Thermal fluctuations propagation in the relativistic Euler regime: a causal appraisal
aa r X i v : . [ g r- q c ] M a y Thermal (cid:29)u tuations propagation in the relativisti Eulerregime: a ausal appraisalA. Sandoval-Villalbazo and D. BrunUniversidad Iberoameri anaProlonga ión Paseo de la Reforma 880Méxi o, D.F. 01219, Méxi oO tober 9, 2018Abstra tIt is shown that thermal (cid:29)u tuations present in a simple non-degenerate relativisti (cid:29)uidsatisfy a wave equation in the Euler regime. The hara teristi propagation speeds are al- ulated and the lassi al expression for the speed of sound is re overed at the non-relativisti limit. Impli ations and generalizations of this work are analyzed.1 Introdu tionRelativisti transport theory has dramati ally in reased its interest due to the dete tion of hightemperature plasmas produ ed in the Relativisti Heavy Ion Collider (RHIC). In this ontext,an Euler (cid:29)uid des ription provides a good approximation for events involving Au-Au ollisions[1℄. Generalizations involving dissipative e(cid:27)e ts have been developed taking into a ount re entexperimental data [2℄. The (cid:28)rst works regarding relativisti hydrodynami s an be tra ked downto the pioneering 1940 E kart's monographs [3℄, and to the relativisti (cid:29)uids se tion in luded inthe Landau-Lifshitz (cid:29)uid me hani s textbook [4℄. The expli it form of the linearized transportequations obtained within E kart's framework raised serious doubts on erning the stability and ausality properties of the system [5℄ [6℄. Indeed, it was only re ently observed that the so- alledstability problem in relativisti hydrodynami s is due to the heat-a eleration oupling introdu edin E kart's work [7℄. Following these ideas, it be ame pertinent to revise the ausality propertiesin this kind of systems, fo using in the possibility of generating a hyperboli partial di(cid:27)erentialequation des ribing temperature (cid:29)u tuations. This work ta kles the problem for an Euler (cid:29)uid andsuggests some new insights for this issue while examining the linearized equations in the Navier-Stokes regime without resorting to extended formalisms.In se tion 2 the basi formalism is presented on the basis of relativisti kineti theory for aninert dilute (cid:29)uid, emphasizing the role of the Enskog transport equation. Se tion 3 is devoted tothe analysis of the linearized transport equations in the Euler regime, by means of a derivationof a wave equation des ribing thermal (cid:29)u tuations in the relativisti ase, and its orrespondingnon-relativisti limit. Some (cid:28)nal thoughts regarding the ausal properties of relativisti (cid:29)uids inthe dissipative ase are in luded in the (cid:28)nal se tion of this work.1 Kineti foundations and transport equationsFor de ades, relativisti kineti theory has been su essfully applied in the study of high temperature(cid:29)uids [8℄. The starting point here is the relativisti Boltzmann equation for a simple (cid:29)uid in theabsen e of external for es: v α f ,α = J ( f f ′ ) (1)In Eq. (1), f is the distribution fun tion in the phase spa e, J ( f f ′ ) is the ollisional kernel, andthe mole ular four velo ity, v α is given by v α = ( γw l , γc ) (2)where w l is the mole ular velo ity (three spatial omponents). As usual, γ = (cid:16) − w l w l c (cid:17) − / . Alllatin indi es run from 1 to 3 and the greek ones run up to 4. A signature (1 , , , − is taken, sothat u α u α = − c . The relativisti generalization of Enskog's transport equation an be asted inthe form [8℄ [9℄: ∂∂t ( n h Ψ i ) + (cid:0) n (cid:10) w l Ψ (cid:11)(cid:1) ; l = 0 (3)where n is the parti le number density and the average of the ollisional invariant h Ψ i is de(cid:28)ned as h Ψ i = 1 n Z γ Ψ f dv ∗ (4)with dv ∗ = γ cd wv [10℄.In the Euler regime, all averages are al ulated using the equilibrium (Juttner) distributionfun tion, valid for a non-degenerate gas [8℄: f (0) = n πc K ( z ) e uβvβzc (5)where u β = h v β i is the hydrodynami velo ity, z = kTmc is the relativisti parameter and K isthe modi(cid:28)ed Bessel fun tion of the se ond kind. Derivatives with respe t to u β an be expli itlyevaluated in Eq.(5). After this operation, for the sake of simpli ity, all al ulations will be performedin the omoving frame of the (cid:29)uid.Now, the ollisional invariants are Ψ = 1 (a onstant), mw l γ (the three-momentum) and mc γ (the me hani al energy). For Ψ = 1 the ontinuity equation follows immediately ∂∂t ( n ) + ( nu l ) ; l = 0 (6)Substituting Ψ = mw l γ , the momentum balan e equation is obtained: ∂∂t (cid:0) n (cid:10) mw k γ (cid:11)(cid:1) + (cid:0) nm (cid:10) w l w k γ (cid:11)(cid:1) ; l = 0 (7)2he use of Eqs. (4) and (5) allows to rewrite Eq. (7) in terms of the lo al thermodynami variables: c ( nε + p ) ∂∂t (cid:0) u l (cid:1) + kn ∂T∂x l + kT ∂n∂x l = 0 (8)where the internal energy per parti le ε reads: ε = 3 nkT + nmc K (1 /z ) K (1 /z ) (9)and the pressure satis(cid:28)es the sate equation p = nkT (10)Finally, for Ψ = mc γ , the resulting balan e equation reads: ∂∂t (cid:0) n (cid:10) γmc (cid:11)(cid:1) + (cid:0) n (cid:10) w l ( γmc ) (cid:11)(cid:1) ; l = 0 (11)or, in terms of the thermodynami variables: ∂ ( nε ) ∂t + pθ = 0 (12)in Eq. (12) we have de(cid:28)ned θ = u α ; α . The set of equations (6,8,12) is highly nonlinear and its fulltreatment is rather omplex. For a system lose to equilibrium we shall linearize this set in orderto perform a (cid:29)u tuation analysis for the thermodynami al variables.3 Linearized equations and ausality analysisIn order to pro eed with the analysis of the Euler system (6,8,12) lose to equilibrium, we de omposeany thermodynami al variable X into a onstant average value X o and a spa e and time dependent(cid:29)u tuation δX , so that X = X o + δX (13)A ording to this de(cid:28)nition, negle ting se ond order terms, the linearized ontinuity equation,obtained from (6) reads: ∂∂t ( δn ) + n o δθ = 0 (14)Analogously, the linearized momentum balan e for the longitudinal mode δθ be omes: ˜ ρ o ∂∂t ( δθ ) + n o k ∇ ( δT ) + kT o ∇ ( δn ) = 0 (15)where we have de(cid:28)ned ˜ ρ o = n o ε o + p o c . For the linearized energy balan e equation we get: n o c v ∂ ( δT ) ∂t + n o kT o δθ = 0 (16)3ere, the heat apa ity (per parti le) is given by c v = (cid:18) ∂ε o ∂T (cid:19) n (17)In order to de ouple the system and establish a partial di(cid:27)erential equation for δT , we (cid:28)rst solvefor δθ in both sides of Eqs. (14) and (16). Equating the results we obtain the useful relation: n o (cid:18) ∂∂t δn (cid:19) − c v kT o (cid:18) ∂∂t δT (cid:19) = 0 (18)We derive with respe t to time in both sides of Eq. (15) : ˜ ρ o ∂ ∂t δθ + n o k ∇ ∂∂t ( δT ) + kT o ∇ ∂∂t ( δn ) = 0 (19)so that, inserting the expression for ∂∂t δn from equation (18) we obtain: ˜ ρ o ∂ ∂t δθ + n o k ∇ ∂∂t ( δT ) + kT o ∇ (cid:18) n o c v kT o (cid:19) ∂∂t ( δT ) = 0 (20)One (cid:28)rst time derivative is immaterial in ea h term of Eq. (20), so that after an elementaryarrangement of terms we an write the wave equation for thermal (cid:29)u tuations: ∇ ( δT ) − ˜ ρ o c v kn o T o ( c v + k ) ∂ ∂t ( δT ) = 0 (21)Thus, the propagation speed ( C s ) of a thermal wave in a relativisti Euler (cid:29)uid is: C s = n o T o ( c v + k ) k ˜ ρ o c v (22)In the non-relativisti limit, z → , c v → k and ˜ ρ o → n o m , so that (22) yields C s → zc (23)whi h is the non-relativisti speed of sound. Also, the relativisti propagation speed (22) an berewritten in terms of z as: C s = ( c v + k ) zc v (cid:16) z + K (1 /z ) K (1 /z ) (cid:17) c (24)It is interesting to noti e that some authors perform a similar analysis for δn negle ting tem-perature (cid:29)u tuations, and only taking into a ount Eqs. (6) and (8), in order to establish a waveequation for density (cid:29)u tuations [11℄. In that ase it is immediate to (cid:28)nd out that the orrespondingpropagation speed is: C T = z (cid:16) z + K (1 /z ) K (1 /z ) (cid:17) c (25)4igure 1: Comparison of (cid:29)u tuation propagation speeds for the full relativisti ase(solid),non-relativisti ase (long dashed), δn (cid:29)u tuations negle ting thermal (cid:29)u tuations(short dashed) and δT (cid:29)u tuations negle ting number density (cid:29)u tuations (dotted). H C x (cid:144) c L In the same order of ideas, one an make a simple analysis negle ting the number density(cid:29)u tuations and taking into a ount only Eqs. (8) and (16). In this ase, the expression for a waveequation for thermal (cid:29)u tuations reads: C n = z c v k (cid:16) z + K (1 /z ) K (1 /z ) (cid:17) c (26)Figure 1 shows a omparison of the hara teristi speeds for in reasing z .4 Final remarksIt has re ently been proved the nonexisten e of generi instabilities in the linearized transportequations at the Navier-Stokes regime [9℄. In this paper it is shown that, in the Euler regime, thereis no ausality problem. The linearized transport equations be ome a hyperboli system and, forfurther resear h, it an be taken as a starting point for a simpli(cid:28)ed al ulation and for validationof numeri al work in the non-linear ase.The non-relativisti limit has been re overed, as expe ted, and thermal (cid:29)u tuations also satisfya hyperboli partial di(cid:27)erential equation. In most textbooks, the establishment of the (paraboli )heat equation is based on an extension of Eq.(12) in luding heat ondu tion, negle ting velo ity(cid:29)u tuations. On the other hand, if the linearized equation of motion (15) is taken as the basis of thedes ription of thermal (cid:29)u tuations, then a ausal equation is obtained for the non-dissipative (cid:29)uid.Thus, for the dissipative ase it is suggested that the suitable generalization of the whole linearizedsystem (6,8,12) should be taken into a ount, emphasizing the role of Eq.(8) when analyzing ausalproperties of the system. Negle ting velo ity (cid:29)u tuations learly leads to non- ausality. It anbe noted, also, that density (cid:29)u tuations (negle ting the thermal ones) and thermal (cid:29)u tuations(negle ting the density ones) present di(cid:27)erent propagation speeds, satisfying the relation C n + C T = C s . Moreover, taking δθ = 0= 0