# Relativistic hydrodynamic fluctuations

## Abstract

We present a general systematic formalism for describing dynamics of fluctuations in an arbitrary relativistic hydrodynamic flow, including their feedback (known as long-time hydrodynamic tails). The fluctuations are described by two-point equal-time correlation functions. We introduce a definition of equal time in a situation where the local rest frame is determined by the local flow velocity and a method of taking derivatives and Wigner transforms of such equal-time correlation functions, which we call confluent. We find that not only do the equations for confluent Wigner functions resemble kinetic equations but also the kinetic equation for phonons propagating on an arbitrary background nontrivially matches the equations for Wigner functions, including relativistic inertial and Coriolis forces due to acceleration and vorticity of the flow. We also describe the procedure of renormalization of short-distance singularities which eliminates cutoff dependence, allowing efficient numerical implementation of these equations.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Illinois at Urbana-Champaign, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1558701

- Alternate Identifier(s):
- OSTI ID: 1609365

- Grant/Contract Number:
- FG0201ER41195; SC0018209; FG02-01ER41195

- Resource Type:
- Published Article

- Journal Name:
- Physical Review C

- Additional Journal Information:
- Journal Name: Physical Review C Journal Volume: 100 Journal Issue: 2; Journal ID: ISSN 2469-9985

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Fluctuations & noise; Relativistic heavy-ion collisions; Relativistic hydrodynamics

### Citation Formats

```
An, Xin, Başar, Gökçe, Stephanov, Mikhail, and Yee, Ho-Ung. Relativistic hydrodynamic fluctuations. United States: N. p., 2019.
Web. doi:10.1103/PhysRevC.100.024910.
```

```
An, Xin, Başar, Gökçe, Stephanov, Mikhail, & Yee, Ho-Ung. Relativistic hydrodynamic fluctuations. United States. doi:10.1103/PhysRevC.100.024910.
```

```
An, Xin, Başar, Gökçe, Stephanov, Mikhail, and Yee, Ho-Ung. Thu .
"Relativistic hydrodynamic fluctuations". United States. doi:10.1103/PhysRevC.100.024910.
```

```
@article{osti_1558701,
```

title = {Relativistic hydrodynamic fluctuations},

author = {An, Xin and Başar, Gökçe and Stephanov, Mikhail and Yee, Ho-Ung},

abstractNote = {We present a general systematic formalism for describing dynamics of fluctuations in an arbitrary relativistic hydrodynamic flow, including their feedback (known as long-time hydrodynamic tails). The fluctuations are described by two-point equal-time correlation functions. We introduce a definition of equal time in a situation where the local rest frame is determined by the local flow velocity and a method of taking derivatives and Wigner transforms of such equal-time correlation functions, which we call confluent. We find that not only do the equations for confluent Wigner functions resemble kinetic equations but also the kinetic equation for phonons propagating on an arbitrary background nontrivially matches the equations for Wigner functions, including relativistic inertial and Coriolis forces due to acceleration and vorticity of the flow. We also describe the procedure of renormalization of short-distance singularities which eliminates cutoff dependence, allowing efficient numerical implementation of these equations.},

doi = {10.1103/PhysRevC.100.024910},

journal = {Physical Review C},

number = 2,

volume = 100,

place = {United States},

year = {2019},

month = {8}

}

DOI: 10.1103/PhysRevC.100.024910

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Works referenced in this record:

##
Effective action for relativistic hydrodynamics: fluctuations, dissipation, and entropy inflow

journal, October 2018

- Haehl, Felix M.; Loganayagam, R.; Rangamani, Mukund
- Journal of High Energy Physics, Vol. 2018, Issue 10

##
Thermally fluctuating second-order viscous hydrodynamics and heavy-ion collisions

journal, April 2015

- Young, C.; Kapusta, J. I.; Gale, C.
- Physical Review C, Vol. 91, Issue 4

##
Stochastic hydrodynamics and long time tails of an expanding conformal charged fluid

journal, May 2019

- Martinez, M.; Schäfer, T.
- Physical Review C, Vol. 99, Issue 5

##
Theory of dynamic critical phenomena

journal, July 1977

- Hohenberg, P. C.; Halperin, B. I.
- Reviews of Modern Physics, Vol. 49, Issue 3

##
Zum Paradoxon der W�rmeleitungstheorie

journal, August 1967

- M�ller, Ingo
- Zeitschrift f�r Physik, Vol. 198, Issue 4

##
Relativistic theory of hydrodynamic fluctuations with applications to heavy-ion collisions

journal, May 2012

- Kapusta, J. I.; Müller, B.; Stephanov, M.
- Physical Review C, Vol. 85, Issue 5

##
Thermal conductivity and chiral critical point in heavy ion collisions

journal, November 2012

- Kapusta, Joseph I.; Torres-Rincon, Juan M.
- Physical Review C, Vol. 86, Issue 5

##
Transient relativistic thermodynamics and kinetic theory

journal, April 1979

- Israel, W.; Stewart, J. M.
- Annals of Physics, Vol. 118, Issue 2

##
Numerical integration of thermal noise in relativistic hydrodynamics

journal, February 2014

- Young, Clint
- Physical Review C, Vol. 89, Issue 2

##
Bulk viscosity from hydrodynamic fluctuations with relativistic hydrokinetic theory

journal, February 2018

- Akamatsu, Yukinao; Mazeliauskas, Aleksas; Teaney, Derek
- Physical Review C, Vol. 97, Issue 2

##
Hydrodynamic fluctuations, long-time tails, and supersymmetry

journal, July 2003

- Kovtun, Pavel; Yaffe, Laurence G.
- Physical Review D, Vol. 68, Issue 2

##
Effective field theory of dissipative fluids

journal, September 2017

- Crossley, Michael; Glorioso, Paolo; Liu, Hong
- Journal of High Energy Physics, Vol. 2017, Issue 9

##
Stickiness of sound: An absolute lower limit on viscosity and the breakdown of second-order relativistic hydrodynamics

journal, July 2011

- Kovtun, Pavel; Moore, Guy D.; Romatschke, Paul
- Physical Review D, Vol. 84, Issue 2

##
Introduction to hydrodynamics

journal, October 2015

- Jeon, Sangyong; Heinz, Ulrich
- International Journal of Modern Physics E, Vol. 24, Issue 10

##
Sign of Kurtosis near the QCD Critical Point

journal, July 2011

- Stephanov, M. A.
- Physical Review Letters, Vol. 107, Issue 5

##
Non-Gaussian Fluctuations near the QCD Critical Point

journal, January 2009

- Stephanov, M. A.
- Physical Review Letters, Vol. 102, Issue 3

##
Relativistic theories of dissipative fluids

journal, August 1995

- Geroch, Robert
- Journal of Mathematical Physics, Vol. 36, Issue 8

##
Hydrodynamics with parametric slowing down and fluctuations near the critical point

journal, August 2018

- Stephanov, M.; Yin, Y.
- Physical Review D, Vol. 98, Issue 3

##
Event-by-event fluctuations in heavy ion collisions and the QCD critical point

journal, November 1999

- Stephanov, M.; Rajagopal, K.; Shuryak, E.
- Physical Review D, Vol. 60, Issue 11

##
Kinetic regime of hydrodynamic fluctuations and long time tails for a Bjorken expansion

journal, January 2017

- Akamatsu, Yukinao; Mazeliauskas, Aleksas; Teaney, Derek
- Physical Review C, Vol. 95, Issue 1

##
Signatures of the Tricritical Point in QCD

journal, November 1998

- Stephanov, M.; Rajagopal, K.; Shuryak, E.
- Physical Review Letters, Vol. 81, Issue 22