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.