bare bulk viscosity to determine the physical bulk viscosity of thefluid. We can view this result as arising from a renormalization group procedure, wherefluid dynamics is matched to a microscopic theory at the scale Λ, and then the evolution of13 / s T / T F FIG. 2: Fluctuation bound (blue line) on ζ/s for the dilute Fermi gas as a function of
T /T F . Weshow the regime T > T c with T c /T F (cid:39) .
17. As explained in the text we estimate the equation ofstate and transport properties using results in the high temperature limit. We have also chosen a s /λ = 1. The error band corresponds to a 50% error in Λ T and Λ Γ . For comparison we show thekinetic theory result for ζ/s as the green line. G R ( ω ) below the scale Λ is computed using stochastic fluid dynamics. For this procedureto be consistent the bare viscosity at the cutoff scale must be positive, and the the physicalviscosity must be larger than ζ Λ . This bound increases with the cutoff scale Λ. The largestpossible Λ is determined by the breakdown scale of fluid dynamics, because above that scalestochastic fluid dynamics is not reliable. Of course, the viscosity at the cutoff scale mustdepend on Λ, so that the physical viscosity ζ (0) is cutoff independent. The same conclusionalso follows from the spectral density given in equ. (48). We observe that the non-analytic √ ω term is negative. If this term is the dominant correction to the spectral density belowthe breakdown scale of fluid dynamics, ω ∼ < ω br , then spectral positivity implies that ζ (0)cannot be arbitrarily small.In order to determine the maximum momentum where fluid dynamics can be trusted wecan study the dispersion relation of diffusive heat modes and propagating sound waves, anddetermine the maximum momentum for which higher order corrections are small comparedto leading order terms.1. Diffusive modes: Heat modes are characterized by ω ∼ D T k . Corrections arise from14 ( ω ) / s ω / T FIG. 3: Schematic form of the bulk viscosity spectral function. This figure shows ζ/s as a functionof the frequency ω in units of T . We have chosen a s /λ = 1 and T /T F = 0 .
2. The low frequencypart shows the function ζ ( ω ) = ζ min − c √ ω , where ζ min is the bound in equ. (57), and c is theuniversal coefficient given in equ. (48). The high frequency part is the function given in equ. (49).The green dotted line shows a model for the low frequency spectral function where we have added amicroscopic contribution ζ micro /s = 0 .
04 to the hydrodynamic result. The microscopic contributionwas chosen to smoothly match the high frequency tail. higher order terms in the derivative expansion. For non-zero frequency the leading correctionis due to the relaxation time. We get ω ∼ D T k (cid:28) τ − κ . For this relation to be maintainedfor all k < Λ we have to require that Λ ∼ < Λ T with Λ T = ( τ κ D T ) − / . In kinetic theory τ κ = ( mκ ) / ( c P T ) and Λ T (cid:39) D T (cid:18) Tm (cid:19) / . (55)Equation (55) implies that the expansion parameter of the low frequency expansion, (cid:15) = ω/ ( D T Λ ), is of order (cid:15) ∼ ( mD T )( ω/T ). For a nearly perfect fluid D T ∼ m − [38] andthe low frequency expansion is valid all the way up to ω ∼ T . In the case of a poor fluid D T (cid:29) m − and the range of validity of the low frequency expansion is smaller. We also notethat equ. (55) ensures that the expansion parameter ( D T Λ /c s ) is indeed small.2. Sound channel: In the sound channel we have ω ∼ c s k (cid:28) Γ k . This implies k ∼ < Λ Γ Γ (cid:39) (cid:32) ∂P∂ρ (cid:33) / s/n . (56)For a weakly interacting gas we get ( ∂P ) / ( ∂ρ ) s/n (cid:39) (5 T ) / (3 m ). We can either use the twoestimates equ. (55, 56) in the respective channels, or use the smaller of the two values. Inthe weak coupling limit, where Pr ∼
1, these two estimates are numerically very similar.Using the first method, we obtain the bound ζ min = (cid:32) A T D T + √ A Γ √ (cid:33) (cid:115) Tm . (57)We observe that there is a minimum value of ζ that is solely controlled by (∆ Tr P/D T ) and (∆ Tr P/ Γ) . This implies that if there is scale breaking in the equation of state, andif the shear viscosity and thermal conductivity are finite, then the bulk viscosity cannot bezero. Fluctuation bounds on the shear viscosity were studied in [11, 12]. We observe thatthe bound on ζ has the same structure as the bound on η , but is suppressed by a factor(∆ Tr P/P ) .Finally, we provide some numerical estimates. For this purpose we assume that thebare bulk viscosity is zero, and that the shear viscosity and thermal conductivity are de-scribed by kinetic theory, η = η ( mT ) / and κ = κ m / T / with η = 15 / (32 √ π ) and κ = 225 / (128 √ π ) [30, 39]. In the case of the shear viscosity this is known to be a goodapproximation even close to the critical temperature [40]. We also use the results for c s , c P and ∆ c P in the dilute limit, see Appendix A. The bound on ζ/s as a function of T /T F isshown in Fig. 2. The width of the band reflects a 50% error related to the choice of Λ. Forcomparison we also show the kinetic theory result ζ/n = z / (24 √ πλ )( λ/a s ) [21]. At hightemperature the fluctuation bound is very small, but near T c (cid:39) . T F [41] the bound iscomparable to the kinetic theory result, indicating that the bulk viscosity must be at leastas big as predicted by kinetic theory. Note that we have extrapolated the bound on ζ/s allthe way to T c , despite the fact that several estimates involve approximations that are onlyreliable for T (cid:29) T c . Similar to the kinetic theory estimates discussed above, it is knownthat in the case of η/s this procedure provides a numerically accurate estimate of the boundnear T c .We note that ζ/s is given in units of ¯ h/k B . Both the hydrodynamic and the kinetic theorycalculation are completely classical. Planck’s constant enters the hydrodynamic calculation16ia the equation of state, and it appears in the kinetic theory calculation in terms of boththe equation of state and the quasi-particle dispersion relation.In Fig. 3 we summarize the available information on the spectral function ζ ( ω ). We plot ζ ( ω ) /s as a function of ω/T . For small ω we show the hydrodynamic prediction in equ. (48)where ζ (0) is assumed to be the fluctuation bound. For large ω we show the tail predictedby the operator product expansion, see equ. (49). We have chosen T /T F = 0 . λ/a s = 1.We conclude that a smooth extrapolation of the large frequency tail to ω = 0 is consistentwith a bulk viscosity ζ (0) which is somewhat larger than the fluctuation bound. As anexample we show the green dotted line which corresponds to ζ = ζ min + ζ micro − c √ ω with ζ micro /s = 0 .
04 and c given by equ. (49). This function smoothly matches the high frequencytail. Integrating the low frequency model and the high frequency tail over the entire range ω ∈ [0 , ∞ ] saturates 65% of the sum rule in equ. (53). We conclude that a reasonable modelof the bulk viscosity spectral function can be obtained by matching the high frequency tailto the hydrodynamic spectral function combined with a small microscopic viscosity. VII. OUTLOOK
In this work we have studied the role of hydrodynamic fluctuations in the bulk stresscorrelation function. We have shown that fluctuations provide a lower bound on the bulkviscosity that only depends on the thermal conductivity and shear viscosity as well as scalebreaking in the equation of state. The physical mechanism for the bound can be understoodin terms of the rate of equilibration of thermal fluctuations. Consider a fluid in equilibriumat density ρ and temperature T . Fluctuations in this fluid are controlled by the entropyfunctional in equ. (13). If the fluid is compressed then the equilibrium density and temper-ature change, and as result the mean square fluctuations in ρ, T, v have to change as well.However, the mechanism for fluctuations to adjust involves diffusion of heat and momentum,and does not take place instantaneously. As a consequence the fluid is slightly out of equi-librium, entropy increases, and the effective bulk viscosity is not zero. This mechanism isparticularly relevant in fluids which have no significant microscopic sources of bulk viscosity.An example of a very good fluid that does not have a simple microscopic mechanism forgenerating bulk viscosity is the dilute Fermi gas near unitarity. Our estimates indicate thatthe ratio of bulk viscosity to entropy density near the phase transition and for | λ/a s | ∼ > ζ/s ∼ > .
1. This is within reach of experiments involving hydrodynamic expansion [42].The effects might be even more significant in two dimensional gases. In these systems bulkviscosity has been studied using the damping of monopole oscillations [43, 44]. It may alsobe possible to observe the non-analyticity of the spectral function or the long time tail inthe Kubo integrand using numerical simulations [45].Our work can be extended in several directions. One interesting question is the role ofcritical fluctuations in the vicinity of a second order phase transition [5, 9]. In that caseloop diagrams similar to the graphs studied in this work lead to an enhancement of the bulkviscosity near the critical point. Another important problem is the study of fluctuations inrelativistic fluids, see [6, 10, 46, 47]. In that case it has been conjectured that the quarkgluon plasma phase transition has a critical end point which is in the universality class ofmodel H [26, 48], and that critical fluctuations can be observed in the relativistic heavy ioncollisions [49].Acknowledgments: This work was supported in part by the US Department of Energygrant DE-FG02-03ER41260 and by the BEST (Beam Energy Scan Theory) DOE TopicalCollaboration. This work was completed while T. S. was a visitor at the Aspen Center forPhysics, which is supported by National Science Foundation grant PHY-1607611.
Appendix A: Thermodynamic quantities
We assume that the equation of state is given in the form P = P ( µ, T ). A specificexample is the virial expansion which provides the equation of state in the form P = νTλ (cid:16) z + b ( T ) z + . . . (cid:17) , (A1)where ν is the number of degrees of freedom ( ν = 2 in the unitary Fermi gas), λ =[(2 π ) / ( mT )] / is the thermal wave length, and z = exp( µ/T ) is the fugacity. Note that wehave set ¯ h = k B = 1. Near unitarity b = b + δb where b = − / (4 √
2) is due to quantumstatistics and [21] δb = 1 √ (cid:32) √ πmT a s + . . . (cid:33) . (A2)Derivatives of the pressure with respect to µ and T determine the entropy density andpressure s = ∂P∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ , n = ∂P∂µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T . (A3)18he energy density is determined by the relation E = µn + sT − P , (A4)and the enthalpy per particle is h = ( E + P ) /n . In order to compute the specific heat atconstant volume we use V = N/n and write c V = TV ∂S∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V = ∂ ( s, V ) ∂ ( T, V ) = ∂ ( s, V ) /∂ ( T, µ ) ∂ ( T, V ) /∂ ( T, µ )= T ∂s∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ − [( ∂n/∂T ) | µ ] ( ∂n/∂µ ) | T , (A5)where we have defined the Jacobian ∂ ( u, v ) ∂ ( x, y ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂u∂x ∂u∂y∂v∂x ∂v∂y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A6)In order to compute c P we make use of the relation between c P − c V and the thermalexpansion coefficient α = (1 /V )( ∂V /∂T ) | P . This relation is given by c P − c V = − TV [( ∂V /∂T ) | P ] ( ∂V /∂P ) | T . (A7)The partial derivatives are 1 V ∂V∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P = 1 n sn ∂n∂µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T − ∂n∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ , V ∂V∂P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = − n ∂n∂µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T . (A8)The second of these relations defines the bulk modulus κ − T = − V − ( ∂V ) / ( ∂P ) | T . We get c P = c V + T (cid:104) sn ( ∂n/∂µ ) | T − ( ∂n/∂T ) | µ (cid:105) ( ∂n/∂µ ) | T . (A9)The isothermal and the adiabatic speed of sound are defined by c T = ∂P∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T , c s = ∂P∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s/n . (A10)We have c T = nm (cid:34) ∂n∂µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T (cid:35) − , c s = c P c V c T , (A11)19nd the thermal expansion coefficient can be written as α = 1 T (cid:34) c T Tm c P − c V n (cid:35) / . (A12)Finally, we can determine the first order derivatives that appear in the expansion in equ. (14).We get ∂P∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = c T , ∂ E ∂ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = hm − ακ T Tρ , ∂P∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ακ T , ∂ E ∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = c V , (A13)where h = ( E + P ) /n is the enthalpy per particle. Partial derivatives of these results withrespect to T and ρ determine the second order coefficients in equ. (15-17). [1] L. Kadanoff, P. Martin, “Hydrodynamic Equations and Correlation Function,” Ann.˜Phys.
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