Causal Baryon Diffusion and Colored Noise
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Causal Baryon Diffusion and Colored Noise Joe Kapusta University of Minnesota Collaborator: Clint Young
description
Causal Baryon Diffusion and Colored Noise. Joe Kapusta University of Minnesota. Collaborator: Clint Young . Sources of Fluctuations in High Energy Nuclear Collisions. Initial state fluctuations Hydrodynamic fluctuations due to finite particle number - PowerPoint PPT Presentation
Transcript of Causal Baryon Diffusion and Colored Noise
Causal Baryon Diffusion and Colored Noise
Joe KapustaUniversity of Minnesota
Collaborator: Clint Young
Sources of Fluctuations in High Energy Nuclear Collisions
• Initial state fluctuations• Hydrodynamic fluctuations due
to finite particle number• Energy and momentum
deposition by jets traversing the medium
• Freeze-out fluctuations
Why the need for causality?
The diffusion equation propagates informationinstantaneously. No good for hydrodynamicmodeling of high energy nuclear collisions!
)( ,)(
uuTJ
JnuJ
)/(
02
nD
nDt What should the current
be modified to?
Why colored noise?
Add noise to the current; in the local rest frame:
ijjiμ tTt(IIt(I )()(2), ,0), xxx
This is white noise (Fourier transform is a constant). It is OK for hydrodynamics if noise is treated as a perturbation, but createshavoc if it is treated nonperturbatively as there will be a dependence on the coarse-grained cell size.
0'
order 3rd - (1968) equation Pipkin-Gurtin
0
order 2nd -(1948) equation Cattaneo
0
order1st - equation diffusionOrdinary
233
3222
2
12
2
2
12
2
nt
Dtt
Dt
nt
Dt
nDt
Descriptions of heat conduction
The Associated Baryon Current
433
432
4321
23
2221
4
' Here
0 :equation Cattaneo
0 :equation diffusionOrdinary
)()(1)(1
Duu
uTJ
AnGR
),(k
23
2224
42
1)1(),(Dki
iiDkA
k
*
11),(AA
niTnn
k
*2 ),(31 AAniTIIk ll
k
Response function
Density correlator
Noise
AA
of poles noise of zeros correlatordensity
Fluctuation-Dissipation Theorem
Ordinary Diffusion Equation
DtrDt
nTtnn 4/exp4
1),( 22/3
x
ijji tTt(II )()(2), xx
Cattaneo Equation
1022
0
12
, velocity Group
.4/1 wherefor poles
complex ofpair a and ,for polesimaginary ofpair a is therecorrelatordensity For the
Dvkkkvv
Dkkk
kk
c
g
cc
c
(Infinite group velocity is not an issue; Brillouin.)
ijji tTt(II
)/||exp()(), 1
1
xx
Dimensionless baryon correlator
.2/ and 2/ where),(
)(')()2/1()exp(2
),(
110
ttvrrtrf
trtrtrttrf
reg
5.1,0.1,5.0t
Gurtin-Pipkin Equation
' requires planefrequency complex
halflower thein lie always poles theorder that In
/' is velocity group
asymptotic thefunction ncorrelatio baryon For the
3122
2230
Dv
Dimensionless baryon correlator
21 3 21
.3/1 and 0 with),(
)()exp()exp()(),(
204
vtrf
trrtbBtaAtrf
reg
x
Gurtin-Pipkin Equation
./ where
withpolescomplex ofpair a are there2For
. withfor poles
complex ofpair a and for polesimaginary ofpair a are there2for and noise For the
22302
02
0
21
220
21
Dvkkkvv
kkkvvkk
kk
g
c
gc
c
Dimensionless noise
.3/1 and 0 with
),()()/exp(2
),(
204
21
v
trgtrrttrg reg
wake.no is there2 When 21
21 3
21
Summary
• We have studied and compared the baryon current in 1st, 2nd and 3rd order dissipative fluid dynamics.• We computed the response function, baryon autocorrelation function, and noise.• One needs at least 2nd order for finite
propagation speed, and at least 3rd order for finite correlation lengths and times for noise.
• These results can readily be implemented in numerical hydrodynamic codes. (I-44)
• Baryon transport and noise should be important even if the net baryon number is zero, and may play a crucial role near a critical point.
• Microscopic calculations are needed to compute the time constants (probably functions of temperature and density).
• Noise happens!Supported by the Office Science, U.S. Department of Energy.
