PART 2 Formalism of relativistic ideal/viscous hydrodynamics.
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Transcript of PART 2 Formalism of relativistic ideal/viscous hydrodynamics.
PART 2
Formalism ofrelativistic ideal/viscous
hydrodynamics
Advertisement: Lecture NotesChapter 4Chapter 4
Hydrodynamics
• Framework to describe space-time evolution of thermodynamic variables
• Balance equations (equations of motion, conservation law)
+ equation of state (matter property) + constitutive equations (phenomenology)
Relativistic Hydrodynamics
Energy-momentum conservation
Current conservationEnergy-Momentum tensor
The i-th conserved current
In H.I.C., Ni = NB
(net baryon current)
Equations of motion in relativistic hydrodynamics
Tensor/Vector Decomposition
Tensor decomposition with a given time-likeand normalized four-vector u
where,
“Projection” Tensor/Vector
•u is perpendicular to .
•u is local four flow velocity. More precisemeaning will be given later.
•Naively speaking, u () picks up time-(space-)like component(s).
•Local rest frame (LRF):
Intuitive Picture of Projection
time likeflow vectorfield
Decomposition of T
:Energy density
:(Hydrostatic+bulk) pressure P = Ps +
:Energy (Heat) current
:Shear stress tensor
<…>: Symmetric, traceless and transverse to u & u
Decomposition of N
:charge density
:charge currentQ. Count the number of unknowns
in the above decomposition and confirmthat it is 10(T)+4k(Ni
).Here k is the number of independent currents.
Note: If you consider u as independent variables,you need additional constraint for them. If you also consider Ps as an independent
variable, you need the equation of state Ps=Ps(e,n).
Ideal and Dissipative PartsEnergy Momentum tensor
Charge current
Ideal part
Dissipative part
Meaning of u
u is four-velocity of “flow”. What kind of flow?Two major definitions of flow are
2. Flow of conserved charge (Eckart)
1. Flow of energy (Landau)
Meaning of u (contd.)Landau
(W=0, uLV=0)
Eckart(V=0,uE
W=0)
uL
V
uE
W
Just a choice of local reference frame.Landau frame can be relevant in H.I.C.
Relation btw. Landau and Eckart
Relation btw. Landau and Eckart (contd.)
Entropy Conservationin Ideal Hydrodynamics
Neglect “dissipative part” of energy momentumtensor to obtain “ideal hydrodynamics”.
Q. Derive the above equation.
Therefore,
Assumption (1st order theory):Non-equilibrium entropy current vector haslinear dissipative term(s) constructed from(V, , , (u).
Entropy Current
Thus, = 0 since N = 0, W = 0 since consideringthe Landau frame, and = 0 since u S shouldbe maximum in equilibrium (stability condition).
(Practical) Assumption:•Landau frame (omitting subscript “L”).•No charge in the system.
The 2nd Law of Thermodynamics and Constitutive Equations
The 2nd thermodynamic law tells us
Q. Check the above calculation.
Constitutive Equations (contd.)
Thermodynamic force
Transport
coefficient“Current”
tensor shear
scalar bulk
Newton
Stokes
Equation of Motion
: Expansion scalar (Divergence)
: Lagrange (substantial) derivative
Equation of Motion (contd.)
Q. Derive the above equations of motion from energy-momentum conservation.
Note: We have not used the constitutive equations to obtain the equations of motion.
Intuitive Interpretation of EoM
Work done by pressure
Production of entropy
Change of volume•Dilution•Compression
Conserved Current Case
Lessons from (Non-Relativistic) Navier-Stokes Equation
Assuming incompressible fluids such that , Navier-Stokes eq. becomes
Final flow velocity comes from interplaybetween these two effects.
Source of flow (pressure gradient)
Diffusion of flow(Kinematic viscosity, /,plays a role of diffusionconstant.)
Generation of Flow
x
P
ExpandExpand
Pressure gradient
Source of flow
Flow phenomenaare important in H.I.Cto understand EOS
Diffusion of Flow
Heat equation(: heat conductivity~diffusion constant)
For illustrative purpose, one discretizes the equation in (2+1)D space:
Diffusion ~ Averaging ~ Smoothing
R.H.S. of descretized heat/diffusion eq.
i
j
i
j
x x
y y
subtract
Suppose i,j is larger (smaller) than an average value around the site, R.H.S. becomes negative (positive).2nd derivative w.r.t. coordinates Smoothing
Shear Viscosity Reduces Flow Difference
Shear flow(gradient of flow) Smoothing of flow
Next time stepMicroscopic interpretation can be made.Net momentum flow in space-like direction. Towards entropy maximum state.
Necessity of Relaxation TimeNon-relativistic case (Cattaneo(1948))
Fourier’s law
: “relaxation time”
Parabolic equation (heat equation) ACAUSAL! Finite Hyperbolic equation (telegraph equation)
Balance eq.:
Constitutive eq.:
Heat Kernel
x x
causalityperturbationon top of
background
Heattransportation
Entropy Current (2nd)Assumption (2nd order theory):Non-equilibrium entropy current vector haslinear + quadratic dissipative term(s)constructed from (V, , , (u)).
Stability condition O.K.
The 2nd Law of Thermodynamics: 2nd order case
Same equation, but different definition of and .
Sometimesomitted, but needed.
Generalization of thermodynamic force!?
Summary:Constitutive Equations
•Relaxation terms appear ( and are relaxation time).•No longer algebraic equations! Dissipative currents become dynamical quantities like thermodynamic variables.•Employed in recent viscous fluid simulations. (Sometimes the last term is neglected.)
: vorticity
PART 3
Bjorken’s Scaling Solution with Viscosity
“Bjorken” Coordinate
0z
t
Boost parallel shiftBoost invariant Independent of s
Bjorken’s Scaling Solution
Hydrodynamic equation for perfect fluids with a simple EoS,
Assuming boost invariance for thermodynamic variables P=P() and 1D Hubble-like flow
Conserved and Non-Conserved Quantity in Scaling Solution
expansion
pdV work
Bjorken’s Equation in the 1st Order Theory
(Bjorken’s solution) = (1D Hubble flow)
Q. Derive the above equation.
Viscous Correction
Correction from shear viscosity(in compressible fluids)
Correction frombulk viscosity
If these corrections vanish, the above equationreduces to the famous Bjorken equation.Expansion scalar = theta = 1/tau in scaling solution
• is obtained from Super Yang-Mills theory.
• is obtained from lattice.
•Bulk viscosity has a prominent peak around Tc.
Recent Topics on Transport Coefficients
Need microscopic theory (e.g., Boltzmann eq.)to obtain transport coefficients.
Kovtun, Son, Starinet,…
Nakamura, Sakai,…
Kharzeev, Tuchin, Karsch, Meyer…
Bjorken’s Equationin the 2nd Order Theory
New terms appear in the 2nd order theory. Coupled differential equationsSometimes, the last terms are neglected.Importance of these terms
where
Why only 00-zz?In EoM of energy density,
appears in spite of constitutive equations.According to the Bjorken solution,
Relaxation Equation?
Digression: Full 2nd order equation?
Beyond I-S equation, see R.Baier et al., JHEP 0804,100 (2008); Tsumura-Kunihiro; D. Rischke, talk at SQM 2008; A.Monnai and TH, in preparation. According to Rischke’s talk, constitutive equations with vanishing heat flow are
Digression (contd.): Bjorken’s Equationin the “full” 2nd order theory
See also, R.Fries et al.,PRC78,034913(2008). Note that the equation for shear is valid only for conformalEOS and that no 2nd and 3rd terms for bulk.
Model EoS (crossover)
Crossover EoS: Tc = 0.17GeV = Tc/50gH = 3, gQ = 37
Relativistic Ideal Gas
Thermodynamic potential for relativistic ideal gases
Energy-Momentum Tensorat 0 in Comoving Frame
In what follows, bulk viscosity is omitted.
Numerical Results (Temperature)
Same initial condition(Energy momentumtensor is isotropic)
T0 = 0.22 GeV0 = 1 fm/c/s = 1/4 = 3/4p
Numerical code (C++) is available upon request.
Numerical Results (Temperature)
Same initial condition(Energy momentumtensor is anisotropic)
T0 = 0.22 GeV0 = 1 fm/c/s = 1/4 = 3/4p
Numerical code (C++) is available upon request.
Numerical Results (Temperature)
Numerical code (C++) is available upon request.
Numerical Results (Entropy)
T0 = 0.22 GeV0 = 1 fm/c/s = 1/4 = 3/4p
Same initial condition(Energy momentumtensor is isotropic)
Numerical code (C++) is available upon request.
Numerical Results (Entropy)
T0 = 0.22 GeV0 = 1 fm/c/s = 1/4 = 3/4p
Same initial condition(Energy momentumtensor is anisotropic)
Numerical code (C++) is available upon request.
Numerical Results (Entropy)
Numerical code (C++) is available upon request.
Numerical Results(Shear Viscosity)
Numerical code (C++) is available upon request.
Numerical Results(Initial Condition Dependence
in the 2nd order theory)
Numerical code (C++) is available upon request.
Numerical Results(Relaxation Time dependence)
Saturated valuesnon-trivial
Relaxation time largerMaximum is smallerRelaxation time smallerSuddenly relaxes to 1st order theory
Remarks
• Sometimes results from ideal hydro are compared with the ones from 1st order theory. But initial conditions must be different.
• Be careful what is attributed for the difference between two results.
• Sensitive to initial conditions and new parameters (relaxation time for stress tensor)
PART 4
Effect of Viscosity on Particle Spectra
Particle Spectra in Hydrodynamic Model
• How to compare with experimental data (particle spectra)?
• Free particles (/L>>1) eventually stream to detectors.
• Need prescription to convert hydrodynamic (thermodynamic) fields (/L<<1) into particle picture.
• Need kinetic (or microscopic) interpretation of hydrodynamic behavior.
Cooper-Frye Formula
•No dynamics of evaporation.•Just counting the net number of particles (out-going particles) - (in-coming particles) through hypersurface •Negative contribution can appear at some space-like hyper surface elements.
Matter in (Kinetic) Equilibrium
u
Kinetically equilibratedmatter at rest
Kinetically equilibratedmatter at finite velocity
px
py
px
py
Isotropic distribution Lorentz-boosted distribution
Relativistic Transport Theory
Boltzmann equationTime evolution of phase space dist. for a rarefied gas:
One-particle phase space distributionCollision termLorentz invariant transition rate
gain loss
Free Streaming, Gain and Loss
x
p
Note: p is mass-on-shell.
gain
loss
loss
gain
H-theorem“Entropy current”:
H-theorem (contd.)
H-theorem (cond.2)
H-theorem (contd.3)
=0
H-theorem (contd.4)
1
1
0
One can identifys with entropy current
Collision InvariantBinary collisions satisfy energy-momentum consevation
So
where(collision invariant)
Collision Invariant (contd.)
Q. Check this identity
Conservation law
5 conservedcurrents(quantities)
Thus should be collision invariant
Equilibrium Distribution
and can depend on position xand are to be /T and u/T
Quantum StatisticsBose enhancement or Pauli blocking
Collision invariant
Some Remarks
• Collision term vanishes if f0 is plugged in.
• If and is constant globally, f0 can be a solution of Boltzmann equation.
• Local equilibrium distribution is NOT a solution of Boltzmann equation.
• Deviation from local equilibrium distribution can be obtained from Boltzmann equation.
Microscopic Interpretation
Single particle phase space density in local equilibrium (no entropy production in Boltzmann eq.):
Kinetic definition of current and energy momentumtensor are
1st Moment
u is normalized, so we can always choose a such that
1st Moment (contd.)
Vanishing for = i due to odd function in integrant.
Q. Go through all steps in the above derivation.
2nd Moment
2nd Moment (contd.)
where
Deviation from Equilibrium Distribution
trace part scalar
Unknowns: 4 + 10 = 14
Important in a multi-component gas(A.Monnai and TH, (2009))
Taylor Expansion around Equilibrium Distribution
Taylor Expansion around Equilibrium Distribution (contd.)
14 Conditions
Stability conditions (2)
Viscosities (12) Epsilons can beexpressed bydissipative currents
Stability Condition
Q. Check the above derivation.
Stability Condition (contd.)
Relation btw. Coefficients and Dissipative Currents
Moments
Bosons do not contribute…
Solutions
Solutions (contd.)
Epsilons are expressed using hydrodynamic variablesAble to calculate particle spectra
Cooper-Frye Formulain Viscous Case
Transverse Momentum Spectra for Pions
Bulk pressure Isotropic = -Shear stress tensor Traceless = 4/3
Summary
• Effect of viscosity on particle spectra can be calculated using hydrodynamic variables.
• Important to constrain equation of state and transport coefficients from experimental data