Investigating the domain of validity of the Gubser solution to the Boltzmann equation
IInvestigating the domain of validity of the Gubser solution to theBoltzmann equation
Ulrich Heinz and Mauricio Martinez
Department of Physics, The Ohio State University,Columbus, OH 43210 United States (Dated: August 15, 2018)
Abstract
We study the evolution of the one particle distribution function that solves exactly the relativisticBoltzmann equation within the relaxation time approximation for a conformal system undergoingsimultaneously azimuthally symmetric transverse and boost-invariant longitudinal expansion. Weshow, for arbitrary values of the shear viscosity to entropy density ratio, that the distributionfunction can become negative in certain kinematic regions of the available phase space dependingon the boundary conditions. For thermal equilibrium initial conditions, we determine numericallythe physical boundary in phase space where the distribution function is always positive definite.The requirement of positivity of this particular exact solution restricts its domain of validity, andit imposes physical constraints on its applicability.
PACS numbers: 51.10.+y, 52.27.Ny,05.20.Dd,47.45.AbKeywords: Boltzmann Equation, Statistical Mechanics, Relativistic Kinetic Theory, Relativistic transport a r X i v : . [ h e p - ph ] S e p . INTRODUCTION The dynamics and transport properties of rarefied gases and fluids are usually describedin terms of the Boltzmann equation. The Boltzmann equation is a partial integro-differentialequation for the distribution function f ( x, p ). In general this equation is solved numerically,and very few exact solutions are known in the literature. Nevertheless, for highly symmetricsystems it is possible to solve this equation analytically under certain approximations for thecollisional kernel [1]. Using a relaxation time approximation (RTA) for the collisional kernel,the relativistic Boltzmann equation has been solved exactly for systems undergoing Bjorkenflow [2] and Gubser flow [3, 4]. The exact solution for the Bjorken flow has been usefulto understand aspects of the isotropization and thermalization problem of a plasma formedby quarks and gluons (QGP) [2–10]. In contrast to Bjorken flow [11], where the systemexpands in boost invariant fashion only along one direction (the “longitudinal” direction),Gubser flow [12, 13] describes systems that undergo additionally simultaneous azymuthallysymmetric expansion in the transverse directions. The solution of the Boltzmann equationfor the Gubser flow [3, 4] was found by exploiting the SO (3) ⊗ SO (1 , ⊗ Z symmetry ofthe flow velocity profile [12, 13] which becomes manifest when mapping Minkowski space, R ⊗ R , conformally onto de Sitter space times a line, dS ⊗ R [12, 13]. In Refs. [3, 4] it wasnoticed that the resulting solutions for moments of the distribution function, such as theenergy density or temperature, can become complex and therefore physically meaninglesswhen propagating backwards in the de Sitter time (see discussion in Appendix B of Ref. [4]).In this work we revisit this issue and find that the unphysical behavior of the moments ofthe distribution function found in Ref. [4] is rooted in a violation of the positivity of thedistribution function in some regions of phase space when propagating the solution of theBoltzmann equation for equilibrium initial conditions backward in the de Sitter time. InMinkowski coordinates, this translates to the distribution function becoming negative incertain momentum ranges at the outer edge of the spatial density profile at fixed time.This work is organized as follows: in Sect. II we review the procedure used in Refs. [3, 4]to find the exact solution of the Boltzmann equation [3, 4]. In Sect. III we show numeri-cal results for the phase space evolution of the distribution function. Our conclusions arepresented in Sect. IV. 2 I. THE ANALYTICAL SOLUTION OF THE RTA BOLTZMANN EQUATIONFOR THE GUBSER EXPANSION
In this section we briefly review the derivation of the exact solution to the RTA Boltzmannequation that is invariant under the group of symmetries of the Gubser flow, i.e., under the SO (3) q ⊗ SO (1 , ⊗ Z group (“Gubser group”) [12, 13]. For additional technical details ofthe method discussed here we refer the reader to Ref. [4].We use the following notation. The metric signature is taken to be the “mostly plus”convention. In Minkowski space the line element is written in Milne coordinates x µ =( τ, r, φ, ς ) as ds = g µν dx µ dx ν = − dτ + dr + r dφ + τ dς . (1)where the longitudinal proper time τ , the spacetime rapidity ς , the transverse radius r andthe azymuthal angle φ are defined in terms of the usual cartesian coordinates ( t, x, y, z ) as τ = √ t − z , ς = tanh − (cid:16) zt (cid:17) r = (cid:112) x + y , φ = arctan (cid:16) yx (cid:17) . (2)The flow velocity u µ is normalized as u µ u µ = − A. The Gubser flow
The dynamics of an expanding conformal fluid in Minkowski space can be understoodin terms of a static conformal fluid defined in a particular curved space. In Minkowskispace, the Gubser flow describes a system which expands azymuthally symmetrically in thetransverse plane and at the same time in a boost invariant manner along the longitudinaldirection. After applying a conformal map between Minkowski space and the de Sitter spacetimes a line, dS ⊗ R , the Gubser fluid velocity becomes static in this curved space. Thus,it is easiest to first study the dynamics in de Sitter space, and then map the solution backto Minkowski space [12, 13].In the de Sitter space, the Gubser symmetry can be made manifest by a suitable choice ofcoordinates. One relates the Milne coordinates x µ = ( τ, r, φ, ς ) defined in Minkowski spaceto a coordinate system ˆ x µ = ( ρ, θ, φ, ς ) in dS ⊗ R as follows [13]: One first performs a Weyl3escaling of the metric: d ˆ s = ds τ = − dτ + dr + r dφ τ + dς . (3)Next, we make the following change of variables: ρ (˜ τ , ˜ r ) = − arcsinh (cid:18) − ˜ τ + ˜ r τ (cid:19) , ρ ∈ ( −∞ , ∞ ) , (4a) θ (˜ τ , ˜ r ) = arctan (cid:18) r τ − ˜ r (cid:19) , θ ∈ (0 , π ) , (4b)where ˜ τ = qτ and ˜ r = qr with q being an energy scale which sets the typical transverse sizeof the system [12, 13]. Substituting Eq. (4) into Eq. (3) one finds d ˆ s = − dρ + cosh ρ (cid:0) dθ + sin θ dφ (cid:1) + dς . (5)In these coordinates the metric in the curved dS ⊗ R space is then given by ˆ g µν =diag( − , cosh ρ, cosh ρ sin θ, ρ is a time-like variable.The Gubser flow is defined in dS ⊗ R as the unit vector ˆ u µ = (1 , , , x µ in dS ⊗ R to the coordinates x µ in R ⊗ R . This involves a Weyl rescaling ofthe fluid velocity components [12, 13]. One finds that in Milne coordinates the fluid velocityin Minkowski space is u µ = (cosh κ (˜ τ , ˜ r ) , sinh κ (˜ τ , ˜ r ) , , κ (˜ τ , ˜ r ) = tanh − (cid:18) τ ˜ r τ + ˜ r (cid:19) . (6)This gives rise to the radial velocity profile v (˜ τ , ˜ r ) = tanh κ (˜ τ , ˜ r ) = 2˜ τ ˜ r τ + ˜ r . (7) B. The exact solution of the RTA Boltzmann equation
The invariance of a system under a particular group of transformations imposes con-straints on the number of independent variables of the distribution function f ( x µ , p i ). Forthe Gubser group one finds that f (ˆ x µ ; ˆ p i ) = f ( ρ ; ˆ p , ˆ p ς ) [3, 4] where ˆ p = ˆ p θ + ˆ p φ / sin θ andˆ p θ , ˆ p φ and ˆ p ς are the momenta conjugate to the coordinates θ , φ and ς in Eq. (5). Thus,4he relativistic RTA Boltzmann equation in de Sitter space reduces to a one-dimensionalordinary differential equation in de Sitter space [3, 4], ∂∂ρ f ( ρ ; ˆ p , ˆ p ς ) = − ˆ T ( ρ ) c (cid:34) f ( ρ ; ˆ p , ˆ p ς ) − f eq (cid:32) ˆ p ρ ˆ T ( ρ ) (cid:33)(cid:35) , (8)where ˆ p ρ = ˆ p ρ ( ρ, ˆ p Ω , ˆ p ς ) = (cid:113) ˆ p / cosh ρ + ˆ p ς , f eq ( x ) = e − x is the Boltzmann equilibriumdistribution function, and ˆ T = c/ ˆ τ rel ( ρ ) and ˆ τ rel ( ρ ) = τ rel ( τ ) /τ are the Weyl-rescaled (unit-less) temperatures and local relaxation time. The constant c is related to the specific shearviscosity of the system by c = 5 η/ S [5, 6, 14, 15] where η is the shear viscosity and S is theentropy density. The formal solution of this equation is [1, 2, 16] f ( ρ ; ˆ p , ˆ p ς ) = D ( ρ, ρ ) f ( ρ ; ˆ p , ˆ p ς ) + 1 c (cid:90) ρρ dρ (cid:48) D ( ρ, ρ (cid:48) ) ˆ T ( ρ (cid:48) ) f eq (ˆ p ρ / ˆ T ( ρ )) , (9)where D ( ρ, ρ ) = exp (cid:104) − (cid:82) ρρ dρ (cid:48) ˆ T ( ρ (cid:48) ) /c (cid:105) is the damping function, and f ( ρ ; ˆ p , ˆ p ς ) is theinitial distribution function at ρ which we choose to be the Boltzmann equilibrium distri-bution, f ( ρ , ˆ p , ˆ p ς ) = f eq (cid:16) ˆ p ρ ( ρ ) / ˆ T ( ρ ) (cid:17) .The temperature ˆ T is related to the solution for f by the dynamical Landau matchingcondition, i.e., by the requirement that ˆ ε ( ρ ) = ˆ ε eq ( ρ ) ∼ ˆ T ( ρ ) where ˆ ε ( ρ ) is the energydensity computed from the non-equilibrium distribution function f ( ρ, ˆ p ) , ˆ p ς . This matchingcondition allows to rewrite Eq. (9) as [3, 4]ˆ T ( ρ ) = D ( ρ, ρ ) H (cid:18) cosh ρ cosh ρ (cid:19) ˆ T ( ρ ) + 1 c (cid:90) ρρ dρ (cid:48) D ( ρ, ρ (cid:48) ) H (cid:18) cosh ρ (cid:48) cosh ρ (cid:19) ˆ T ( ρ (cid:48) ) , (10)where H ( x ) = 12 (cid:32) x + x tanh − (cid:0) √ − x (cid:1) √ − x (cid:33) . (11)The numerical solution of (10) is uniquely determined by the initial de Sitter time ρ , theinitial value ˆ T = ˆ T ( ρ ), and (through c ) the chosen value of η/ S . In order to solve for f inEq. (9), we first find the temperature ˆ T ( ρ ) by iteratively solving the integral equation (10)as in [5, 6], and then plug this ˆ T ( ρ ) into Eq. (9) for every de Sitter time-step ρ , and performthe ρ (cid:48) integral on the r.h.s. Since Eq. (9) is diagonal in the momentum variables ˆ p Ω , ˆ p ς ,the evolution of f with ρ can be studied separately for each point in momentum-space. InRef. [3, 4] we showed that all macroscopic moments of the distribution function f , (i.e., allthe components of the energy-momentum tensor ˆ T µν ) can be directly obtained as ρ (cid:48) -integrals5ver the solution ˆ T ( ρ ) from Eq. (10), with different weight functions, without even studyingthe distribution function f in Eq. (9) itself.In Ref. [4] (App.B) we observed for a few specific choices of ρ that the solution ofEq. (10) leads to complex-valued temperatures at large enough negative values of ρ − ρ .This observation was generic and held for any value of η/ S . The problem appeared toarise only for ρ < ρ , i.e., only in the de Sitter past; for ρ > ρ the solution for ˆ T ( ρ ) wasalways real. Obviously, a complex-valued temperature is physically meaningless. In practicethis issue can be resolved by imposing boundary conditions in the infinitely distant pastof the de Sitter time, i.e. for ρ → −∞ [4]. It was argued in Ref. [4] that the complexvalues of the temperature might be related with the violation of the positivity condition ofthe distribution function in some regions of phase space. Clearly, if this condition is notsatisfied, the probabilistic meaning of the distribution function is lost.In this work we investigate the evolution of the distribution function (9) in phase spacefor arbitrary values of ρ , ˆ T and η/ S . Results of our studies are presented in the followingsection. Before going into the discussion of our numerical results, though, we make thefollowing analytical observation: Expanding the solution (9) around ρ for small ρ − ρ it isstraightforward to see, for any choice of (ˆ p , ˆ p ς ), that f ( ρ, ˆ p , ˆ p ς ) increases for ρ > ρ anddecreases for ρ < ρ . The rate of increase/decrease depends on (ˆ p , ˆ p ς ). From this we seethat, as the system evolves forward in de Sitter time ρ , f will remain positive if it is so at ρ = ρ , but that f will decrease, and can for any choice of (ˆ p , ˆ p ς ) turn negative, as wefollow the system backward in ρ to sufficiently large negative values of ρ − ρ . III. RESULTS
In Fig. 1 and 2 we show contour plots of the distribution function f ( ρ, ˆ p , ˆ p ς ) as a functionof ˆ p Ω and ˆ p ς for fixed values of the de Sitter time ρ , for the specific parameter choices4 πη/ S = 3 and ˆ T = 1. In Fig. 1 we impose equilibrium boundary conditions at ˆ ρ = 0 andthen study f at ρ = − ρ = − ρ , imposing the same initial conditions at ρ = − f at ρ = − ρ = − ρ . While in both figures we observe that thedistribution function becomes negative at large values of ˆ p Ω and small values of ˆ p ς , and that6 IG. 1: (Color online) Snapshots of the two-dimensional slice of the ˆ p Ω and ˆ p ς evolution of thephase space distribution f ( ρ, ˆ p Ω , ˆ p ς ) as a function of ˆ p Ω and ˆ p ς for fixed values of ρ = − ρ = −
6. In both figures we consider ρ = 0, 4 πη/ S = 3 and ˆ T = 1. this problem becomes more severe as | ρ − ρ | increases, i.e. as ρ becomes more negative, theproblem is clearly, for the same value of | ρ − ρ | , more serious for smaller initial values ρ .The fact that the distribution function goes negative in some region of phase space isindependent of the value of η/S . This is seen in Fig. 3 where we show f = const. contoursin the ˆ p Ω − ˆ p ς plane for two different values of η/ S , namely 4 πη/ S = 3 (left panel) and4 πη/ S = 10 (right panel). In both panels we imposed equilibrium initial conditions withˆ T = 1 at ρ = 0 and plotted f at ρ = −
6. For fixed | ρ − ρ | , as η/ S increases the line f ( ρ, ˆ p , ˆ p ς ) = 0 separating physical ( f >
0) from unphysical behavior ( f <
0) is seen to movecloser to the ˆ p ς = 0 axis and towards larger ˆ p Ω values. In other words, for larger specificshear viscosity, the ρ -evolution that eventually drives f negative proceeds more slowly.7 IG. 2: (Color online) Snapshots of the two-dimensional slice of the ˆ p Ω and ˆ p ς evolution of thephase space distribution f ( ρ, ˆ p Ω , ˆ p ς ) as a function of ˆ p Ω and ˆ p ς for fixed values of ρ = − ρ = −
8. In both figures we consider ρ = −
2, 4 πη/ S = 3 and ˆ T = 1. While the positivity condition of the phase space distribution is violated for any valueof the temperature ˆ T , the speed with which this happens as ρ evolves backwards from ρ depends strongly on the choice of ˆ T even if η/ S is held constant. This is perhapsnot unexpected since ˆ T also controls the scattering rate (due to the conformal relationˆ T ˆ τ =const.). In Fig. 4 we present contours of the distribution function in the ˆ p Ω − ˆ p ς planefor two additional values of the initial temperature, ˆ T = 2 (left panel) and ˆ T = 3 (rightpanel) (in addition to the case ˆ T = 1 shown in Fig. 3), all for the same value 4 πη/ S = 3.As in Fig. 3, equilibrium initial conditions are implemented at ρ = 0, and f is plotted for ρ = −
6. As ˆ T is increased, the line f = 0 separating the physical from the unphysical regionmoves to smaller ˆ p Ω and larger ˆ p ς values, i.e. the unphysical region grows. Increasing the8 IG. 3: (Color online) Contour plots of f ( ρ, ˆ p , ˆ p ς ) for fixed values ρ = 0 and ρ = − π ) η/ S = 3 (left panel) and (4 π ) η/ S = 10 (right panel). In both panels ˆ T = 1.FIG. 4: (Color online) Contour plots of f ( ρ, ˆ p , ˆ p ς ) for a fixed values of ρ = 0 and ρ = − T = 2 (left panel) and ˆ T = 3 (right panel). In both panels (4 π ) η/ S = 3. initial temperature and energy density obviously speeds up the evolution towards unphysicalbehavior as de Sitter times evolves backward.In Fig. 5 we show this ρ evolution by plotting the f = 0 surface separating the physicalfrom the unphysical regions in the 3-dimensional ρ − ˆ p Ω − ˆ p ς space, for initial conditionsimposed at ρ = 0 (left panel) and ρ = −
1, respectively. In this figure we do not show theregion where ˆ p ς < p ς variable by construction [3, 4]. Clearly the unphysical region only appears for ρ < ρ , first9 IG. 5: (Color online) 3D surface defined by f ( ρ, ˆ p Ω , ˆ p ς ) = 0 for two different initial conditions ρ = 0 (left panel) and ρ = − π ) η/ S = 3 and ˆ T = 1.The gray lines drawn over the physical boundary condition f = 0 correspond to constant values of ρ − ˆ p Ω − ˆ p ς over the surface. at large ˆ p Ω and later (i.e. for more negative values of ρ − ρ ), also for smaller ˆ p Ω values,and it is largest when ˆ p ς is small. As the initial time ρ is decreased, the unphysical regionappears more quickly as a function of ρ , but covers a smaller region in ˆ p Ω and ˆ p ς .The non-positive behavior of f ( ρ, ˆ p Ω , ˆ p ς ) in the de Sitter space imposes limitations ofthe validity of this particular solution when the information is mapped back to Minkowskispace. For massless particles the distribution function is a Lorentz and Weyl scalar [4], soone only needs to transform the phase space coordinates ( ρ, ˆ p Ω , ˆ p ς ) to the correspondingones in Minkowski space ( ρ ( τ, r ) , p ( τ, r, p τ , p r ) , p ς ). When transforming the momentumcoordinates ˆ p µ in de Sitter to the corresponding ones in Minkowski space one must performthe associated Weyl rescaling (ˆ p µ = τ p µ ) together with the covariant transformation of the10omentum components ˆ p ρ ˆ p θ ˆ p φ ˆ p φ = ∂ρ∂τ ∂ρ∂r ∂θ∂τ ∂θ∂r τ p τ τ p r τ p φ τ p ς . (12)By using the coordinate transformations (4) we obtain explicitlyˆ p ρ = τ γ ( p τ − v ( τ, r ) p r ) , (13a)ˆ p θ = τ cosh ρ ( τ, r ) γ ( p r − v ( τ, r ) p τ ) , (13b)ˆ p φ = τ p φ , (13c)ˆ p ς = τ p ς = p ς , (13d)where v ( τ, r ) is the radial Gubser flow velocity in Minkowski space, given in Eq. (7), and γ = 1 / √ − v . Eqs. (13a) and (13b) show that up to a Jacobian factor, related to theWeyl-rescaling, (ˆ p ρ , ˆ p θ ) are related with the Minkowski-space components ( p τ , p r ) by a radialboost with velocity v ( τ, r ). Eqs. (13) are inverted by p τ = γτ (cid:0) ˆ p ρ + v ( τ, r ) cosh ρ ˆ p θ (cid:1) , (14a) p r = γτ (cid:0) cosh ρ ˆ p θ + v ( τ, r ) ˆ p ρ (cid:1) , (14b) p φ = r p φ = r τ ˆ p φ , (14c) p ς = ˆ p ς , (14d)Thus, we can understand the momentum p µ in the lab frame in terms of the momentum inde Sitter space boosted by radial flow.In order to develop some intuition what the break-down of the solution for the distributionfunction in de Sitter space implies in Minkowski space we make the following considerations: • The scale of energy q is determined by the transverse size R ⊥ of the nucleus; we assume R ⊥ = 5 fm and thus set q = 1 /R ⊥ = 0.04 GeV. • We study the system in Minkowski space at a fixed longitudinal proper time τ forwhich we take ˜ τ = qτ = 0 .
3, i.e. τ = 1.5 fm/ c .11 The rotational invariance of the Gubser flow about the longitudinal axis allows us tochoose ˆ p φ = 0. Moreover, this lets us identify the radial component of the momentumin Milne coordinates with the transverse momentum of the particle. • ˆ p ς = p ς is related to the longitudinal component of the momentum in the boostedframe, i.e. p ς = p z t − E z ≡ w . It is an invariant under longitudinal boosts, i.e. underthe subgroup SO (1 ,
1) of the Gubser group [3, 4]. Thus we can evaluate p ς at z = 0where t and τ coincide and thus p ς = τ p z . • For z = 0 and ˆ p φ = 0 we can write the SO (3) q invariant ˆ p asˆ p = ˆ p θ + ˆ p φ sin θ , = (cid:0) ˆ p θ cosh ρ ( τ, r ) (cid:1) = cosh ρ ( τ, r ) τ [ γ ( p T − v ( τ, r ) p τ )] , (15)where p τ = (cid:112) p T + p z and p T = p r . • The Minkowski-space temperature T ( τ, r ) and its de Sitter analog ˆ T ( ρ ) are related byWeyl rescaling as T ( τ, r ) = ˆ T ( ρ ( τ, r )) τ . (16)For τ = 1 . c and r = 0, Eq. (4a) gives us that ρ ≈ − .
2. When using for theinitial condition of the temperature in de Sitter space the natural scale ˆ T ( ρ ) = 1, weobtain from the numerical solution of Eq. (10) that ˆ T ( ρ = − .
2) = 0 .
22. Thus thecorresponding central temperature T ( τ = 1 . c , r = 0) = 0 .
029 GeV.In order to visualize the distribution function in Minkowski space, we write f ( τ, r, z = 0 , p T , p z ) = f (cid:0) ρ ( τ, r ) , ˆ p ( τ, r, p T , p z ) , ˆ p ς ( τ, p z ) (cid:1) . (17) In polar coordinates the momentum component p φ is p φ = p T r sin( φ − φ p ) . where p T is the transverse momentum of the particle and φ p is the angle between the two-dimensionaltransverse momentum vector p T and the horizontal axis in the transverse plane. Therefore fixing p φ = 0is equivalent to saying that the radial unit vector in the transverse plane is aligned with p T , i.e., φ = φ p .It is straightforward to conclude that in this case p r = p T . τ = 1 . c , and z = p φ = 0we find r = 5 (cid:112) − . ρ − .
91 fm , (18a) p z = 0 .
13 ˆ p ς GeV , (18b) p T = . γ (cid:18) ˆ p Ω cosh ρ + v (cid:113) ˆ p cosh ρ + ˆ p ς (cid:19) GeV if p T > γv | p z | , . γ (cid:18) − ˆ p Ω cosh ρ + v (cid:113) ˆ p cosh ρ + ˆ p ς (cid:19) GeV if p T < γv | p z | (18c)where v = √− . ρ − . . − sinh ρ ,γ = 0 . − sinh ρ cosh ρ (19)Note that r > ρ < τ .In order to reconstruct the physical boundary in the r − p T − p z phase space, we firstdetermine numerically the set of coordinates ( ρ, ˆ p Ω , ˆ p ς ) where f = 0 for thermal initialconditions at ρ = 0, using (4 π ) η/S = 3 from Fig. 5 (left panel). Once this informationis determined in de Sitter space, it is simply mapped to Minkowski space by evaluatingEqs. (18) for each selected data point ( ρ, ˆ p Ω , ˆ p ς ). Note that the range of ( ρ, ˆ p Ω , ˆ p ς ) covered inFig. 5 limits the range in ( r, p T , p z ) that we can access in this way. To extend the ( r, p T , p z )range requires extending the numerical results shown in Fig. 5 to a larger range in ( ρ, ˆ p , ˆ p ς )which is numerically costly. For this reason we show in Fig. 6 only a rather limited sectionof the surface in ( r, p T , p z ) space that separates regions of positive and negative values forthe distribution function f .In Fig. 6, the region where f < f > p T from (18c) we need to check the signof p T − γv | p z | . All points shown in Fig. 6 correspond to p T > γvp z , i.e. to the upper signin Eq. (18c). We checked numerically that the opposite sign applies in regions of large r ,beyond the range shown in Fig. 6.From Fig. 6, we observe that in the shown range r ∈ [2 fm ,
10 fm] the distributionfunction becomes negative for sufficiently large values of p T and sufficiently small values of p z . As we move out to larger r values (i.e. towards the tail of the density distribution)the unphysical region moves towards smaller p T values and thus covers a larger fraction of13 IG. 6: (Color online) 3D surface defined by f ( r, p T , p z ) = 0 for (4 π ) η/ S = 3 at τ = 1 . q = 0 .
04 GeV. The gray lines drawn over the physical boundary condition f = 0 correspond toconstant values of r − p T − p z over the surface. See text for further details regarding the initialconditions. momentum space. For larger r , the solution with thermal boundary conditions at ρ = 0thus remains physical only for very small p T values combined with sufficiently large | p z | . Atfixed r , the critical surface separating f > f < p T and p z andapproximately given by | p crit z | = α ( r )( p T − p T ( r )) where p T ( r ) is the line where the criticalsurface intersects the p z = 0 plane. We find α (2 fm) ≈ α (10 fm) ≈ τ and all the other chosen parameters ( ˆ T ≡ ˆ T ( ρ ), η/ S and q ). For p z > | p crit z | the solution is physical (i.e. f >
0) while for | p z | < | p crit z | it isunphysical ( f < IV. CONCLUSIONS
In this work we have studied the dynamics in phase space of the recently found exactsolution to the conformal RTA Boltzmann equation which undergoes Gubser expansion [3, 4].We determined the distribution function by first solving equation (10) for the temperatureˆ T ( ρ ) and then evaluating Eq. (9) point by point as a function of the variables ( ρ, ˆ p Ω , ˆ p ς ) that14efine the phase space of the system in de Sitter space. We assumed thermal equilibriumboundary conditions at de Sitter time ρ .We observe that this exact solution becomes negative, and therefore physically meaning-less, for sufficiently large negative values for ρ − ρ . This proves the conjecture in [4] thatthe unphysical behaviour of the moments of the distribution function is likely related withthe violation of positivity of the distribution function in certain phase space regions.The non-physical behaviour of the distribution function depends strongly on the initialvalue ρ . If the evolution of f ( ρ, ˆ p Ω , ˆ p ς ) starts at ρ = −∞ the system is always evolving inthe forward direction in the de Sitter time ρ and the distribution function remains alwayspositive. When imposing thermal equilibrium initial conditions at some finite value ρ , thesystem can evolve not only forward but also backward in de Sitter time ρ . In this case thedistribution function f ( ρ, ˆ p Ω , ˆ p ς ) is always positive definite for ρ − ρ ≥ ρ − ρ < p ς andlarge ˆ p Ω , qualitatively independent of the value of the shear viscosity over entropy densityratio η/ S and the initial temperature ˆ T . The generic shape of the surface separating thephysical ( f >
0) from the unphysical region is shown in de Sitter coordinates in Fig. 5 andin Minkowski coordinates in Fig. 6. In Minkowski space, the exact solution for f becomesunphysical at large r , large p T and small p z ; this corresponds to large negative values of ρ − ρ , large ˆ p Ω and small ˆ p ς . The example studied in Sect. III shows that when choosingvalues for the parameters q , ˆ T and η/ S that are natural for heavy ion collisions, problemsof non-positivity of f arise already at moderately small values of r and p T when p z = 0.This renders the analytical solution found in [3, 4], which assumes local thermal equilibriumon a surface ρ , unsuitable for heavy-ion phenomenology.This does not mean, however, that this exact solution cannot be used to test differenthydrodynamic approximation methods, as done in Refs. [3, 4]. There is no problem withsuch tests as long as the comparison is performed (either in de Sitter or Minkowski space)in the physically allowed region where f >
0. An alternative approach which guaranteesalways the positivity of the exact solution is to fix the initial condition at sufficiently largenegative ρ values such that the region ρ > ρ includes all of the interesting range in τ and r covered by the evolution of a heavy-ion collision. In this case, with a suitable choice forthe initial form f ( ρ , ˆ p Ω , ˆ p ς ) that is presumably not of equilibrium form, the exact solution15f Eq. (9) might become relevant for heavy-ion phenomenology. Acknowledgments
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