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Universal Phenomena at Strong Coupling and Gravity Ayan Mukhopadhyay Harish-Chandra Research Institute, Chhatnag Road, Jhusi Allahabad, India 211019 ayan@mri.ernet.in September 28, 2009 Universal Phenomena at Strong Coupling and Gravity Ayan

  1. Introduction Our Results We show that the relativistic semiclassical Boltzmann equation has “conservative solutions” which could be determined by the energy-momentum tensor alone. We can justify our study of Boltzmann equation at weak coupling because previous work of Arnold, Yaffe and others have demonstrated that an effective Boltzmann equation is as good as perturbative gauge theory to study, for example, transport phenomena in high temperature QCD. We argue that these conservative solutions exist also in the exact microscopic theory. We naturally identify the conservative solutions with the universal sector at strong coupling and large N . We find the right method of extrapolating the conservative condition at weak coupling to regularity condition in gravity at strong coupling. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 4 / 29

  2. Introduction Outline of the rest of the talk (a) Recap of Boltzmann equation and how it includes hydrodynamics Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 5 / 29

  3. Introduction Outline of the rest of the talk (a) Recap of Boltzmann equation and how it includes hydrodynamics (b) Conservative Solutions of the Boltzmann Equation Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 5 / 29

  4. Introduction Outline of the rest of the talk (a) Recap of Boltzmann equation and how it includes hydrodynamics (b) Conservative Solutions of the Boltzmann Equation (c) Phenomena beyond hydrodynamics like local relaxation as features of conservative solutions Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 5 / 29

  5. Introduction Outline of the rest of the talk (a) Recap of Boltzmann equation and how it includes hydrodynamics (b) Conservative Solutions of the Boltzmann Equation (c) Phenomena beyond hydrodynamics like local relaxation as features of conservative solutions Beyond Kinetic Theory : The Israel-Stewart-Muller Formalism Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 5 / 29

  6. Introduction Outline of the rest of the talk (a) Recap of Boltzmann equation and how it includes hydrodynamics (b) Conservative Solutions of the Boltzmann Equation (c) Phenomena beyond hydrodynamics like local relaxation as features of conservative solutions Beyond Kinetic Theory : The Israel-Stewart-Muller Formalism (a) Proposal for the regularity condition for pure gravity in AdS 5 : the right way to extrapolate the conservative condition at weak coupling to the regularity condition at strong coupling Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 5 / 29

  7. Introduction Outline of the rest of the talk (a) Recap of Boltzmann equation and how it includes hydrodynamics (b) Conservative Solutions of the Boltzmann Equation (c) Phenomena beyond hydrodynamics like local relaxation as features of conservative solutions Beyond Kinetic Theory : The Israel-Stewart-Muller Formalism (a) Proposal for the regularity condition for pure gravity in AdS 5 : the right way to extrapolate the conservative condition at weak coupling to the regularity condition at strong coupling (b) Predictions and Consistency Checks of our Proposal Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 5 / 29

  8. Introduction Outline of the rest of the talk (a) Recap of Boltzmann equation and how it includes hydrodynamics (b) Conservative Solutions of the Boltzmann Equation (c) Phenomena beyond hydrodynamics like local relaxation as features of conservative solutions Beyond Kinetic Theory : The Israel-Stewart-Muller Formalism (a) Proposal for the regularity condition for pure gravity in AdS 5 : the right way to extrapolate the conservative condition at weak coupling to the regularity condition at strong coupling (b) Predictions and Consistency Checks of our Proposal (c) Our proposal and Quasinormal Modes Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 5 / 29

  9. Introduction Outline of the rest of the talk (a) Recap of Boltzmann equation and how it includes hydrodynamics (b) Conservative Solutions of the Boltzmann Equation (c) Phenomena beyond hydrodynamics like local relaxation as features of conservative solutions Beyond Kinetic Theory : The Israel-Stewart-Muller Formalism (a) Proposal for the regularity condition for pure gravity in AdS 5 : the right way to extrapolate the conservative condition at weak coupling to the regularity condition at strong coupling (b) Predictions and Consistency Checks of our Proposal (c) Our proposal and Quasinormal Modes Discussion : Open Issues in how Irreversibility (decoherence/thermalization) emerges at Long Time Scales Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 5 / 29

  10. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 1/3 The Boltzmann equation gives a very successful description of non-equilibrium phenomena in rarefied monoatomic gases. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 6 / 29

  11. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 1/3 The Boltzmann equation gives a very successful description of non-equilibrium phenomena in rarefied monoatomic gases. It describes the evolution of one particle phase space distribution f when the molecules interact by a central force law. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 6 / 29

  12. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 1/3 The Boltzmann equation gives a very successful description of non-equilibrium phenomena in rarefied monoatomic gases. It describes the evolution of one particle phase space distribution f when the molecules interact by a central force law. It describes phenomena at length scales even less than the mean free path and at time scales even less than the local relaxation time. However it fails at molecular length and time scales. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 6 / 29

  13. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 1/3 The Boltzmann equation gives a very successful description of non-equilibrium phenomena in rarefied monoatomic gases. It describes the evolution of one particle phase space distribution f when the molecules interact by a central force law. It describes phenomena at length scales even less than the mean free path and at time scales even less than the local relaxation time. However it fails at molecular length and time scales. Boltzmann’s H -theorem states that the phase space integral of f ln f monotonically decreases with time and further the final state where the time derivative vanishes is that of global equilibrium. We will call irreversibility of this type ”irreversibility at all time-scales." Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 6 / 29

  14. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 1/3 The Boltzmann equation gives a very successful description of non-equilibrium phenomena in rarefied monoatomic gases. It describes the evolution of one particle phase space distribution f when the molecules interact by a central force law. It describes phenomena at length scales even less than the mean free path and at time scales even less than the local relaxation time. However it fails at molecular length and time scales. Boltzmann’s H -theorem states that the phase space integral of f ln f monotonically decreases with time and further the final state where the time derivative vanishes is that of global equilibrium. We will call irreversibility of this type ”irreversibility at all time-scales." The origin of irreversibility is chiefly due to the assumption made in the Boltzmann equation that the two particle velocity distribution locally factorises. This is called the ergodic hypothesis . There is a very rigorous modern understanding of how the ergodic hypothesis emerges for “good” multiparticle phase space distributions which do not distinguish between precollisional and postcollisional configurations [Lanford, Cercignani, Spohn, etc] so that the Boltzmann equation can be rigorously derived from reversible Hamiltonian mechanics in large classical systems. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 6 / 29

  15. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 2/3 The Boltzmann equation for the one particle phase space distribution f ( x ,ξ ) is ( ∂ ∂ t + ξ. ∂ ∂ x ) f ( x ,ξ ) = J ( f , f )( x ,ξ ) (1) where, � ′ ) g ( x ,ξ ∗ ′ ) − f ( x ,ξ ) g ( x ,ξ ∗ )) B ( θ, V ) d ξ ∗ d ǫ d θ J ( f , g ) = ( f ( x ,ξ (2) is the change in phase space distribution due to binary collisions Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 7 / 29

  16. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 2/3 The Boltzmann equation for the one particle phase space distribution f ( x ,ξ ) is ( ∂ ∂ t + ξ. ∂ ∂ x ) f ( x ,ξ ) = J ( f , f )( x ,ξ ) (1) where, � ′ ) g ( x ,ξ ∗ ′ ) − f ( x ,ξ ) g ( x ,ξ ∗ )) B ( θ, V ) d ξ ∗ d ǫ d θ J ( f , g ) = ( f ( x ,ξ (2) is the change in phase space distribution due to binary collisions The collison variables are explained in the figure below: r n V’ rdrd ε θ r V Figure: The collision variables Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 7 / 29

  17. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 3/3 The collisional kernel B ( θ, V ) is defined as below B ( θ, V ) = Vr ∂ r ( θ, V ) (3) ∂θ Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 8 / 29

  18. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 3/3 The collisional kernel B ( θ, V ) is defined as below B ( θ, V ) = Vr ∂ r ( θ, V ) (3) ∂θ The velocities of the target and bullet molecule are related to the initial velocities of the target and bullet molecule as below ′ i = ξ i − n i ( n . V ) (4) ξ ξ ∗ ′ = ξ ∗ i + n i ( n . V ) i ′ . n = V . n . so that V Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 8 / 29

  19. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Boltzmann Equation : Brief Description 3/3 The collisional kernel B ( θ, V ) is defined as below B ( θ, V ) = Vr ∂ r ( θ, V ) (3) ∂θ The velocities of the target and bullet molecule are related to the initial velocities of the target and bullet molecule as below ′ i = ξ i − n i ( n . V ) (4) ξ ξ ∗ ′ = ξ ∗ i + n i ( n . V ) i ′ . n = V . n . so that V Let φ ( ξ ) be a function of ξ . We will call it a collisional invariant if ′ ) − φ ( ξ ∗ ′ ) = 0. Clearly the collisional invariants are five in φ ( ξ )+ φ ( ξ ∗ ) − φ ( ξ number and they are 1 ,ξ i ,ξ 2 . We will collectively denote them as ψ α . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 8 / 29

  20. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Hydrodynamic Equations from the Boltzmann Equation (1/2) Using symmetry one can easily prove that: � φ ( ξ )( J ( f , g )+ J ( g , f )) d ξ = (5) 1 � ′ ) − φ ( ξ ∗ ′ ))( J ( f , g )+ J ( g , f )) d ξ ( φ ( ξ )+ φ ( ξ ∗ ) − φ ( ξ 4 Therefore if φ ( ξ ) is a collisional invariant, i.e if φ ( ξ ) = ψ α ( ξ ) , � ψ α ( ξ ) J ( f , f ) d ξ = 0 (6) Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 9 / 29

  21. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Hydrodynamic Equations from the Boltzmann Equation (1/2) Using symmetry one can easily prove that: � φ ( ξ )( J ( f , g )+ J ( g , f )) d ξ = (5) 1 � ′ ) − φ ( ξ ∗ ′ ))( J ( f , g )+ J ( g , f )) d ξ ( φ ( ξ )+ φ ( ξ ∗ ) − φ ( ξ 4 Therefore if φ ( ξ ) is a collisional invariant, i.e if φ ( ξ ) = ψ α ( ξ ) , � ψ α ( ξ ) J ( f , f ) d ξ = 0 (6) So, the Boltzmann equation implies ∂ρ α ∂ t + ∂ � ( ξ i ψ α fd ξ ) = 0 (7) ∂ x i where ρ α = � ψ α fd ξ are the locally conserved quantities. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 9 / 29

  22. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Hydrodynamic Equations from the Boltzmann Equation (1/2) Using symmetry one can easily prove that: � φ ( ξ )( J ( f , g )+ J ( g , f )) d ξ = (5) 1 � ′ ) − φ ( ξ ∗ ′ ))( J ( f , g )+ J ( g , f )) d ξ ( φ ( ξ )+ φ ( ξ ∗ ) − φ ( ξ 4 Therefore if φ ( ξ ) is a collisional invariant, i.e if φ ( ξ ) = ψ α ( ξ ) , � ψ α ( ξ ) J ( f , f ) d ξ = 0 (6) So, the Boltzmann equation implies ∂ρ α ∂ t + ∂ � ( ξ i ψ α fd ξ ) = 0 (7) ∂ x i where ρ α = � ψ α fd ξ are the locally conserved quantities. Instead of using ρ α , we will use the hydrodynamic variables: � fd ξ, u i = 1 � ξ i fd ξ, p = 1 � ξ 2 fd ξ ρ = (8) ρ 3 Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 9 / 29

  23. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Hydrodynamic Equations from the Boltzmann Equation (2/2) Now, our hydrodynamic equations are as below: ∂ρ ∂ t + ∂ ( ρ u r ) = 0 (9) ∂ x r ∂ u i ∂ u i + 1 ∂ ( p δ ir + p ir ) ∂ t + u r = 0 ∂ x r ρ ∂ x r ∂ p ∂ t + ∂ ( u r p )+ 2 3 ( p δ ir + p ir ) ∂ u i + 1 ∂ S r = 0 ∂ x r ∂ x r 3 ∂ x r Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 10 / 29

  24. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Hydrodynamic Equations from the Boltzmann Equation (2/2) Now, our hydrodynamic equations are as below: ∂ρ ∂ t + ∂ ( ρ u r ) = 0 (9) ∂ x r ∂ u i ∂ u i + 1 ∂ ( p δ ir + p ir ) ∂ t + u r = 0 ∂ x r ρ ∂ x r ∂ p ∂ t + ∂ ( u r p )+ 2 3 ( p δ ir + p ir ) ∂ u i + 1 ∂ S r = 0 ∂ x r ∂ x r 3 ∂ x r We can also define a local temperature T using the ideal gas equation of state p /ρ = RT locally ( R is Boltzmann constant/ mass of molecule). Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 10 / 29

  25. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Hydrodynamic Equations from the Boltzmann Equation (2/2) Now, our hydrodynamic equations are as below: ∂ρ ∂ t + ∂ ( ρ u r ) = 0 (9) ∂ x r ∂ u i ∂ u i + 1 ∂ ( p δ ir + p ir ) ∂ t + u r = 0 ∂ x r ρ ∂ x r ∂ p ∂ t + ∂ ( u r p )+ 2 3 ( p δ ir + p ir ) ∂ u i + 1 ∂ S r = 0 ∂ x r ∂ x r 3 ∂ x r We can also define a local temperature T using the ideal gas equation of state p /ρ = RT locally ( R is Boltzmann constant/ mass of molecule). The shear stress tensor, p ij , is defined as follows � p ij = ( c i c j − RT δ ij ) fd ξ (10) where c i = ξ i − u i . One can easily see that, from definition, p ij δ ij = 0. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 10 / 29

  26. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Hydrodynamic Equations from the Boltzmann Equation (2/2) Now, our hydrodynamic equations are as below: ∂ρ ∂ t + ∂ ( ρ u r ) = 0 (9) ∂ x r ∂ u i ∂ u i + 1 ∂ ( p δ ir + p ir ) ∂ t + u r = 0 ∂ x r ρ ∂ x r ∂ p ∂ t + ∂ ( u r p )+ 2 3 ( p δ ir + p ir ) ∂ u i + 1 ∂ S r = 0 ∂ x r ∂ x r 3 ∂ x r We can also define a local temperature T using the ideal gas equation of state p /ρ = RT locally ( R is Boltzmann constant/ mass of molecule). The shear stress tensor, p ij , is defined as follows � p ij = ( c i c j − RT δ ij ) fd ξ (10) where c i = ξ i − u i . One can easily see that, from definition, p ij δ ij = 0. Also we define, S ijk as below: � S ijk = c i c j c k fd ξ (11) and the heat flow vector S i through S i = S ijk δ ij . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 10 / 29

  27. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Moment Equations Let us now define the n-th moment of f to be � f ( n ) = c n fd ξ (12) We note that f ( 2 ) = p δ ij + p ij , f ( 3 ) ijk = S ijk , etc ij Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 11 / 29

  28. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Moment Equations Let us now define the n-th moment of f to be � f ( n ) = c n fd ξ (12) We note that f ( 2 ) = p δ ij + p ij , f ( 3 ) ijk = S ijk , etc ij The equation satisfied by the moments f ( n ) ’s for n ≥ 2 are as follows: ρ f ( n − 1 ) ∂ f ( 2 ) ∂ f ( n ) + ∂ )+ ∂ u − 1 ( u i f ( n ) + f ( n + 1 ) f ( n ) i = J ( n ) (13) i i ∂ t ∂ x i ∂ x i ∂ x i where � J ( n ) = ′ f ′ c n B ( f 1 − ff 1 ) d θ d ǫ d ξ d ξ 1 (14) Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 11 / 29

  29. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Moment Equations Let us now define the n-th moment of f to be � f ( n ) = c n fd ξ (12) We note that f ( 2 ) = p δ ij + p ij , f ( 3 ) ijk = S ijk , etc ij The equation satisfied by the moments f ( n ) ’s for n ≥ 2 are as follows: ρ f ( n − 1 ) ∂ f ( 2 ) ∂ f ( n ) + ∂ )+ ∂ u − 1 ( u i f ( n ) + f ( n + 1 ) f ( n ) i = J ( n ) (13) i i ∂ t ∂ x i ∂ x i ∂ x i where � J ( n ) = ′ f ′ c n B ( f 1 − ff 1 ) d θ d ǫ d ξ d ξ 1 (14) It can be shown that ∞ J ( n ) � B ( n , p , q ) ( ρ, T ) f ( p ) f ( q ) = (15) µ µνρ ν ρ p , q = 0 , p ≥ q Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 11 / 29

  30. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases The Moment Equations Let us now define the n-th moment of f to be � f ( n ) = c n fd ξ (12) We note that f ( 2 ) = p δ ij + p ij , f ( 3 ) ijk = S ijk , etc ij The equation satisfied by the moments f ( n ) ’s for n ≥ 2 are as follows: ρ f ( n − 1 ) ∂ f ( 2 ) ∂ f ( n ) + ∂ )+ ∂ u − 1 ( u i f ( n ) + f ( n + 1 ) f ( n ) i = J ( n ) (13) i i ∂ t ∂ x i ∂ x i ∂ x i where � J ( n ) = ′ f ′ c n B ( f 1 − ff 1 ) d θ d ǫ d ξ d ξ 1 (14) It can be shown that ∞ J ( n ) � B ( n , p , q ) ( ρ, T ) f ( p ) f ( q ) = (15) µ µνρ ν ρ p , q = 0 , p ≥ q It can be shown that B ( 2 , 2 , 0 ) ( ρ, T ) = B ( 2 ) ( ρ, T ) δ ik δ jl . ijkl Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 11 / 29

  31. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (1/4) We will first describe the conservative solutions of the Boltzmann equation for nonrelativistic monoatomic gases . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 12 / 29

  32. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (1/4) We will first describe the conservative solutions of the Boltzmann equation for nonrelativistic monoatomic gases . The energy momentum tensor can always be parametrised by (a) the five hydrodynamic variables ( ρ , u i , p ) and (b) the shear stress tensor ( p ij ) in a comoving locally inertial frame. We have seen how these can be defined through the first ten velocity moments of f . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 12 / 29

  33. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (1/4) We will first describe the conservative solutions of the Boltzmann equation for nonrelativistic monoatomic gases . The energy momentum tensor can always be parametrised by (a) the five hydrodynamic variables ( ρ , u i , p ) and (b) the shear stress tensor ( p ij ) in a comoving locally inertial frame. We have seen how these can be defined through the first ten velocity moments of f . Let f ( n ) , a tensor of rank n be the n -th velocity moment of f ( x ,ξ ) so that f ( n ) = � c n fd ξ , where c i = ξ i − u i . At equilibrium all these f ( n ) ’s vanish. However in conservative solutions these do not vanish and in fact can be very large. These f ( n ) ’s are determined functionally in terms of the ten independent variables of the energy-momentum tensor. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 12 / 29

  34. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (1/4) We will first describe the conservative solutions of the Boltzmann equation for nonrelativistic monoatomic gases . The energy momentum tensor can always be parametrised by (a) the five hydrodynamic variables ( ρ , u i , p ) and (b) the shear stress tensor ( p ij ) in a comoving locally inertial frame. We have seen how these can be defined through the first ten velocity moments of f . Let f ( n ) , a tensor of rank n be the n -th velocity moment of f ( x ,ξ ) so that f ( n ) = � c n fd ξ , where c i = ξ i − u i . At equilibrium all these f ( n ) ’s vanish. However in conservative solutions these do not vanish and in fact can be very large. These f ( n ) ’s are determined functionally in terms of the ten independent variables of the energy-momentum tensor. For instance, the heat-flow vector S i is given in terms of the ten variables as below: S i = 15 pR ∂ T 2 B ( 2 ) ( 2 RT ∂ p ir 3 ∂ T − 2 p ir ∂ p + + 7 Rp ir )+ ... (16) 2 B ( 2 ) ∂ x i ∂ x r ∂ x r ρ ∂ x r where B ( 2 ) is a specific function of the molecular mass, radius, local density and temperature and can be determined from the Boltzmann equation. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 12 / 29

  35. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (1/4) We will first describe the conservative solutions of the Boltzmann equation for nonrelativistic monoatomic gases . The energy momentum tensor can always be parametrised by (a) the five hydrodynamic variables ( ρ , u i , p ) and (b) the shear stress tensor ( p ij ) in a comoving locally inertial frame. We have seen how these can be defined through the first ten velocity moments of f . Let f ( n ) , a tensor of rank n be the n -th velocity moment of f ( x ,ξ ) so that f ( n ) = � c n fd ξ , where c i = ξ i − u i . At equilibrium all these f ( n ) ’s vanish. However in conservative solutions these do not vanish and in fact can be very large. These f ( n ) ’s are determined functionally in terms of the ten independent variables of the energy-momentum tensor. For instance, the heat-flow vector S i is given in terms of the ten variables as below: S i = 15 pR ∂ T 2 B ( 2 ) ( 2 RT ∂ p ir 3 ∂ T − 2 p ir ∂ p + + 7 Rp ir )+ ... (16) 2 B ( 2 ) ∂ x i ∂ x r ∂ x r ρ ∂ x r where B ( 2 ) is a specific function of the molecular mass, radius, local density and temperature and can be determined from the Boltzmann equation. All the higher moments similarly can be systematically determined for such solutions in unique functional forms of the ten independent variables. These functional forms have systematic expansions in two parameters, the derivative expansion parameter which is (typical scale of variation/ mean free path) and amplitude expansion parameter (typical value of non-hydrodynamic shear stress/ hydrostatic pressure). Only spatial derivatives and no time derivative appear in the functional forms of f ( n ) . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 12 / 29

  36. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (2/4) Since all the velocity moments of f are unique local functions of the ten variables and their spatial derivatives, it follows that f is also uniquely determined by the ten variables. Once f is determined, any observable can also be determined through f . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 13 / 29

  37. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (2/4) Since all the velocity moments of f are unique local functions of the ten variables and their spatial derivatives, it follows that f is also uniquely determined by the ten variables. Once f is determined, any observable can also be determined through f . The ten variables satisfy the following equations of motion closed amongst themselves ∂ρ ∂ t + ∂ ( ρ u r ) = 0 (17) ∂ x r ∂ u i ∂ u i + 1 ∂ ( p δ ir + p ir ) ∂ t + u r = 0 ∂ x r ρ ∂ x r ∂ p ∂ t + ∂ ( u r p )+ 2 3 ( p δ ir + p ir ) ∂ u i + 1 ∂ S r = 0 ∂ x r ∂ x r 3 ∂ x r ∂ p ij ∂ t + ∂ ( u r p ij )+ ∂ S ijr − 1 ∂ S r 3 δ ij ∂ x r ∂ x r ∂ x r + ∂ u j p ir + ∂ u i p jr − 2 ∂ u r 3 δ ij p rs ∂ x r ∂ x r ∂ x s + p ( ∂ u i + ∂ u j − 2 ∂ u r ) = B ( 2 ) ( ρ, T ) p ij 3 δ ij ∂ x j ∂ x i ∂ x r ∞ B ( 2 , p , q ) � ( ρ, T ) f ( p ) f ( q ) + ν ρ ij νρ p , q = 0 ; p ≥ q ; ( p , q ) � ( 2 , 0 ) Above all the higher moments, including S i , has been determined in terms of the hydrodynamic variables, the shear stress tensor and their spatial derivatives. Since spatial derivatives of arbitrary orders are present in these functional forms, we need analytic data as initial conditions for these equations Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 13 / 29

  38. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (2/4) Since all the velocity moments of f are unique local functions of the ten variables and their spatial derivatives, it follows that f is also uniquely determined by the ten variables. Once f is determined, any observable can also be determined through f . The ten variables satisfy the following equations of motion closed amongst themselves ∂ρ ∂ t + ∂ ( ρ u r ) = 0 (17) ∂ x r ∂ u i ∂ u i + 1 ∂ ( p δ ir + p ir ) ∂ t + u r = 0 ∂ x r ρ ∂ x r ∂ p ∂ t + ∂ ( u r p )+ 2 3 ( p δ ir + p ir ) ∂ u i + 1 ∂ S r = 0 ∂ x r ∂ x r 3 ∂ x r ∂ p ij ∂ t + ∂ ( u r p ij )+ ∂ S ijr − 1 ∂ S r 3 δ ij ∂ x r ∂ x r ∂ x r + ∂ u j p ir + ∂ u i p jr − 2 ∂ u r 3 δ ij p rs ∂ x r ∂ x r ∂ x s + p ( ∂ u i + ∂ u j − 2 ∂ u r ) = B ( 2 ) ( ρ, T ) p ij 3 δ ij ∂ x j ∂ x i ∂ x r ∞ B ( 2 , p , q ) � ( ρ, T ) f ( p ) f ( q ) + ν ρ ij νρ p , q = 0 ; p ≥ q ; ( p , q ) � ( 2 , 0 ) Above all the higher moments, including S i , has been determined in terms of the hydrodynamic variables, the shear stress tensor and their spatial derivatives. Since spatial derivatives of arbitrary orders are present in these functional forms, we need analytic data as initial conditions for these equations Any solution of the above equations of motion of the ten variables can be uniquely lifted to a full solution of the Boltzmann equation for f through the functional forms for f ( n ) already determined. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 13 / 29

  39. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (3/4) There are two special kinds of conservative solutions (a) normal or purely-hydrodynamic solutions [Enskog(1917), Burnett(1935), Chapman(1939)] where f is determined as functional of the five hydrodynamic variables and their spatial derivatives only Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 14 / 29

  40. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (3/4) There are two special kinds of conservative solutions (a) normal or purely-hydrodynamic solutions [Enskog(1917), Burnett(1935), Chapman(1939)] where f is determined as functional of the five hydrodynamic variables and their spatial derivatives only (b) homogenous non-hydrodynamic solutions where all hydrodynamic variables are constants and the shear stress tensor (therefore all the higher moments) is a function of time only, describing dynamics in only velocity space and hence relaxation. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 14 / 29

  41. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (3/4) There are two special kinds of conservative solutions (a) normal or purely-hydrodynamic solutions [Enskog(1917), Burnett(1935), Chapman(1939)] where f is determined as functional of the five hydrodynamic variables and their spatial derivatives only (b) homogenous non-hydrodynamic solutions where all hydrodynamic variables are constants and the shear stress tensor (therefore all the higher moments) is a function of time only, describing dynamics in only velocity space and hence relaxation. The normal solutions can be found by noting that the equation for p ij has a special algebraic solution given in terms of hydrodynamic variables only. This solution is unique. Upto two derivatives this solution is as below: η 2 η 2 p ( D Dt σ ij − 2 ( σ ik σ kj − 1 p ij = ησ ij + λ 1 p ( ∂. u ) σ ij + λ 2 3 δ ij σ lm σ lm )) (18) η 2 η 2 ρ T ( ∂ i ∂ j T − 1 p ρ T ( ∂ i p ∂ j T + ∂ j p ∂ i T − 2 + λ 3 3 δ ij � T )+ λ 4 3 δ ij ∂ l p ∂ l T ) η 2 p ρ T ( ∂ i T ∂ j T − 1 + λ 5 3 δ ij ∂ l T ∂ l T )+ ...... where σ ij = ∂ i u j + ∂ j u i − ( 2 / 3 ) δ ij ∂ l u l , η = ( p / B 2 ) and the λ ’s which are pure numbers can be determined from the Boltzmann equation. Note all time-derivatives can be replaced by spatial derivatives through hydrodynamic equations of motion. This matches with the second order expression for p ij for normal solutions [Chapman and Cowling, Chapter 15] Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 14 / 29

  42. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (4/4) Interestingly, the homogenous non-hydrodynamic solutions has singularities. For instance f ( 4 ) ijkl = ( 2 B ( 2 ) δ ( klmn )( ijtu ) − B ( 4 , 4 , 0 ) ( klmn )( ijtu ) ) − 1 B ( 4 , 2 , 2 ) ( ijtu )( pqrs ) p pq p rs + ..... and this becomes indeterminate when ( 2 B ( 2 ) δ ( klmn )( ijtu ) − B ( 4 , 4 , 0 ) ( klmn )( ijtu ) ) regarded as an 81 × 81 matrix fails to be invertible. Such singularities also appear in normal solutions in kinetic theory of liquids (to be discussed later) and has the interpretation of local nucleation of solid phase and so here the singularities probably signal local condensation of the liquid phase. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 15 / 29

  43. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (4/4) Interestingly, the homogenous non-hydrodynamic solutions has singularities. For instance f ( 4 ) ijkl = ( 2 B ( 2 ) δ ( klmn )( ijtu ) − B ( 4 , 4 , 0 ) ( klmn )( ijtu ) ) − 1 B ( 4 , 2 , 2 ) ( ijtu )( pqrs ) p pq p rs + ..... and this becomes indeterminate when ( 2 B ( 2 ) δ ( klmn )( ijtu ) − B ( 4 , 4 , 0 ) ( klmn )( ijtu ) ) regarded as an 81 × 81 matrix fails to be invertible. Such singularities also appear in normal solutions in kinetic theory of liquids (to be discussed later) and has the interpretation of local nucleation of solid phase and so here the singularities probably signal local condensation of the liquid phase. Any generic solution of the Boltzmann Equation at sufficiently late times is approximated by an appropriate conservative solution. Since the maximum of the propagation speeds of the linear modes increases as more and more moments are included [Boillat, Muller], we can argue that, at a sufficiently late time, the part of the higher moments functionally independent of the hydrodynamic variables and the shear stress tensor becomes irrelevant, so that the dynamics is well approximated by an appropriate conservative solution. Thus ten variables suffice to capture systematically a whole range of phenomena which includes hydrodynamics and relaxation. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 15 / 29

  44. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Conservative Solutions (4/4) Interestingly, the homogenous non-hydrodynamic solutions has singularities. For instance f ( 4 ) ijkl = ( 2 B ( 2 ) δ ( klmn )( ijtu ) − B ( 4 , 4 , 0 ) ( klmn )( ijtu ) ) − 1 B ( 4 , 2 , 2 ) ( ijtu )( pqrs ) p pq p rs + ..... and this becomes indeterminate when ( 2 B ( 2 ) δ ( klmn )( ijtu ) − B ( 4 , 4 , 0 ) ( klmn )( ijtu ) ) regarded as an 81 × 81 matrix fails to be invertible. Such singularities also appear in normal solutions in kinetic theory of liquids (to be discussed later) and has the interpretation of local nucleation of solid phase and so here the singularities probably signal local condensation of the liquid phase. Any generic solution of the Boltzmann Equation at sufficiently late times is approximated by an appropriate conservative solution. Since the maximum of the propagation speeds of the linear modes increases as more and more moments are included [Boillat, Muller], we can argue that, at a sufficiently late time, the part of the higher moments functionally independent of the hydrodynamic variables and the shear stress tensor becomes irrelevant, so that the dynamics is well approximated by an appropriate conservative solution. Thus ten variables suffice to capture systematically a whole range of phenomena which includes hydrodynamics and relaxation. It can be shown that the relativistic semiclassical Boltzmann equation has conservative solutions as well. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 15 / 29

  45. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Multi-Component Systems In order to generalize conservative solutions of Boltzmann equation to relativistic gauge theories we also need to understand how to construct such solutions for multi-component systems. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 16 / 29

  46. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Multi-Component Systems In order to generalize conservative solutions of Boltzmann equation to relativistic gauge theories we also need to understand how to construct such solutions for multi-component systems. In N = 4 SYM theory, all the particles form a multiplet whose internal degrees of freedom are spin and ( SO ( 6 ) R ) charge along with the color indices. From the point of view of gravity, since in the universal sector we have pure gravity on the dual side, not only local and global charges and currents, but also the higher multipole moments of these charge distrubutions are absent at the boundary. So, the most natural reflection of this on the conservative solutions is that there is equipartition at every point in phase space over the internal, i.e the spin, charge and color degrees of freedom. Then we can easily construct an effective single component Boltzmann equation by summing over interactions in all spin, charge and color channels. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 16 / 29

  47. Conservative Solutions of the Boltzmann Equation Non-relativistic classical monoatomic gases Multi-Component Systems In order to generalize conservative solutions of Boltzmann equation to relativistic gauge theories we also need to understand how to construct such solutions for multi-component systems. In N = 4 SYM theory, all the particles form a multiplet whose internal degrees of freedom are spin and ( SO ( 6 ) R ) charge along with the color indices. From the point of view of gravity, since in the universal sector we have pure gravity on the dual side, not only local and global charges and currents, but also the higher multipole moments of these charge distrubutions are absent at the boundary. So, the most natural reflection of this on the conservative solutions is that there is equipartition at every point in phase space over the internal, i.e the spin, charge and color degrees of freedom. Then we can easily construct an effective single component Boltzmann equation by summing over interactions in all spin, charge and color channels. For other conformal gauge theories with gravity duals, we may also do the same even though all particles do not form a multiplet. This is possible because of mass degeneracy. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 16 / 29

  48. Non-equilibrium Energy-Momentum Tensor in Conformal Theories Definitions of the Nine Parameters The equilibrium energy momentum tensor for a conformal theory is t ( 0 ) µν = ( π T ) 4 ( 4 u µ u ν + η µν ) . Going to the comoving inertial frame where t ( 0 ) µν is diag ( ǫ, p , p , p ) , we find that the energy density ǫ and the pressure p are ǫ 3 = p = ( π T ) 4 . Let π µν be the non-equilibrium part of the energy momentum tensor so that the total energy-momentum tensor is t µν = t ( 0 ) µν + π µν . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 17 / 29

  49. Non-equilibrium Energy-Momentum Tensor in Conformal Theories Definitions of the Nine Parameters The equilibrium energy momentum tensor for a conformal theory is t ( 0 ) µν = ( π T ) 4 ( 4 u µ u ν + η µν ) . Going to the comoving inertial frame where t ( 0 ) µν is diag ( ǫ, p , p , p ) , we find that the energy density ǫ and the pressure p are ǫ 3 = p = ( π T ) 4 . Let π µν be the non-equilibrium part of the energy momentum tensor so that the total energy-momentum tensor is t µν = t ( 0 ) µν + π µν . We define u µ as the velocity of energy transport and u µ u ν t µν = ǫ as the energy density, so π µν should be such that u µ π µν = 0 as the energy density remains uncorrected and in the local inertial comoving frame the energy-momentum tensor can be corrected only in the spatial block perpendicular to u µ . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 17 / 29

  50. Non-equilibrium Energy-Momentum Tensor in Conformal Theories Definitions of the Nine Parameters The equilibrium energy momentum tensor for a conformal theory is t ( 0 ) µν = ( π T ) 4 ( 4 u µ u ν + η µν ) . Going to the comoving inertial frame where t ( 0 ) µν is diag ( ǫ, p , p , p ) , we find that the energy density ǫ and the pressure p are ǫ 3 = p = ( π T ) 4 . Let π µν be the non-equilibrium part of the energy momentum tensor so that the total energy-momentum tensor is t µν = t ( 0 ) µν + π µν . We define u µ as the velocity of energy transport and u µ u ν t µν = ǫ as the energy density, so π µν should be such that u µ π µν = 0 as the energy density remains uncorrected and in the local inertial comoving frame the energy-momentum tensor can be corrected only in the spatial block perpendicular to u µ . The total energy momentum tensor is traceless, while the equilibrium part is traceless by itself. So, π µν is also traceless and since u µ π µν = 0, π µν has five independent components. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 17 / 29

  51. Non-equilibrium Energy-Momentum Tensor in Conformal Theories Definitions of the Nine Parameters The equilibrium energy momentum tensor for a conformal theory is t ( 0 ) µν = ( π T ) 4 ( 4 u µ u ν + η µν ) . Going to the comoving inertial frame where t ( 0 ) µν is diag ( ǫ, p , p , p ) , we find that the energy density ǫ and the pressure p are ǫ 3 = p = ( π T ) 4 . Let π µν be the non-equilibrium part of the energy momentum tensor so that the total energy-momentum tensor is t µν = t ( 0 ) µν + π µν . We define u µ as the velocity of energy transport and u µ u ν t µν = ǫ as the energy density, so π µν should be such that u µ π µν = 0 as the energy density remains uncorrected and in the local inertial comoving frame the energy-momentum tensor can be corrected only in the spatial block perpendicular to u µ . The total energy momentum tensor is traceless, while the equilibrium part is traceless by itself. So, π µν is also traceless and since u µ π µν = 0, π µν has five independent components. Thus the total energy-momentum tensor can be parametrised by nine variables u µ , T and the five independent components of π µν . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 17 / 29

  52. Non-equilibrium Energy-Momentum Tensor in Conformal Theories Definitions of the Nine Parameters The equilibrium energy momentum tensor for a conformal theory is t ( 0 ) µν = ( π T ) 4 ( 4 u µ u ν + η µν ) . Going to the comoving inertial frame where t ( 0 ) µν is diag ( ǫ, p , p , p ) , we find that the energy density ǫ and the pressure p are ǫ 3 = p = ( π T ) 4 . Let π µν be the non-equilibrium part of the energy momentum tensor so that the total energy-momentum tensor is t µν = t ( 0 ) µν + π µν . We define u µ as the velocity of energy transport and u µ u ν t µν = ǫ as the energy density, so π µν should be such that u µ π µν = 0 as the energy density remains uncorrected and in the local inertial comoving frame the energy-momentum tensor can be corrected only in the spatial block perpendicular to u µ . The total energy momentum tensor is traceless, while the equilibrium part is traceless by itself. So, π µν is also traceless and since u µ π µν = 0, π µν has five independent components. Thus the total energy-momentum tensor can be parametrised by nine variables u µ , T and the five independent components of π µν . The conservation of the energy-momentum tensor ∂ µ t µν = 0, gives us the forced relativistic Euler equation which shows that π µν is the relativistic shear stress tensor. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 17 / 29

  53. Non-equilibrium Energy-Momentum Tensor in Conformal Theories The Conformal Hydrodynamic Energy-Momentum Tensor (1/2) The most general form of the traceless hydrodynamic conformal shear stress tensor upto two orders in the derivative expansion, when all conserved currents vanish and with our definitions of u µ and T is as below [Baier, Romatschke, Son, Starinets, Stephanov (2007)] π µν − ησ µν = (19) � ( u · ∂ ) σ µν + 1 � 3 σ µν ( ∂ · u ) − ( u ν σ µβ + u µ σ νβ )( u · ∂ ) u β + α 1 � α − 1 � σ αµ σ ν 3 P µν σ αβ σ αβ + α 3 ( σ αµ ω ν α + σ αµ ω ν + α 2 α ) � α − 1 � ω αµ ω ν 3 P µν ω αβ ω αβ + O ( ∂ 3 u ) + α 4 where P µν is the projection tensor orthogonal to u µ P µν = u µ u ν + η µν (20) σ µν is the hydrodynamic strain rate σ µν = 1 2 P µα P νβ ( ∂ α u β + ∂ β u α ) − 1 3 P µν ( ∂. u ) (21) ω µν is the hydrodynamic vorticity tensor ω µν = 1 2 P µα P νβ ( ∂ α u β − ∂ α u β ) (22) Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 18 / 29

  54. Non-equilibrium Energy-Momentum Tensor in Conformal Theories The Conformal Hydrodynamic Energy-Momentum Tensor (2/2) The semiclassical Boltzmann equation can be used to determine hydrodynamic transport coefficients in high temperature QCD and it is as good as the full perturbative description [Arnold and Yaffe]. In fact, we only need to use only tree level S-matrices and ignoring bare quark masses the hydrodynamic energy-momentum tensor at high temperature is conformal. At weak coupling, η/ s is parametrically O ( 1 / ( g 4 ln ( 1 / g ))) , in fact all ln ( 1 / g ) terms can be resummed [Arnold, Moore, Yaffe]. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 19 / 29

  55. Non-equilibrium Energy-Momentum Tensor in Conformal Theories The Conformal Hydrodynamic Energy-Momentum Tensor (2/2) The semiclassical Boltzmann equation can be used to determine hydrodynamic transport coefficients in high temperature QCD and it is as good as the full perturbative description [Arnold and Yaffe]. In fact, we only need to use only tree level S-matrices and ignoring bare quark masses the hydrodynamic energy-momentum tensor at high temperature is conformal. At weak coupling, η/ s is parametrically O ( 1 / ( g 4 ln ( 1 / g ))) , in fact all ln ( 1 / g ) terms can be resummed [Arnold, Moore, Yaffe]. The Boltzmann equation has been used to find out the higher order hydrodynamic transport coefficients at the leading order and as mentioned this is as good as the full perturbative description [Moore and York (2009)]. The results for 3-quark QCD at leading order are Ts α 1 = 5 . 9 to 5 . 0 (varies with g ), Ts α 2 = 5 . 2 to 4 . 1 η 2 η 2 (varies with g ), Ts α 3 = 2 Ts α 1 and Ts α 4 6 π ( m D 1 T ) 2 . = 0. Here, the effective coupling constant is defined as g = η 2 η 2 η 2 Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 19 / 29

  56. Non-equilibrium Energy-Momentum Tensor in Conformal Theories The Conformal Hydrodynamic Energy-Momentum Tensor (2/2) The semiclassical Boltzmann equation can be used to determine hydrodynamic transport coefficients in high temperature QCD and it is as good as the full perturbative description [Arnold and Yaffe]. In fact, we only need to use only tree level S-matrices and ignoring bare quark masses the hydrodynamic energy-momentum tensor at high temperature is conformal. At weak coupling, η/ s is parametrically O ( 1 / ( g 4 ln ( 1 / g ))) , in fact all ln ( 1 / g ) terms can be resummed [Arnold, Moore, Yaffe]. The Boltzmann equation has been used to find out the higher order hydrodynamic transport coefficients at the leading order and as mentioned this is as good as the full perturbative description [Moore and York (2009)]. The results for 3-quark QCD at leading order are Ts α 1 = 5 . 9 to 5 . 0 (varies with g ), Ts α 2 = 5 . 2 to 4 . 1 η 2 η 2 (varies with g ), Ts α 3 = 2 Ts α 1 and Ts α 4 6 π ( m D 1 T ) 2 . = 0. Here, the effective coupling constant is defined as g = η 2 η 2 η 2 At strong coupling for conformal gauge theories with gravity duals the hydrodynamic transport coefficients in absence of charged currents are universal as they can be obtained by looking at regular linear or non-linear perturbations (of Einstein’s equation in five dimensions with a negative cosmological constant) about the black brane solution which are slowly varying both spatially and temporally with respect to the temperature. The famous result is that η/ s = 1 / 4 π [Kovtun, Son, Starinets]. The results [Baier, Romatschke, Son, Starinets, Stephanov; Bhattacharya, Hubeny, Minwalla, Rangamani (2007)] for higher order transport coefficients are : α 1 η = 2 − ln 2 2 π T , α 2 2 π T , α 3 1 η = ln 2 η = 2 π T and α 4 = 0. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 19 / 29

  57. Beyond Kinetic Theory Do Conservative Solutions exist in the Exact Microscopic Theory? The untruncated BBGKY heirarchy for non-relativistic systems is equivalent to the exact microscopic theory. Normal/purely hydrodynamic solutions have been constructed for the untruncated heirarchy [Born and Green (1949)]. These also exist if semiclassical corrections are included. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 20 / 29

  58. Beyond Kinetic Theory Do Conservative Solutions exist in the Exact Microscopic Theory? The untruncated BBGKY heirarchy for non-relativistic systems is equivalent to the exact microscopic theory. Normal/purely hydrodynamic solutions have been constructed for the untruncated heirarchy [Born and Green (1949)]. These also exist if semiclassical corrections are included. � ν ( r ) φ ′ ( r ) r 3 dr + the gaseous part The viscosity, for instance receives corrections as in η = 1 15 ( ν ( r ) is given by the two-body phase space distribution function where r is the relative separation), and increases with temperature unlike gases. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 20 / 29

  59. Beyond Kinetic Theory Do Conservative Solutions exist in the Exact Microscopic Theory? The untruncated BBGKY heirarchy for non-relativistic systems is equivalent to the exact microscopic theory. Normal/purely hydrodynamic solutions have been constructed for the untruncated heirarchy [Born and Green (1949)]. These also exist if semiclassical corrections are included. � ν ( r ) φ ′ ( r ) r 3 dr + the gaseous part The viscosity, for instance receives corrections as in η = 1 15 ( ν ( r ) is given by the two-body phase space distribution function where r is the relative separation), and increases with temperature unlike gases. It is certainly plausible that the conservative solutions of the untruncated BBGKY heirarchy also exist. We are investigating this currently. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 20 / 29

  60. Beyond Kinetic Theory Do Conservative Solutions exist in the Exact Microscopic Theory? The untruncated BBGKY heirarchy for non-relativistic systems is equivalent to the exact microscopic theory. Normal/purely hydrodynamic solutions have been constructed for the untruncated heirarchy [Born and Green (1949)]. These also exist if semiclassical corrections are included. � ν ( r ) φ ′ ( r ) r 3 dr + the gaseous part The viscosity, for instance receives corrections as in η = 1 15 ( ν ( r ) is given by the two-body phase space distribution function where r is the relative separation), and increases with temperature unlike gases. It is certainly plausible that the conservative solutions of the untruncated BBGKY heirarchy also exist. We are investigating this currently. Recent experimental evidences at RHIC suggests that second order hydrodynamics is indeed relevant to explain the expansion of the quark-gluon plasma. Moreover, the dynamics can be approximated quite well by an appropriate purely hydrodynamic equation involving corrections to the Navier-Stokes. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 20 / 29

  61. Beyond Kinetic Theory Do Conservative Solutions exist in the Exact Microscopic Theory? The untruncated BBGKY heirarchy for non-relativistic systems is equivalent to the exact microscopic theory. Normal/purely hydrodynamic solutions have been constructed for the untruncated heirarchy [Born and Green (1949)]. These also exist if semiclassical corrections are included. � ν ( r ) φ ′ ( r ) r 3 dr + the gaseous part The viscosity, for instance receives corrections as in η = 1 15 ( ν ( r ) is given by the two-body phase space distribution function where r is the relative separation), and increases with temperature unlike gases. It is certainly plausible that the conservative solutions of the untruncated BBGKY heirarchy also exist. We are investigating this currently. Recent experimental evidences at RHIC suggests that second order hydrodynamics is indeed relevant to explain the expansion of the quark-gluon plasma. Moreover, the dynamics can be approximated quite well by an appropriate purely hydrodynamic equation involving corrections to the Navier-Stokes. So, it is likely that normal and conservative solutions exist in the exact relativistic quantum guage theories like QCD such that a generic state at sufficient late times can be approximated by an appropriate conservative solution. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 20 / 29

  62. Beyond Kinetic Theory Do Conservative Solutions exist in the Exact Microscopic Theory? The untruncated BBGKY heirarchy for non-relativistic systems is equivalent to the exact microscopic theory. Normal/purely hydrodynamic solutions have been constructed for the untruncated heirarchy [Born and Green (1949)]. These also exist if semiclassical corrections are included. � ν ( r ) φ ′ ( r ) r 3 dr + the gaseous part The viscosity, for instance receives corrections as in η = 1 15 ( ν ( r ) is given by the two-body phase space distribution function where r is the relative separation), and increases with temperature unlike gases. It is certainly plausible that the conservative solutions of the untruncated BBGKY heirarchy also exist. We are investigating this currently. Recent experimental evidences at RHIC suggests that second order hydrodynamics is indeed relevant to explain the expansion of the quark-gluon plasma. Moreover, the dynamics can be approximated quite well by an appropriate purely hydrodynamic equation involving corrections to the Navier-Stokes. So, it is likely that normal and conservative solutions exist in the exact relativistic quantum guage theories like QCD such that a generic state at sufficient late times can be approximated by an appropriate conservative solution. The higher order transport coefficients could be exactly defined (at least implicitly) if we can construct normal solutions of the exact relativistic quantum gauge theories. We are investigating this currently as well. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 20 / 29

  63. Beyond Kinetic Theory The Israel-Stewart-Muller Formalism and Gauge/Gravity duality The “philosophy” of the ISM formalism is to do phenomenology of irreversible transient phenomena using kinetic moments as independent variables even beyond the weak coupling regime where any kinetic theory can be constructed. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 21 / 29

  64. Beyond Kinetic Theory The Israel-Stewart-Muller Formalism and Gauge/Gravity duality The “philosophy” of the ISM formalism is to do phenomenology of irreversible transient phenomena using kinetic moments as independent variables even beyond the weak coupling regime where any kinetic theory can be constructed. The formalism employs the restriction that one can construct an entropy current of the form su µ whose local divergence is positive definite. This formalism also restricts second order hydrodynamics, as the higher moments have solutions which are “purely hydrodynamic” in nature. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 21 / 29

  65. Beyond Kinetic Theory The Israel-Stewart-Muller Formalism and Gauge/Gravity duality The “philosophy” of the ISM formalism is to do phenomenology of irreversible transient phenomena using kinetic moments as independent variables even beyond the weak coupling regime where any kinetic theory can be constructed. The formalism employs the restriction that one can construct an entropy current of the form su µ whose local divergence is positive definite. This formalism also restricts second order hydrodynamics, as the higher moments have solutions which are “purely hydrodynamic” in nature. However, the “regularity condition” in gravity is equivalent to an exact microscopic description, so we should not expect irreversibility at time scales less than the decoherence time scale or the relaxation time scale. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 21 / 29

  66. Beyond Kinetic Theory The Israel-Stewart-Muller Formalism and Gauge/Gravity duality The “philosophy” of the ISM formalism is to do phenomenology of irreversible transient phenomena using kinetic moments as independent variables even beyond the weak coupling regime where any kinetic theory can be constructed. The formalism employs the restriction that one can construct an entropy current of the form su µ whose local divergence is positive definite. This formalism also restricts second order hydrodynamics, as the higher moments have solutions which are “purely hydrodynamic” in nature. However, the “regularity condition” in gravity is equivalent to an exact microscopic description, so we should not expect irreversibility at time scales less than the decoherence time scale or the relaxation time scale. The second order hydrodynamic behaviour obtained from gauge/gravity duality has been shown to violate the ISM formalism [Baier, et al; Loganayagam, etc] Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 21 / 29

  67. Beyond Kinetic Theory The Israel-Stewart-Muller Formalism and Gauge/Gravity duality The “philosophy” of the ISM formalism is to do phenomenology of irreversible transient phenomena using kinetic moments as independent variables even beyond the weak coupling regime where any kinetic theory can be constructed. The formalism employs the restriction that one can construct an entropy current of the form su µ whose local divergence is positive definite. This formalism also restricts second order hydrodynamics, as the higher moments have solutions which are “purely hydrodynamic” in nature. However, the “regularity condition” in gravity is equivalent to an exact microscopic description, so we should not expect irreversibility at time scales less than the decoherence time scale or the relaxation time scale. The second order hydrodynamic behaviour obtained from gauge/gravity duality has been shown to violate the ISM formalism [Baier, et al; Loganayagam, etc] In our extrapolation of conservative solutions to the proposal for the “regularity condition” in gravity we will not restrict ourselves to the tenets of the ISM formalism but follow its broader “philosophy”. We will show how this extrapolation can be done unambiguously and systematically. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 21 / 29

  68. Beyond Kinetic Theory The Israel-Stewart-Muller Formalism and Gauge/Gravity duality The “philosophy” of the ISM formalism is to do phenomenology of irreversible transient phenomena using kinetic moments as independent variables even beyond the weak coupling regime where any kinetic theory can be constructed. The formalism employs the restriction that one can construct an entropy current of the form su µ whose local divergence is positive definite. This formalism also restricts second order hydrodynamics, as the higher moments have solutions which are “purely hydrodynamic” in nature. However, the “regularity condition” in gravity is equivalent to an exact microscopic description, so we should not expect irreversibility at time scales less than the decoherence time scale or the relaxation time scale. The second order hydrodynamic behaviour obtained from gauge/gravity duality has been shown to violate the ISM formalism [Baier, et al; Loganayagam, etc] In our extrapolation of conservative solutions to the proposal for the “regularity condition” in gravity we will not restrict ourselves to the tenets of the ISM formalism but follow its broader “philosophy”. We will show how this extrapolation can be done unambiguously and systematically. Interestingly, Ilya Prigogine also made an attempt to rewrite exact microscopic Hamiltonian dynamics in a “proto-thermodynamic” language. So our approach also conforms with his vision. In fact, it could be the first instance, where his vision could be concretely formulated and understood. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 21 / 29

  69. The Full Universal Sector as Defined by Pure Gravity A Proposal for the Regularity Condition in Gravity (1/2) It is natural to identify our conservative solutions at weak coupling with the universal sector at strong coupling and large N as that will explain why all observables get determined by the energy-momentum tensor alone. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 22 / 29

  70. The Full Universal Sector as Defined by Pure Gravity A Proposal for the Regularity Condition in Gravity (1/2) It is natural to identify our conservative solutions at weak coupling with the universal sector at strong coupling and large N as that will explain why all observables get determined by the energy-momentum tensor alone. Now the “conservative” condition on the energy-momentum tensor becomes the “regularity” condition in gravity such that the dual solutions have smooth future horizons. So, on top of the conservation equation ∂ µ [( π T ) 4 ( 4 u µ u ν + η µν )+ π µν ] = 0 which gives us the forced Euler equation with π µν as an independent variable; the regularity condition must involve five independent equations which tells us how any analytic initial data on π µν evolves with time. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 22 / 29

  71. The Full Universal Sector as Defined by Pure Gravity A Proposal for the Regularity Condition in Gravity (1/2) It is natural to identify our conservative solutions at weak coupling with the universal sector at strong coupling and large N as that will explain why all observables get determined by the energy-momentum tensor alone. Now the “conservative” condition on the energy-momentum tensor becomes the “regularity” condition in gravity such that the dual solutions have smooth future horizons. So, on top of the conservation equation ∂ µ [( π T ) 4 ( 4 u µ u ν + η µν )+ π µν ] = 0 which gives us the forced Euler equation with π µν as an independent variable; the regularity condition must involve five independent equations which tells us how any analytic initial data on π µν evolves with time. When π µν is given in terms of hydrodynamic variables only, we will have “normal” solutions of the microscopic theory and the gravity duals at strong coupling could be easily identified with the “tubewise black brane solutions” found by Bhattachaya, et al. In a radial tube from every point at the boundary these solutions can be parametrised by the local hydrodynamic variables which from the gravity viewpoint are the Goldstone-like fields corresponding to boost and scale invariance, the maximally commuting broken symmetries present in the asymptotic geometry. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 22 / 29

  72. The Full Universal Sector as Defined by Pure Gravity A Proposal for the Regularity Condition in Gravity (1/2) It is natural to identify our conservative solutions at weak coupling with the universal sector at strong coupling and large N as that will explain why all observables get determined by the energy-momentum tensor alone. Now the “conservative” condition on the energy-momentum tensor becomes the “regularity” condition in gravity such that the dual solutions have smooth future horizons. So, on top of the conservation equation ∂ µ [( π T ) 4 ( 4 u µ u ν + η µν )+ π µν ] = 0 which gives us the forced Euler equation with π µν as an independent variable; the regularity condition must involve five independent equations which tells us how any analytic initial data on π µν evolves with time. When π µν is given in terms of hydrodynamic variables only, we will have “normal” solutions of the microscopic theory and the gravity duals at strong coupling could be easily identified with the “tubewise black brane solutions” found by Bhattachaya, et al. In a radial tube from every point at the boundary these solutions can be parametrised by the local hydrodynamic variables which from the gravity viewpoint are the Goldstone-like fields corresponding to boost and scale invariance, the maximally commuting broken symmetries present in the asymptotic geometry. We propose the regularity condition as the most general equation for π µν which can reproduce the correct purely hydrodynamic energy-momentum tensor known exactly upto second order in derivatives as a special solution. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 22 / 29

  73. The Full Universal Sector as Defined by Pure Gravity A Proposal for the Regularity Condition in Gravity (2/2) Therefore, our regularity condition for pure gravity in AdS 5 is: � ( u · ∂ ) π µν + 4 � � π µβ u ν + π νβ u µ � 3 π µν ( ∂ · u ) − ( 1 − λ 3 ) ( u · ∂ ) u β (23) 2 π T ( 2 − ln2 )[ π µν + 2 ( π T ) 3 σ µν = − � ( u · ∂ ) σ µν + 1 � � u ν σ µβ + u µ σ νβ � − λ 3 ( 2 − ln2 )( π T ) 2 3 σ µν ( ∂ · u ) − ( u · ∂ ) u β − λ 4 ( ln2 )( π T ) 2 ( σ αµ ω ν α + σ αµ ω ν α ) � α − 1 � − 2 λ 1 ( π T ) 2 σ αµ σ ν 3 P µν σ αβ σ αβ ] ln 2 α ω αν + π ν ( 2 − ln2 )( π µ α ω αµ ) − ( 1 − λ 4 ) 2 λ 2 � 1 α ) − 1 � 2 ( π µα σ ν α + π να σ µ 3 P µν π αβ σ αβ − ( 2 − ln2 ) 1 − λ 1 − λ 2 � α − 1 � π µα π ν 3 P µν π αβ π αβ + + ( 2 − ln2 )( π T ) 3 � π 3 ,π∂π,∂ 2 π,π 2 ∂ u ,π∂ 2 u ,∂ 2 π,∂ 3 u , ( ∂ u )( ∂ 2 u ) , ( ∂ u ) 3 � O where the O ( π 3 ,π∂π,... ) term indicates that the corrections to our proposal can include terms of the structures displayed or those with more derivatives or containing more powers of π µν or both only. Also, the four λ i ’s ( i = 1 , 2 , 3 , 4) are pure numbers. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 23 / 29

  74. The Full Universal Sector as Defined by Pure Gravity A Simple Prediction of our Proposal A simple predction of our proposal is that the universal sector consists of three branches of linearized fluctuations which could be easily identified with their weak coupling counterparts in the conservative solutions. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 24 / 29

  75. The Full Universal Sector as Defined by Pure Gravity A Simple Prediction of our Proposal A simple predction of our proposal is that the universal sector consists of three branches of linearized fluctuations which could be easily identified with their weak coupling counterparts in the conservative solutions. By the logic of our proposal, two of the three branches consists of the hydrodynamic sound and shear branches. It can be shown that these branches are exactly the same as in the quasinormal mode spectrum obtained by solving the linearized fluctuations of the black brane with infalling boundary condition at the horizon. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 24 / 29

  76. The Full Universal Sector as Defined by Pure Gravity A Simple Prediction of our Proposal A simple predction of our proposal is that the universal sector consists of three branches of linearized fluctuations which could be easily identified with their weak coupling counterparts in the conservative solutions. By the logic of our proposal, two of the three branches consists of the hydrodynamic sound and shear branches. It can be shown that these branches are exactly the same as in the quasinormal mode spectrum obtained by solving the linearized fluctuations of the black brane with infalling boundary condition at the horizon. The third branch in the spectrum can be seen as follows. We put all the hydrodynamic variables u µ and T to be spatio-temporal constants and also the flow at rest so that u µ = ( 1 , 0 , 0 , 0 ) . Then the conservation of energy-momentum tensor requires ∂ µ π µν = 0. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 24 / 29

  77. The Full Universal Sector as Defined by Pure Gravity A Simple Prediction of our Proposal A simple predction of our proposal is that the universal sector consists of three branches of linearized fluctuations which could be easily identified with their weak coupling counterparts in the conservative solutions. By the logic of our proposal, two of the three branches consists of the hydrodynamic sound and shear branches. It can be shown that these branches are exactly the same as in the quasinormal mode spectrum obtained by solving the linearized fluctuations of the black brane with infalling boundary condition at the horizon. The third branch in the spectrum can be seen as follows. We put all the hydrodynamic variables u µ and T to be spatio-temporal constants and also the flow at rest so that u µ = ( 1 , 0 , 0 , 0 ) . Then the conservation of energy-momentum tensor requires ∂ µ π µν = 0. The conservation of the energy-momentum tensor can be achieved if we put π 00 = π 0 i = 0 and π ij ’s are functions of time t only. The linearized equation π µν is solved if π ij = A ij exp ( − t /τ π ) with τ π = ( 1 − λ 3 )( 2 − ln 2 ) and A ij is a spatio-temporal constant and traceless matrix. 2 π T Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 24 / 29

  78. The Full Universal Sector as Defined by Pure Gravity A Simple Prediction of our Proposal A simple predction of our proposal is that the universal sector consists of three branches of linearized fluctuations which could be easily identified with their weak coupling counterparts in the conservative solutions. By the logic of our proposal, two of the three branches consists of the hydrodynamic sound and shear branches. It can be shown that these branches are exactly the same as in the quasinormal mode spectrum obtained by solving the linearized fluctuations of the black brane with infalling boundary condition at the horizon. The third branch in the spectrum can be seen as follows. We put all the hydrodynamic variables u µ and T to be spatio-temporal constants and also the flow at rest so that u µ = ( 1 , 0 , 0 , 0 ) . Then the conservation of energy-momentum tensor requires ∂ µ π µν = 0. The conservation of the energy-momentum tensor can be achieved if we put π 00 = π 0 i = 0 and π ij ’s are functions of time t only. The linearized equation π µν is solved if π ij = A ij exp ( − t /τ π ) with τ π = ( 1 − λ 3 )( 2 − ln 2 ) and A ij is a spatio-temporal constant and traceless matrix. 2 π T We note that the third branch is the branch that contains the above mode ω = − i τ − 1 π , k = 0. This mode at weak coupling was associated with relaxation or local equilibriation in the quasiparticle-velocity space, so we will call this branch as the relaxation branch. Such a branch is not present in the quasinormal mode spectrum. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 24 / 29

  79. The Full Universal Sector as Defined by Pure Gravity A Simple Prediction of our Proposal A simple predction of our proposal is that the universal sector consists of three branches of linearized fluctuations which could be easily identified with their weak coupling counterparts in the conservative solutions. By the logic of our proposal, two of the three branches consists of the hydrodynamic sound and shear branches. It can be shown that these branches are exactly the same as in the quasinormal mode spectrum obtained by solving the linearized fluctuations of the black brane with infalling boundary condition at the horizon. The third branch in the spectrum can be seen as follows. We put all the hydrodynamic variables u µ and T to be spatio-temporal constants and also the flow at rest so that u µ = ( 1 , 0 , 0 , 0 ) . Then the conservation of energy-momentum tensor requires ∂ µ π µν = 0. The conservation of the energy-momentum tensor can be achieved if we put π 00 = π 0 i = 0 and π ij ’s are functions of time t only. The linearized equation π µν is solved if π ij = A ij exp ( − t /τ π ) with τ π = ( 1 − λ 3 )( 2 − ln 2 ) and A ij is a spatio-temporal constant and traceless matrix. 2 π T We note that the third branch is the branch that contains the above mode ω = − i τ − 1 π , k = 0. This mode at weak coupling was associated with relaxation or local equilibriation in the quasiparticle-velocity space, so we will call this branch as the relaxation branch. Such a branch is not present in the quasinormal mode spectrum. So, our proposal predicts that we should have such a linearized regular perturbation of the black brane unless λ 3 = 1. If λ 3 = 1, to the orders that we have written our equation describing regular non-linear perturbations about global equilibrium we have two possibilities, firstly there is a strange kind of gapless zero mode or there is no third branch at all and the non-hydrodynamic solutions arise only when the full non-linear equation is considered. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 24 / 29

  80. The Full Universal Sector as Defined by Pure Gravity Consistency Checks of Our Proposal (1/2) The internal consistency of our proposal can be checked, by determining λ ’s by various independent means. Here we will first look at two independent means of determining λ 3 and λ 1 + λ 2 . The two independent means will be considering two different kinds of fluctuations about the linearized homogenous non-hydrodynamic solution we discussed before. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 25 / 29

  81. The Full Universal Sector as Defined by Pure Gravity Consistency Checks of Our Proposal (1/2) The internal consistency of our proposal can be checked, by determining λ ’s by various independent means. Here we will first look at two independent means of determining λ 3 and λ 1 + λ 2 . The two independent means will be considering two different kinds of fluctuations about the linearized homogenous non-hydrodynamic solution we discussed before. Configration 1 The hydrodynamic variables are not perturbed, so still are spatio-temporal constants. We keep π 00 = π oi = 0 and maintain spatial translational invariance so that π ij is still a function of time only. Therefore, ∂ µ π µν = 0. π ij obeys the following equation of motion which is exact upto third order terms. ( 1 − λ 3 ) d π ij 2 π T dt = − ( 2 − ln 2 ) π ij (24) ( 2 − ln 2 )( π T ) 3 ( π ik π kj − 1 1 − λ 1 − λ 2 + 3 δ ij π lm π lm ) + O ( d 2 π dt 2 , d π dt π,π 3 ) ′ defined as ( τ/ T ) , where τ is the relaxation time and T is the The expansion parameters are ǫ typical time scale of variation of the solution; and δ defined as | π ij | / ( π T ) 4 . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 25 / 29

  82. The Full Universal Sector as Defined by Pure Gravity Consistency Checks of Our Proposal (1/2) The internal consistency of our proposal can be checked, by determining λ ’s by various independent means. Here we will first look at two independent means of determining λ 3 and λ 1 + λ 2 . The two independent means will be considering two different kinds of fluctuations about the linearized homogenous non-hydrodynamic solution we discussed before. Configration 1 The hydrodynamic variables are not perturbed, so still are spatio-temporal constants. We keep π 00 = π oi = 0 and maintain spatial translational invariance so that π ij is still a function of time only. Therefore, ∂ µ π µν = 0. π ij obeys the following equation of motion which is exact upto third order terms. ( 1 − λ 3 ) d π ij 2 π T dt = − ( 2 − ln 2 ) π ij (24) ( 2 − ln 2 )( π T ) 3 ( π ik π kj − 1 1 − λ 1 − λ 2 + 3 δ ij π lm π lm ) + O ( d 2 π dt 2 , d π dt π,π 3 ) ′ defined as ( τ/ T ) , where τ is the relaxation time and T is the The expansion parameters are ǫ typical time scale of variation of the solution; and δ defined as | π ij | / ( π T ) 4 . Incidentally, we also find how in this case gravity may reproduce quantum coherent behaviour as opposed to the explicitly irreversible case of the Boltzmann equation as a d 2 π/ dt 2 may appear at ǫ ′ 2 δ order. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 25 / 29

  83. The Full Universal Sector as Defined by Pure Gravity Consistency checks of our proposal (2/2) Configration 2 The temperature is not perturbed, so is still a spatio-temporal constant. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 26 / 29

  84. The Full Universal Sector as Defined by Pure Gravity Consistency checks of our proposal (2/2) Configration 2 The temperature is not perturbed, so is still a spatio-temporal constant. We consider linearized fluctuations of u µ and π µν . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 26 / 29

  85. The Full Universal Sector as Defined by Pure Gravity Consistency checks of our proposal (2/2) Configration 2 The temperature is not perturbed, so is still a spatio-temporal constant. We consider linearized fluctuations of u µ and π µν . The expansion parameters are ǫ = ( 1 / ( L T )) where L is the spatio-temporal scale of variation in the solution and δ = | π 0 µν | / ( π T ) 4 . Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 26 / 29

  86. The Full Universal Sector as Defined by Pure Gravity Consistency checks of our proposal (2/2) Configration 2 The temperature is not perturbed, so is still a spatio-temporal constant. We consider linearized fluctuations of u µ and π µν . The expansion parameters are ǫ = ( 1 / ( L T )) where L is the spatio-temporal scale of variation in the solution and δ = | π 0 µν | / ( π T ) 4 . The linearized equations exact at orders ǫ , δ and ǫδ are � ∂ µ δπ µν � ( u 0 .∂ ) δ u ν + u 0 ν ( ∂.δ u ) = (25) 4 ( π T ) 4 � + 4 � ( 1 − λ 3 )( u 0 .∂ ) δπ µν = − ( 1 − λ 3 ) ( δ u .∂ ) π µν 3 π µν 0 ( ∂.δ u ) − ( π µβ 0 + π µβ 0 u ν 0 u ν 0 )( u 0 .∂ ) δ u β 0 2 π T � δπ µν + 2 ( π T ) 3 σ µν � − ( 2 − ln 2 ) ln 2 π µα � α + π να ν 0 ω µ � − ( 1 − λ 4 ) 0 ω α ( 2 − ln 2 ) ( 2 − ln 2 )( 1 2 λ 2 − 1 � π µα 0 σ ν α + π µα 0 σ ν � 3 P µν 0 π αβ − 0 σ αβ ) α 2 1 − λ 1 − λ 2 � � α − 1 0 π 0 αβ − 2 0 δ u ν + u ν π µα 0 δπ ν α + π να 0 δπ µ 3 ( u µ 0 δ u µ ) π αβ 3 P µν 0 π αβ + 0 δπ αβ ( 2 − ln 2 )( π T ) 3 where, π µν is as in the basic configuration and 0 σ µν = 1 0 ( ∂ α δ u β + ∂ β δ u α ) − 1 2 P µα 0 P νβ 3 P µν 0 ( ∂.δ u ) (26) l µ = ǫ αβγµ u α 0 ∂ β δ u γ One may readily see from the first equation in (25) that ∂.δ u = 0 (27) Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 26 / 29

  87. The Full Universal Sector as Defined by Pure Gravity Consistency checks of our proposal (2/2) Configration 2 The temperature is not perturbed, so is still a spatio-temporal constant. We consider linearized fluctuations of u µ and π µν . The expansion parameters are ǫ = ( 1 / ( L T )) where L is the spatio-temporal scale of variation in the solution and δ = | π 0 µν | / ( π T ) 4 . The linearized equations exact at orders ǫ , δ and ǫδ are � ∂ µ δπ µν � ( u 0 .∂ ) δ u ν + u 0 ν ( ∂.δ u ) = (25) 4 ( π T ) 4 � + 4 � ( 1 − λ 3 )( u 0 .∂ ) δπ µν = − ( 1 − λ 3 ) ( δ u .∂ ) π µν 3 π µν 0 ( ∂.δ u ) − ( π µβ 0 + π µβ 0 u ν 0 u ν 0 )( u 0 .∂ ) δ u β 0 2 π T � δπ µν + 2 ( π T ) 3 σ µν � − ( 2 − ln 2 ) ln 2 π µα � α + π να ν 0 ω µ � − ( 1 − λ 4 ) 0 ω α ( 2 − ln 2 ) ( 2 − ln 2 )( 1 2 λ 2 − 1 � π µα 0 σ ν α + π µα 0 σ ν � 3 P µν 0 π αβ − 0 σ αβ ) α 2 1 − λ 1 − λ 2 � � α − 1 0 π 0 αβ − 2 0 δ u ν + u ν π µα 0 δπ ν α + π να 0 δπ µ 3 ( u µ 0 δ u µ ) π αβ 3 P µν 0 π αβ + 0 δπ αβ ( 2 − ln 2 )( π T ) 3 where, π µν is as in the basic configuration and 0 σ µν = 1 0 ( ∂ α δ u β + ∂ β δ u α ) − 1 2 P µα 0 P νβ 3 P µν 0 ( ∂.δ u ) (26) l µ = ǫ αβγµ u α 0 ∂ β δ u γ One may readily see from the first equation in (25) that ∂.δ u = 0 (27) One can determine the other λ ’s through coefficients of various non-linear terms as well and further check consistency. Ayan Mukhopadhyay (HRI) Universal Phenomena at Strong Coupling and Gravity September 28, 2009 26 / 29

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