Transcript スライド 1 - University of Tokyo

Viscous Effects on Distribution Function
and Relativistic Hydrodynamic Equations
Akihiko Monnai
Department of Physics, The University of Tokyo
Collaborator: Tetsufumi Hirano
Flow and Dissipation in Ultrarelativistic Heavy Ion Collisions
September 16th 2009, ECT* Italy
AM and T. Hirano,
arXiv:0903.4436; arXiv:0907.3078
2
Outlook
1.
2.
3.
4.
5.
Introduction
Distortion of Distribution
Numerical Estimation
Viscous Hydrodynamic Equations
Summary and Outlook
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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1. Introduction
Next:
2. Distortion of Distribution
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Introduction
Success of ideal hydrodynamic models
at relativistic heavy ion collisions
Development of viscous hydrodynamic models
(1) to understand the hot QCD matter itself and
(2) to determine the transport coefficients from experimental data
We put emphasize on bulk viscosity
“Large” bulk viscosity near Tc
Mizutani et al. (‘88)
Paech & Pratt (‘06)
Kharzeev & Tuchin (’08) …
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Introduction
• Viscous effects on the particle spectra
variation of the flow/hypersurface
Hydrodynamic equations (Sec. 4)
particles
Cooper & Frye (‘74)
modification of the distribution
Determined in Grad’s method in a
multi-component system (Sec. 2)
freezeout hypersurface Σ
Numerical estimation (Sec. 3)
hadron
resonance
gas
QGP
An interface from a hydrodynamic
model to a cascade model needed
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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2. Distortion of Distribution
Previous: 1. Introduction
Next:
3. Numerical Results
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Distortion of Distribution


for a single-component gas by Israel & Stewart (‘79)
Tensor decomposition and the macroscopic variables:
where
and
.
Bulk pressure :
Energy current:
Charge current:
Shear tensor:
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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The Matching Conditions

Landau matching conditions:
,
= necessary conditions to ensure thermodynamic stability
i.e.
etc.
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Relativistic Kinetic Theory

Kinetic definitions for a multi-component gas:
where gi is the degeneracy and bi the baryon number.

14 equations (= kinetic definitions + matching conditions)
= “Bridges” from the macroscopic variables to the microscopic
distribution
Π, Wμ, Vμ, πμν
δfi
Given
Unknown
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Grad’s 14-Moment Method

Distortion expressed with 14 (= 4+10) unknowns:
where + for bosons and – for fermions.

No scalar, but non-zero trace tensor.
The trace part:
particle species dependent
(mass dependent)
NOT equivalent in a multi-component system!
The scalar term:
particle species independent
(macroscopic quantity)
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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Comments

Why not the quadratic ansatz of
?
Dusling & Teaney (‘08)
thermodynamically unstable

Why not εμi and εμνi ?
 The number of macroscopic equations = 14
No room for additional unknowns
 Introducing more microscopic physics (i.e. cross sections)?
Generality is lost to be consistent
e.g. transport coefficients
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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Decomposition of Moments

Must-items of the tensor decomposition:
*Contributions are
: [baryons] + [anti-baryons] + [mesons]
: [baryons] – [anti-baryons]
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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Determination of Distortion

Insert
into the kinetic relations:
Scalar terms
Vector terms
Tensor term
where

,
,
and
.
The unique form of the deviation is determined:
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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3. Numerical Estimation
Previous: 2. Distortion of Distribution
Next:
4. Viscous Hydrodynamic Equations
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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EoS and Transport Coefficients


Equation of state: 16-component hadron resonance gas
*mesons and baryons with mass up to
under
Transport coefficients:
,
: sound velocity
s: entropy density
: free parameter

.
Kovtun et al.(‘05)
Weinberg (‘71) …
Freezeout temperature: Tf = 0.16(GeV)
and
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
(
).
Akihiko Monnai
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Flow and Dissipative Currents

Profiles of the flow
and the freezeout hypersurface
a (3+1)-dimensional ideal hydrodynamic simulation
:
Hirano et al.(‘06)

Estimation of dissipative currents: Navier-Stokes limit
,

Definition of elliptic flow coefficient v2(pT):
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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Bulk Viscosity and Particle Spectra

Au+Au,
pT -spectra
, b = 7.2(fm), pT -spectra and v2(pT) of
suppressed
v2 (pT)
enhanced
*Possible overestimations for… (i) Navier-Stokes limit (no relaxation effects)
(ii) ideal hydro flow (derivatives are larger)
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Quadratic Ansatz

pT -spectra and v2(pT) of
when
Effects of the bulk viscosity is underestimated in the quadratic ansatz.
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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4. Viscous Hydrodynamic Equations
Previous: 3. Numerical Estimation
Next:
5. Summary and Outlook
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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2nd Order Israel-Stewart Theory
Single component system
 Ideal hydrodynamics
6 unknowns
(1),

(1),
(1),
5 Conservation Eqs. + 1 EoS
(3)
,
and
Viscous hydrodynamics
Constitutive Equations:
9 independent Eqs. for
because
(1),
(3),
(5)
does not count.
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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In Multi-Component System

Particle number conservation does not hold
instead one has baryon number conservation

The trace of the constitutive eqs. is also not 0
which means

.
Two non-trivial consequences:
i) Traceless
is not allowed for a multi-component system
ii) One more constitutive equation is obtained
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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2nd Law of Thermodynamics

The entropy current
obeys
is constrained
If
were traceless, so was
, i.e.,
in the case of a multi-component system
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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Redundant Equation?
Multi-component system
 10 Constitutive Eqs. (tensor-decomposed)
2 independent eqs. for
3 eqs. for
5 eqs. for
because
The uncertainty can be removed by considering the
correct linear combination of the constitutive eqs.
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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Viscous Hydrodynamic Equations

Frame-independent constitutive equations (preliminary)
Bulk pressure
Cf. Betz et al. (‘08)
Heat current
Shear stress tensor
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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5. Summary and Outlook
Previous: 4. Viscous Hydrodynamic Equations
Next:
Appendix
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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Summary and Outlook

Determination of
in a multi-component system
- Viscous correction
has non-zero trace.

Visible effects of
on particle spectra
- pT-spectra is suppressed; v2(pT) is enhanced

Derivation of constitutive eqs. in a multi-component system
- Non-trivialities involved; post I-S terms reconfirmed

Bulk viscosity can be important in constraining the transport
coefficients from experimental data.

Full Viscous hydrodynamics needs to be developed to see
more realistic behavior of the particle spectra.
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Appendix
Previous: 5. Summary and Outlook
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Shear Viscosity and Particle Spectra

pT -spectra and v2(pT) of
with shear viscous correction
Non-triviality of shear viscosity; both pT -spectra and v2(pT) suppressed
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Shear & Bulk Viscosity on Spectra

pT -spectra and v2(pT) of
bulk viscosity
with corrections from shear and
Accidental cancellation in viscous corrections in v2(pT)
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Bjorken Model

pT -spectra and v2(pT) of
geometry:
in Bjorken model with cylindrical
Bulk viscosity suppresses pT-spectra
Shear viscosity enhances pT-spectra
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai
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Blast wave model

pT -spectra and v2(pT) of
Shear viscosity enhances pT-spectra
and suppresses v2(pT).
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
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Single vs. Multi-Component

Distortion
because
depends on all the components in the gas, i.e.
and
in
are “macroscopic” quantities.
Viscous Effects on Distribution Function and Relativistic Hydrodynamic Equations, ECT*, Sep. 16 th 2009
Akihiko Monnai