Transcript Hydrodynamics and Flow

PART 2
Formalism of
relativistic ideal/viscous
hydrodynamics
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Chapter 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
Equations of motion in relativistic hydrodynamics
Energy-momentum conservation
Energy-Momentum tensor
Current conservation
The i-th conserved current
In H.I.C., Nim = NBm (net baryon current)
Tensor/Vector Decomposition
Tensor decomposition with a given time-like
and normalized four-vector um
where,
“Projection” Tensor/Vector
•um is local four flow velocity. More precise
meaning will be given later.
•um is perpendicular to Dmn.
•Local rest frame (LRF):
•Naively speaking, um (Dmn) picks up time(space-)like component(s).
Intuitive Picture of Projection
time like
flow vector
field
Decomposition of Tmu
:Energy density
:Energy (Heat) current
:Shear stress tensor
:(Hydrostatic+bulk) pressure
P = Ps + P
<…>: Symmetric, traceless and transverse to um & un
Decomposition of Nm
:charge density
:charge current
Q. Count the number of unknowns
in the above decomposition and confirm
that it is 10(Tmn)+4k(Nim).
Here k is the number of independent currents.
Note: If you consider um 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 Parts
Energy Momentum tensor
Charge current
Ideal part
Dissipative part
Meaning of um
um is four-velocity of “flow”. What kind of flow?
Two major definitions of flow are
1. Flow of energy (Landau)
2. Flow of conserved charge (Eckart)
Meaning of um (contd.)
Landau
(Wm=0, uLmVm=0)
uLm
Vm
Eckart
(Vm=0,uEmWm=0)
uEm
Just a choice of local reference frame.
Landau frame can be relevant in H.I.C.
Wm
Relation btw.
Landau and Eckart
Relation btw.
Landau and Eckart (contd.)
Entropy Conservation
in Ideal Hydrodynamics
Neglect “dissipative part” of energy momentum
tensor to obtain “ideal hydrodynamics”.
Therefore,
Q. Derive the above equation.
Entropy Current
Assumption (1st order theory):
Non-equilibrium entropy current vector has
linear dissipative term(s) constructed from
(Vm, P, pmn, (um)).
(Practical) Assumption:
•Landau frame (omitting subscript “L”).
•No charge in the system.
Thus, a = 0 since Nm = 0, Wm = 0 since considering
the Landau frame, and g = 0 since um Sm should
be maximum in equilibrium (stability condition).
The 2nd Law of Thermodynamics
and Constitutive Equations
The 2nd thermodynamic law tells us
Q. Check the above calculation.
Constitutive Equations (contd.)
Newton
Stokes
Thermodynamic
force
Transport
coefficient
tensor
shear
scalar
bulk
“Current”
Equation of Motion
: Lagrange (substantial)
derivative
: Expansion scalar (Divergence)
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
Change of volume
•Dilution
•Compression
Work done by pressure
Production of entropy
Conserved Current Case
Lessons from (Non-Relativistic)
Navier-Stokes Equation
Assuming incompressible fluids such that
, Navier-Stokes eq. becomes
Diffusion of flow
Source of flow
(Kinematic viscosity, h/r,
(pressure gradient)
plays a role of diffusion
constant.)
Final flow velocity comes from interplay
between these two effects.
Generation of Flow
P
Pressure gradient
Expand
Source of flow
Expand
x
 Flow phenomena
are important in H.I.C
to understand EOS
Diffusion of Flow
Heat equation
(k: 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.
y
y
j
subtract
i
j
i
x
x
Suppose Ti,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 step
Microscopic interpretation can be made.
Net momentum flow in space-like direction.
 Towards entropy maximum state.
Necessity of Relaxation Time
Non-relativistic case (Cattaneo(1948))
Balance eq.:
Constitutive eq.:
Fourier’s law
t : “relaxation time”
Parabolic equation (heat equation)
ACAUSAL!
Finite t
Hyperbolic equation (telegraph equation)
Heat Kernel
x
perturbation
on top of
background
x
Heat
transportation
causality
Entropy Current (2nd)
Assumption (2nd order theory):
Non-equilibrium entropy current vector has
linear + quadratic dissipative term(s)
constructed from (Vm, P, pmn, (um)).
Stability condition O.K.
The 2nd Law of Thermodynamics:
2nd order case
Sometimes
omitted,
but needed.
 Generalization of thermodynamic force!?
Same equation, but different definition of p and P.
Summary:
Constitutive Equations
w: vorticity
•Relaxation terms appear (tp and tP 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.)
PART 3
Bjorken’s Scaling Solution
with Viscosity
“Bjorken” Coordinate
t
Boost  parallel shift
Boost invariant
 Independent of hs
z
0
Bjorken’s Scaling Solution
Assuming boost invariance for thermodynamic
variables P=P(t) and 1D Hubble-like flow
Hydrodynamic equation for perfect fluids with a
simple EoS,
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 from
bulk viscosity
 If these corrections vanish, the above equation
reduces to the famous Bjorken equation.
Expansion scalar = theta = 1/tau in scaling solution
Recent Topics on Transport
Coefficients
Need microscopic theory (e.g., Boltzmann eq.)
to obtain transport coefficients.
•
is obtained from
Super Yang-Mills theory.
Kovtun, Son, Starinet,…
•
is obtained from lattice.
Nakamura, Sakai,…
•Bulk viscosity has a prominent peak around Tc.
Kharzeev, Tuchin, Karsch, Meyer…
Bjorken’s Equation
in the 2nd Order Theory
where
New terms appear in the 2nd order theory.
 Coupled differential equations
Sometimes, the last terms are neglected.
Importance of these terms
Why only p00-pzz?
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 Equation
in the “full” 2nd order theory
See also, R.Fries et al.,PRC78,034913(2008).
Note that the equation for shear is valid only for conformal
EOS and that no 2nd and 3rd terms for bulk.
Model EoS (crossover)
Crossover EoS:
Tc = 0.17GeV
D = Tc/50
gH = 3, gQ = 37
Relativistic Ideal Gas
Thermodynamic potential for relativistic ideal gases
Energy-Momentum Tensor
at t0 in Comoving Frame
In what follows, bulk viscosity is omitted.
Numerical Results
(Temperature)
T0 = 0.22 GeV
t0 = 1 fm/c
h/s = 1/4p
tp = 3h/4p
Same initial
condition
(Energy momentum
tensor is isotropic)
Numerical code (C++) is available upon request.
Numerical Results
(Temperature)
T0 = 0.22 GeV
t0 = 1 fm/c
h/s = 1/4p
tp = 3h/4p
Same initial
condition
(Energy momentum
tensor is anisotropic)
Numerical code (C++) is available upon request.
Numerical Results
(Temperature)
Numerical code (C++) is available upon request.
Numerical Results (Entropy)
T0 = 0.22 GeV
t0 = 1 fm/c
h/s = 1/4p
tp = 3h/4p
Same initial
condition
(Energy momentum
tensor is isotropic)
Numerical code (C++) is available upon request.
Numerical Results (Entropy)
T0 = 0.22 GeV
t0 = 1 fm/c
h/s = 1/4p
tp = 3h/4p
Same initial
condition
(Energy momentum
tensor 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 values
non-trivial
Relaxation time larger
Maximum p is smaller
Relaxation time smaller
Suddenly 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/L>>1) eventually stream to
detectors.
• Need prescription to convert hydrodynamic
(thermodynamic) fields (l/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 S
•Negative contribution can appear at some
space-like hyper surface elements.
Matter in (Kinetic) Equilibrium
Kinetically equilibrated
matter at rest
Kinetically equilibrated
matter at finite velocity
um
py
py
px
Isotropic distribution
px
Lorentz-boosted distribution
Relativistic Transport Theory
Boltzmann equation
Time evolution of phase space dist. for a rarefied gas:
gain
One-particle phase space distribution
Collision term
Lorentz invariant transition rate
loss
Free Streaming, Gain and Loss
p
loss gain
gain
loss
x
Note: p is mass-on-shell.
H-theorem
“Entropy current”:
H-theorem (contd.)
H-theorem (cond.2)
H-theorem (contd.3)
=0
H-theorem (contd.4)
1
0
1
One can identify
sm with entropy current
Collision Invariant
Binary collisions satisfy energy-momentum consevation
So
where
(collision invariant)
Collision Invariant (contd.)
Q. Check this identity
Conservation law
5 conserved
currents
(quantities)
Equilibrium Distribution
Thus
should be collision invariant
a and bm can depend on position x
and are to be m/T and um/T
Quantum Statistics
Bose enhancement or Pauli blocking
Collision invariant
Some Remarks
• Collision term vanishes if f0 is plugged in.
• If a and b 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 momentum
tensor are
1st Moment
um is normalized, so we can always choose amn such that
1st Moment (contd.)
Vanishing for n= 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
Important in a multi-component gas
(A.Monnai and TH, (2009))
Unknowns: 4 + 10 = 14
Taylor Expansion around
Equilibrium Distribution
Taylor Expansion around
Equilibrium Distribution (contd.)
14 Conditions
Stability conditions (2)
Viscosities (12)
Epsilons can be
expressed by
dissipative
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 variables
Able to calculate particle spectra
Cooper-Frye Formula
in Viscous Case
Transverse Momentum Spectra
for Pions
Bulk pressure
 Isotropic
 P = -zq
Shear stress tensor
 Traceless
 p = 4hq/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