Principles of Hydrodynamics Fluid/Gravity Correspondence The Entropy Current Outlook. The Entropy Current. Yaron Oz (Tel-Aviv University)

Transcription

1 The Entropy Current Yaron Oz (Tel-Aviv University) International Symposium Ahrenshoop on the Theory of Elementary Particles 2012

2 Outline Principles of Hydrodynamics Fluid/Gravity Correspondence The Entropy Current Outlook

3 Fluid Dynamics and Gravity The Fluid/Gravity correspondence relates fluid dynamics to black hole dynamics.

4 Fluid Dynamics and Gravity The essential ingredients needed to relate fluids to gravity are the existence of a horizon in the gravitational background that is related to a thermal equilibrium state in the field theory, and a derivative expansion around it. One can think about the fluid/gravity correspondence as an extension of black hole thermodynamics, where charges are upgraded into local currents, and the black hole entropy into a local entropy current.

5 The Horizon Geometry Hydrodynamics is the deformation of the black hole geometry U (x). T(x)

6 Experiment: Heavy-Ion Collisions The QCD plasma produced at RHIC and LHC seems to exhibit a strong coupling relativistic fluid dynamics α s (T RHIC ) O(1).

7 Luzum, Romatschke: Elliptic Flow

8 Superfluid Hydrodynamics - CFL Phase

9 Non-Relativistic Navier-Stokes Equations The limit of non-relativistic macroscopic motions v c leads to the non-relativistic incompressible Navier-Stokes equations (Fouxon,Y.O.; Bhattacharyya, Minwalla, Wadia). t v i + v j j v i = i P + ν jj v i + f i (1) v i (x, t), i = 1,..., d, (d 2) obeying i v i = 0 is the velocity vector field, P(x, t) is the fluid pressure divided by the density, ν is the (kinematic) viscosity and f i (x, t) are the components of an externally applied (random) force. Black hole dynamics offers a new geometrical framework to study at the two main fundamental problems: Turbulence and Singularities.

10 Turbulence

11 Turbulence Anomalous scaling: ( S n (r) (v(x) v(y)) r ) n r ξ n (2) r r x y. Anomalous Exponents n/3 1 ξ n Figure 1: Anomalous exponents versus the linear Kolomogorov scaling in three space dimensions. The plot is a schematic draw of numerical simulations and experimental data. n

12 Basic Hydrodynamics The effective degrees of freedom are charge densities ρ( x, t), and the hydrodynamics equations are conservation laws t ρ + i j i = 0 (3) The constitutive relations express j i in terms of ρ and its derivatives. For instance j i = D i ρ (Fick s law: 1855), and we get t ρ D i i ρ = 0 (4) Writing ρ( k, t) = d 3 xe i k xρ( x,t) we have ρ( k, t) = e Dk 2t ρ( k, t = 0) (5) This is the characteristic behaviour of hydrodynamic mode. It has a life τ(k) = 1 Dk 2 which is infinite in the limit k 0.

13 The Hydrodynamic Modes The dispersion relation of the mode ω + idk 2 = 0 (6) It appears as a pole in the retarded thermal correlation function (Kadanoff and Martin:1965). S( k,ω) = ρ( k,ω)ρ(0, 0) T (7) In gravity these are the quasi-normal modes of the black hole.

14 Relativistic Hydrodynamics Hydrodynamics applies under the condition that the correlation length of the fluid l cor (l mfp ) is much smaller than the characteristic scale L of variations of the macroscopic fields Kn l cor /L 1 (8) Hydrodynamics equations are conservation laws µ T µν = 0, µ J µ a = 0 (9) The constitutive relation has the form of a series (derivative expansion) in the small parameter Kn 1, T µν (x) = l=0 T µν µν (l) (x), T (l) (Kn)l (10)

15 Relativistic Hydrodynamics The constitutive relation express T µν and J µ a in terms of the energy density ǫ(x), the pressure p(x), the charge densities ρ a (x) and the d-velocity field u µ (x) satisfying u µ u µ = 1. The other local fields can be determined by using the equation of state and the thermodynamic relations dǫ = Tds + µ a dρ a dp = sdt + ρ a dµ a (11) One introduces phenomenologically a local entropy current s µ, whose divergence is required to be non-negative order by order in the derivative expansion µ s µ 0 (12) This leads to non-trivial conditions on the transport coefficients.

16 Ideal Hydrodynamics Keeping only the first term l = 0 in the series gives ideal hydrodynamics, which is essentially determined by thermodynamics. One has where P µν = η µν + u µ u ν. T µν (0) = ǫu µ u ν + pp µν J µ a(0) = ρ a u µ s µ (0) = su µ (13) There is one less conservation equation than the number of fields ǫ, p,ρ a, u µ, and one more equation is needed in order to have a complete description of the system. This is called the equation of state and it allows to choose a set of independent variables. e.g. ǫ,ρ a, u µ.

17 Ideal CFT Hydrodybnamics In CFT hydrodynamics one has T µ µ = 0 and the equation of state reads ǫ = 3p. In this case, there is only one dimensionfull quantity, the temperature T, and ǫ, p T 4. The ideal conformal fluid stress-energy tensor can be recast in the form T µν (0) = T 4 [η µν + 4u µ u ν ] (14) up to and overall constant coefficient.

18 The Entropy Current At the ideal hydrodynamics order there is no dissipation and the hydrodynamic conservation laws imply that µ s µ (0) = 0. The entropy current in the conformal case s µ = su µ T d 1 u µ (15)

19 Viscous Hydrodynamics The first order dissipative hydrodynamics is obtained by keeping also the l = 1 term in the series. Going out of equilibrium, there is an ambiguity in the definition of the fields, which requires a choice of a frame. In one such frame, the Landau frame, the fluid velocity is an eigenvector of the stress energy tensor, and the energy density is its eigenvalue T ν µ u ν = ǫu µ. In this frame, the local rest frame of the flow is where the energy density is at rest. The charge density is ρ a = u µ J µ a. The first order stress energy tensor and the symmetry currents satisfy u µ T µν (1) = 0, u µj µ a(1) = 0 (16)

20 Viscous Hydrodynamics We have where T µν (1) = ησ µν ζ( α u α )P µν J µ a(1) = Tσ ab P µν ν µ b T (17) σ µν = P µα P νβ (α u β) 1 3 αu α P µν (18) is the shear tensor.

21 Viscous Hydrodynamics There are three types of transport coefficients at this order, the shear viscosity η and the bulk viscosity ζ and the conductivity matrix σ ab. In the conformal case ζ = 0. The transport coefficients can be calculated form the retarded Green functions of the microscopic thermal field theory using linear response theory and the Kubo formula.

22 The Entropy Current The entropy current at this order reads s µ (1) = µa T Jµ a(1) (19) It is no longer conserved at the first viscous order and we have µ s µ = η T σ µνσ µν + ζ T ( µu µ ) 2 + Tσ ab P µν µ µ a T ν µ b T 0 (20)

23 Coupling to External Fields A generalization: coupling global symmetry currents to macroscopic gauge fields F a µν. The conservation laws read ν T ν µ = F a µνj ν a µ J µ a = 0 (21) The conductivity term in the charge current is modified: E a µ = F a µνu ν. ( J a (1)µ = σ ab E bµ TP µν µ b ) ν T T (1)µν and s (1)µ remain the same. (22) The presence of gauge fields makes the entropy constraint more powerful.

24 Axial Terms There are additional transport terms if we allow the use of parity-odd Levi-Civita tensor ǫ µνρσ. This has been first discovered in the context of the the fluid/gravity correspondence (Erdmenger et.al, Banerjee et.al.). The charge current may get a contribution proportional to the vorticity ω µ = (1/2)ǫ µνρσ u ν ρ u σ, or to the magnetic field B aµ = (1/2)ǫ µνρσ u ν F a ρσ. The global symmetry current takes the form j µ a = ρ a u µ + σ a b ( E µ b TPµν ν µ b T ) + ξ a ω µ + ξ (B) ab Bbµ (23) The new axial conductivities are related to quantum anomalies. The local entropy increase fixes the vortical and magnetic conductivities ξ a,ξ (B) ab.

25 Axial triangle anomalies: Quantum Anomalies µ J µ a = 1 8 C abcǫ µνρσ F b µνf c ρσ π 2β aǫ µνρσ R µνκλ R ρσ λκ C abc = 1 4π 2 tr{t (a T b T c) }, β a = 1 24 tr{t a}. The anomaly coefficients are (Son,Surowka; Neiman, Y.O,Landsteiner et.al) (24) ξ a = C abc µ b µ c + 2β a T 2 2ρ ( ) a 1 ǫ + p 3 C bcdµ b µ c µ d + 2β b µ b T 2 ξ (B) ab = C abc µ c ρ ( ) a 1 ǫ + p 2 C bcdµ c µ d + β b T 2 (25)

26 Chiral Magnetic and Vortical Effects (Kharzeev, Son) Chiral magnetic effect: charge separation. Chiral vortical effect: baryon number separation. N c µ ( 5 J = 2π 2 tr(vaq) B ) + tr(vab)2µ B ω The electromagnetic current corresponds to V = Q and the baryon current to V = B. The ratio between the baryon asymmetry and the charge asymmetry increases when the center of mass energy is lowered.

27 QCD Axial Anomaly in HIC (Keren-Zur,Y.O.) For off-central collisions we expect enhancement of Ω (in general spin-excited hadrons) production along the rotation axis of the collision.

28 Gravitational Description Consider the (d+1)-dimensional Einstein equations with negative cosmological constant x m = (x µ, r). E mn = R mn + dg mn = 0 (26) These equations have a particular "thermal equilibrium" solution - the boosted black brane ds 2 = 2u µ dx µ dr r 2 f[br]u µ u ν dx µ dx ν + r 2 P µν dx µ dx ν where f(r) = 1 1 r d and the constant T = d 4πb is the temperature. (27)

29 Gravitational Description At r = b 1 there is an event horizon. It is a null hypersurface with induced metric ds 2 H = b 2 P µν dx µ dx ν, P µν u µ = 0 (28) The normal to the horizon The entropy current reads l r = 0, l µ = u µ (29) s µ = 1 4 bd 1 u µ (30)

30 Gravitational Description One looks for a solution of the Einstein equation by the method of variation of constants (Bhattacharya et.al.) g mn = (g 0 ) mn + δg mn (31) (g 0 ) mn dx m dx n = 2u µ (x α )dx µ dr r 2 f[b(x α )r]u µ (x α )u ν (x α )dx µ dx ν + r 2 P µν (x α )dx µ dx ν (32) One requires AdS asymptotics and regularity at the horizon and gets order by order equations for u µ (x α ) and T(x α ). The series for g mn is the series in the Knudsen number of the boundary CFT hydrodynamics.

31 Gravitational Description The constraint equations projected on any hypersurface r = r c are E r m = 0. E r r = 0 is the equation of state. E r µ = 0 are the hydrodynamics equations µ T µν = 0. The constraints equations can be written in terms if the induced metric on the hypersurface and its extrinsic curvature D ν K ν µ µk = 0 (33)

32 Extrinsic Curvature K µ ν = µ l ν (34) n(x) x y n(y) The extrinsic curvature measures the bending of the surface embedded in the higher dimensional space. The normal vector n(x) at the point x is parallel transported to the normal vector n(y) at the point y and differs from the dashed vector at point y.

33 Horizon Dynamics The way the black brane horizon geometry encodes the boundary fluid dynamics is reminiscent of the Membrane Paradigm (Damour) in classical general relativity, according to which any black hole has a fictitious fluid living on its horizon. U (x). T(x)

34 The Area Current s µ = vl µ, where v is a scalar density equal to the horizon area density, is the horizon area current. The expansion θ can be expressed as vθ = µ s µ (35) The area increase theorem (Hawking) seems to guarantee that µ s µ 0 (has been checked up to second order in the derivative expansion). Shear Expansion

35 The Focusing Equation: κ is the surface gravity The Focusing Equation κθ = l µ µ θ θ2 + σ 2 (H) + 8πT µνl µ l ν (36) l m m l n = κl n (37) T mn is the bulk energy-momentum tensor, σ(h) 2 is the square of the horizon s shear tensor, For µ s µ 0 we need the null energy condition T mn l m l n 0. The term l µ µ θ can have either sign. This term is related to the fact that an event horizon is defined not locally, but by boundary conditions at future infinity which ensure that the horizon is eternal.

36 The Bulk Viscosity ζ (Eling,Y.O.) The bulk viscosity is captured by the horizon dynamics - it follows from µ s µ 0. Consider (d + 1)-dimensional gravitational backgrounds holographically describing thermal states in strongly coupled d-dimensional field theories. The (d + 1)-dimensional gravitational action reads I = 1 ( ) gd d+1 x R 1 ( φ i ) 2 V(φ i ) + I gauge 16π 2 V(φ i ) represents the potential for the scalar fields and I gauge represents the action of gauge fields (abelian or non-abelian) A a µ. i (38)

37 The Bulk Viscosity ζ Using focusing equation we derive ζ η = i ( s dφh i ds + ρa dφh i dρ a ) 2 (39) η is the shear viscosity, s is the entropy density, ρ a are the charges associated with the gauge fields A a µ, and φ H i are the values of the scalar fields on the horizon.

38 Wald Entropy In higher curvature theories of gravity, Wald studied quasi-stationary processes in a general diffeomorphism invariant theory of gravity and was able to derive a general formula for the black hole entropy. The relevant quantity is the antisymmetric Noether potential Q mn associated with diffeomorphisms along a vector l m. We consider the case where the gravitational Lagrangian depends only algebraically on the Riemann tensor L(g mn, R mnlp ). Here the potential has the form Q mn = ( g 2L mnlp l l p + 4l p l L mnlp) (40) with L mnlp = L/ R mnlp.

39 Wald Entropy When the black hole solution is stationary, l m is a Killing vector. The total entropy of the horizon is unambiguous and is proportional to an integral of Q mn over any horizon cross-section S wald = 1 Q mn dσ mn (41) T In Einstein gravity, we have L mnlp = ( g/32π)(g ml g np g mp g nl ), and one gets the Bekenstein-Hawking entropy. In general the entropy will depend on the both the intrinsic and the extrinsic geometries of the horizon surface. When calculating the entropy of stationary horizons, the second term in Q mn does not contribute to the integral, and may be omitted. This term is important in non-stationary solutions.

40 Wald Entropy Current When the horizon is dynamical, the Wald formula is subject to ambiguities because there is no longer a preferred choice of the Killing vector and because one is free to add total divergence to the Lagrangian, symplectic potential, and the Noether current. There is no known analog of the area theorem in a general higher curvature theory. In the context of the fluid-gravity correspondence, we propose a definition for a local entropy current (Chapman, Neiman, Y.O.) s m = 2π κ Qmn l n (42) This is to be evaluated on the horizon, so l n is the normal to the horizon and κ is the surface gravity associated with this normal vector.

41 Wald Entropy Current The current is effectively a flux of the Noether potential through the horizon surface. This expression contains an ambiguity in the definition of the vector l m off the horizon surface in the bulk. In particular the current s m depends on the derivative of l m with respect to the bulk radial coordinate. We remove the ambiguities by imposing L l g rm = 0 (43) where L l is the Lie derivative in the direction of l.

42 Wald Entropy Current This choice follows from the standard gauge imposed on the bulk metric in the fluid-gravity correspondence g rr = 0, g rµ u µ, where u µ corresponds to the fluid velocity. The construction of the bulk metric in a derivative expansion is carried out such that this gauge condition is preserved. With this gauge choice, one has a map between the boundary and the horizon coordinates, and the hydrodynamics is realized in the x µ space. The conditions that we propose determines l A, such that it is preserves the gauge condition. This is, in fact, a necessary condition if we want the entropy current constructed from the horizon data to be compatible with the hydrodynamics determined by the boundary stress-energy tensor.

43 Wald Entropy Current However, while this choice removes the ambiguity in the definition of l A, we do not know whether it guarantees that the entropy current has a non-negative divergence. We studied this issue in two setups: Einstein-Maxwell theory with a gravitational Chern-Simons term (Chapman,Neiman,Y.O.), and Rindler Hydrodynamics (Eling, Meyer,Y.O.) with Gauss-Bonnet corrections. In both cases we found that the entropy current has a non-negative divergence. The entropy current in the presence of a gravitational Chern-Simons term reproduced exactly the expected result of anomalous hydrodynamics s µ = su µ 16πTλ a µ a ω µ (44)

44 Outlook Turbulence and Singularities. Anomalous transport in fluids (and superfluids). Entropy increase theorems.