Second-order (2+1) -dimensional anisotropic hydrodynamics

doi:https://doi.org/10.1103/PhysRevC.90.054910

Title: Second-order $(2 + 1)$ -dimensional anisotropic hydrodynamics

Abstract

We present a complete formulation of second-order (2+1)-dimensional anisotropic hydrodynamics. The resulting framework generalizes leading-order anisotropic hydrodynamics by allowing for deviations of the one-particle distribution function from the spheroidal form assumed at leading order. We derive complete second-order equations of motion for the additional terms in the macroscopic currents generated by these deviations from their kinetic definition using a Grad-Israel-Stewart 14-moment ansatz. The result is a set of coupled partial differential equations for the momentum-space anisotropy parameter, effective temperature, the transverse components of the fluid four-velocity, and the viscous tensor components generated by deviations of the distribution from spheroidal form. We then perform a quantitative test of our approach by applying it to the case of one-dimensional boost-invariant expansion in the relaxation time approximation (RTA) in which case it is possible to numerically solve the Boltzmann equation exactly.We demonstrate that the second-order anisotropic hydrodynamics approach provides an excellent approximation to the exact (0+1)-dimensional RTA solution for both small and large values of the shear viscosity.

Authors:

Bazow, Dennis ^[1];

^[1]; Strickland, Michael ^[2]

The Ohio State Univ., Columbus, OH (United States)
Kent State Univ., Kent, OH (United States)

Publication Date:: 2014-11-21

Research Org.:: The Ohio State Univ., Columbus, OH (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Nuclear Physics (NP)

OSTI Identifier:: 1604545

Alternate Identifier(s):: OSTI ID: 1179929

Grant/Contract Number:: SC0004286; SC0004104

Resource Type:: Journal Article: Accepted Manuscript

Journal Name:: Physical Review. C, Nuclear Physics

Additional Journal Information:: Journal Volume: 90; Journal Issue: 5; Journal ID: ISSN 0556-2813

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 73 NUCLEAR PHYSICS AND RADIATION PHYSICS

Citation Formats


                    Bazow, Dennis, Heinz, Ulrich, and Strickland, Michael. Second-order (2+1) -dimensional anisotropic hydrodynamics.  United States: N. p., 2014. 
        Web.  doi:10.1103/PhysRevC.90.054910.

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                    Bazow, Dennis, Heinz, Ulrich, & Strickland, Michael. Second-order (2+1) -dimensional anisotropic hydrodynamics.  United States.  https://doi.org/10.1103/PhysRevC.90.054910

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                    Bazow, Dennis, Heinz, Ulrich, and Strickland, Michael. Fri .  
        "Second-order (2+1) -dimensional anisotropic hydrodynamics".  United States.  https://doi.org/10.1103/PhysRevC.90.054910.  https://www.osti.gov/servlets/purl/1604545.

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@article{osti_1604545,

  title        = {Second-order (2+1) -dimensional anisotropic hydrodynamics},

  author       = {Bazow, Dennis and Heinz, Ulrich and Strickland, Michael},

  abstractNote = {We present a complete formulation of second-order (2+1)-dimensional anisotropic hydrodynamics. The resulting framework generalizes leading-order anisotropic hydrodynamics by allowing for deviations of the one-particle distribution function from the spheroidal form assumed at leading order. We derive complete second-order equations of motion for the additional terms in the macroscopic currents generated by these deviations from their kinetic definition using a Grad-Israel-Stewart 14-moment ansatz. The result is a set of coupled partial differential equations for the momentum-space anisotropy parameter, effective temperature, the transverse components of the fluid four-velocity, and the viscous tensor components generated by deviations of the distribution from spheroidal form. We then perform a quantitative test of our approach by applying it to the case of one-dimensional boost-invariant expansion in the relaxation time approximation (RTA) in which case it is possible to numerically solve the Boltzmann equation exactly.We demonstrate that the second-order anisotropic hydrodynamics approach provides an excellent approximation to the exact (0+1)-dimensional RTA solution for both small and large values of the shear viscosity.},

  doi          = {10.1103/PhysRevC.90.054910},

  url          = {https://www.osti.gov/biblio/1604545},
  journal      = {Physical Review. C, Nuclear Physics},

  issn         = {0556-2813},

  number       = 5,

  volume       = 90,

  place        = {United States},

  year         = {2014},

  month        = {11}

}

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Title: Second-order ( 2 + 1 ) -dimensional anisotropic hydrodynamics

Abstract

Citation Formats

On the kinetic theory of rarefied gases journal, December 1949

Generalized Orthogonal Polynomial Solutions of the Boltzmann Equation journal, January 1965

Transient relativistic thermodynamics and kinetic theory journal, April 1979

Derivation of transient relativistic fluid dynamics from the Boltzmann equation journal, June 2012

Relativistic Grad polynomials journal, July 1974

Constraining relativistic viscous hydrodynamical evolution journal, April 2009

Dissipative dynamics of highly anisotropic systems journal, December 2010

Highly anisotropic and strongly dissipative hydrodynamics for early stages of relativistic heavy-ion collisions journal, March 2011

Non-boost-invariant motion of dissipative and highly anisotropic fluid journal, December 2010

Non-boost-invariant anisotropic dynamics journal, April 2011

Highly anisotropic and strongly dissipative hydrodynamics with transverse expansion journal, November 2011

Projection method for boost-invariant and cylindrically symmetric dissipative hydrodynamics journal, April 2012

Boost-invariant (2+1)-dimensional anisotropic hydrodynamics journal, June 2012

Highly anisotropic hydrodynamics in 3 + 1 space-time dimensions journal, June 2012

Hydrodynamics of anisotropic quark and gluon fluids journal, March 2013

Mixture of Anisotropic Fluids journal, January 2013

Collective modes of an anisotropic quark-gluon plasma journal, August 2003

Nonstationary irreversible thermodynamics: A causal relativistic theory journal, September 1976

Second-Order Dissipative Fluid Dynamics for Ultrarelativistic Nuclear Collisions journal, January 2002

Causal theories of dissipative relativistic fluid dynamics for nuclear collisions journal, March 2004

Dissipative hydrodynamics for viscous relativistic fluids journal, March 2006

Dissipative hydrodynamics in 2 + 1 dimensions journal, October 2006

Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC? journal, October 2007

Suppression of elliptic flow in a minimally viscous quark–gluon plasma journal, January 2008

Causal viscous hydrodynamics in 2 + 1 dimensions for relativistic heavy-ion collisions journal, June 2008

Multiplicity scaling in ideal and viscous hydrodynamics journal, August 2008

Conformal relativistic viscous hydrodynamics: Applications to RHIC results at s NN = 200 GeV journal, September 2008

New Developments in Relativistic Viscous Hydrodynamics journal, January 2010

Elliptic and Triangular Flow in Event-by-Event D = 3 + 1 Viscous Hydrodynamics journal, January 2011

Higher flow harmonics from ( 3 + 1 ) D event-by-event viscous hydrodynamics journal, February 2012

Event-by-Event Anisotropic Flow in Heavy-ion Collisions from Combined Yang-Mills and Viscous Fluid Dynamics journal, January 2013

Collective Flow and Viscosity in Relativistic Heavy-Ion Collisions journal, October 2013

Hydrodynamic Modeling of Heavy-Ion Collisions journal, April 2013

Applicability of causal dissipative hydrodynamics to relativistic heavy ion collisions journal, January 2009

Dissipative Relativistic Fluid Dynamics: A New Way to Derive the Equations of Motion from Kinetic Theory journal, October 2010

Relativistic dynamics of non-ideal fluids: Viscous and heat-conducting fluids. II. Transport properties and microscopic description of relativistic nuclear matter journal, July 2007

Second order dissipative fluid dynamics from kinetic theory journal, January 2011

Complete second-order dissipative fluid dynamics journal, May 2009

The Mathematical Theory of Non-Uniform Gases journal, May 1962

Derivation of fluid dynamics from kinetic theory with the 14-moment approximation journal, November 2012

Relativistic dissipative hydrodynamics from kinetic theory with relaxation-time approximation journal, May 2013

Relativistic third-order dissipative fluid dynamics from kinetic theory journal, August 2013

Extension of relativistic dissipative hydrodynamics to third order journal, April 2010

Macroscopic approximation to relativistic kinetic theory from a nonlinear closure journal, February 2013

Anisotropic hydrodynamics for rapidly expanding systems journal, October 2013

Testing viscous and anisotropic hydrodynamics in an exactly solvable case journal, August 2013

Hydrodynamic Gradient Expansion in Gauge Theory Plasmas journal, May 2013

Matching pre-equilibrium dynamics and viscous hydrodynamics journal, February 2010

Parton equilibration in relativistic heavy ion collisions journal, September 1993

“Bottom-up” thermalization in heavy ion collisions journal, March 2001

Thermalization of gluons in ultrarelativistic heavy ion collisions by including three-body interactions in a parton cascade journal, June 2005

Transport rates and momentum isotropization of gluon matter in ultrarelativistic heavy-ion collisions journal, August 2007

Violation of the Holographic Viscosity Bound in a Strongly Coupled Anisotropic Plasma journal, January 2012

Instabilities of an anisotropically expanding non-Abelian plasma: 1 D + 3 V discretized hard-loop simulations journal, August 2008

Dissipative hydrodynamics and heavy-ion collisions journal, June 2006

Thermal equilibration in ultra-relativistic heavy-ion collisions journal, April 1984

A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems journal, May 1954

Anisotropic flow and jet quenching in relativistic nuclear collisions journal, February 2015

Introduction to hydrodynamics journal, October 2015

Paramagnetic squeezing of a uniformly expanding quark-gluon plasma in and out of equilibrium journal, March 2019

Collective flow and hydrodynamics in large and small systems at the LHC journal, June 2017

Viscous cosmologies journal, October 2019

New theories of relativistic hydrodynamics in the LHC era journal, February 2018

Including off-diagonal anisotropies in anisotropic hydrodynamics journal, July 2019

Exact solutions and attractors of higher-order viscous fluid dynamics for Bjorken flow journal, September 2019

Relativistic third-order viscous corrections to the entropy four-current from kinetic theory journal, February 2015

Pre-equilibrium evolution effects on heavy-ion collision observables journal, June 2015

Event-by-event fluctuations in a perturbative QCD + saturation + hydrodynamics model: Determining QCD matter shear viscosity in ultrarelativistic heavy-ion collisions journal, February 2016

Anisotropic hydrodynamics with a scalar collisional kernel journal, April 2018

Studying the validity of relativistic hydrodynamics with a new exact solution of the Boltzmann equation journal, December 2014

Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation journal, June 2016

Bottomonium suppression using a lattice QCD vetted potential journal, January 2018

Anisotropic nonequilibrium hydrodynamic attractor journal, February 2018

Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow journal, February 2018

Dynamical systems and nonlinear transient rheology of the far-from-equilibrium Bjorken flow journal, June 2019

Prescaling and Far-from-Equilibrium Hydrodynamics in the Quark-Gluon Plasma journal, March 2019

Heavy-flavour and quarkonium production in the LHC era: from proton–proton to heavy-ion collisions journal, February 2016

Title: Second-order $(2 + 1)$ -dimensional anisotropic hydrodynamics

On the kinetic theory of rarefied gases
journal, December 1949

Generalized Orthogonal Polynomial Solutions of the Boltzmann Equation
journal, January 1965

Transient relativistic thermodynamics and kinetic theory
journal, April 1979

Derivation of transient relativistic fluid dynamics from the Boltzmann equation
journal, June 2012

Relativistic Grad polynomials
journal, July 1974

Constraining relativistic viscous hydrodynamical evolution
journal, April 2009

Dissipative dynamics of highly anisotropic systems
journal, December 2010

Highly anisotropic and strongly dissipative hydrodynamics for early stages of relativistic heavy-ion collisions
journal, March 2011

Non-boost-invariant motion of dissipative and highly anisotropic fluid
journal, December 2010

Non-boost-invariant anisotropic dynamics
journal, April 2011

Highly anisotropic and strongly dissipative hydrodynamics with transverse expansion
journal, November 2011

Projection method for boost-invariant and cylindrically symmetric dissipative hydrodynamics
journal, April 2012

Boost-invariant (2+1)-dimensional anisotropic hydrodynamics
journal, June 2012

Highly anisotropic hydrodynamics in $3 + 1$ space-time dimensions
journal, June 2012

Hydrodynamics of anisotropic quark and gluon fluids
journal, March 2013

Mixture of Anisotropic Fluids
journal, January 2013

Collective modes of an anisotropic quark-gluon plasma
journal, August 2003

Nonstationary irreversible thermodynamics: A causal relativistic theory
journal, September 1976

Second-Order Dissipative Fluid Dynamics for Ultrarelativistic Nuclear Collisions
journal, January 2002

Causal theories of dissipative relativistic fluid dynamics for nuclear collisions
journal, March 2004

Dissipative hydrodynamics for viscous relativistic fluids
journal, March 2006

Dissipative hydrodynamics in $2 + 1$ dimensions
journal, October 2006

Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?
journal, October 2007

Suppression of elliptic flow in a minimally viscous quark–gluon plasma
journal, January 2008

Causal viscous hydrodynamics in 2 + 1 dimensions for relativistic heavy-ion collisions
journal, June 2008

Multiplicity scaling in ideal and viscous hydrodynamics
journal, August 2008

Conformal relativistic viscous hydrodynamics: Applications to RHIC results at $\sqrt{s_{NN}} = 200$ GeV
journal, September 2008

New Developments in Relativistic Viscous Hydrodynamics
journal, January 2010

Elliptic and Triangular Flow in Event-by-Event $D = 3 + 1$ Viscous Hydrodynamics
journal, January 2011

Higher flow harmonics from $(3 + 1)$ D event-by-event viscous hydrodynamics
journal, February 2012

Event-by-Event Anisotropic Flow in Heavy-ion Collisions from Combined Yang-Mills and Viscous Fluid Dynamics
journal, January 2013

Collective Flow and Viscosity in Relativistic Heavy-Ion Collisions
journal, October 2013

Hydrodynamic Modeling of Heavy-Ion Collisions
journal, April 2013

Applicability of causal dissipative hydrodynamics to relativistic heavy ion collisions
journal, January 2009

Dissipative Relativistic Fluid Dynamics: A New Way to Derive the Equations of Motion from Kinetic Theory
journal, October 2010

Relativistic dynamics of non-ideal fluids: Viscous and heat-conducting fluids. II. Transport properties and microscopic description of relativistic nuclear matter
journal, July 2007

Second order dissipative fluid dynamics from kinetic theory
journal, January 2011

Complete second-order dissipative fluid dynamics
journal, May 2009

The Mathematical Theory of Non-Uniform Gases
journal, May 1962

Derivation of fluid dynamics from kinetic theory with the 14-moment approximation
journal, November 2012

Relativistic dissipative hydrodynamics from kinetic theory with relaxation-time approximation
journal, May 2013

Relativistic third-order dissipative fluid dynamics from kinetic theory
journal, August 2013

Extension of relativistic dissipative hydrodynamics to third order
journal, April 2010

Macroscopic approximation to relativistic kinetic theory from a nonlinear closure
journal, February 2013

Anisotropic hydrodynamics for rapidly expanding systems
journal, October 2013

Testing viscous and anisotropic hydrodynamics in an exactly solvable case
journal, August 2013

Hydrodynamic Gradient Expansion in Gauge Theory Plasmas
journal, May 2013

Matching pre-equilibrium dynamics and viscous hydrodynamics
journal, February 2010

Parton equilibration in relativistic heavy ion collisions
journal, September 1993

“Bottom-up” thermalization in heavy ion collisions
journal, March 2001

Thermalization of gluons in ultrarelativistic heavy ion collisions by including three-body interactions in a parton cascade
journal, June 2005

Transport rates and momentum isotropization of gluon matter in ultrarelativistic heavy-ion collisions
journal, August 2007

Violation of the Holographic Viscosity Bound in a Strongly Coupled Anisotropic Plasma
journal, January 2012

Instabilities of an anisotropically expanding non-Abelian plasma: $1 D + 3 V$ discretized hard-loop simulations
journal, August 2008

Dissipative hydrodynamics and heavy-ion collisions
journal, June 2006

Thermal equilibration in ultra-relativistic heavy-ion collisions
journal, April 1984

A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems
journal, May 1954

Anisotropic flow and jet quenching in relativistic nuclear collisions
journal, February 2015

Introduction to hydrodynamics
journal, October 2015

Paramagnetic squeezing of a uniformly expanding quark-gluon plasma in and out of equilibrium
journal, March 2019

Collective flow and hydrodynamics in large and small systems at the LHC
journal, June 2017

Viscous cosmologies
journal, October 2019

New theories of relativistic hydrodynamics in the LHC era
journal, February 2018

Including off-diagonal anisotropies in anisotropic hydrodynamics
journal, July 2019

Exact solutions and attractors of higher-order viscous fluid dynamics for Bjorken flow
journal, September 2019

Relativistic third-order viscous corrections to the entropy four-current from kinetic theory
journal, February 2015

Pre-equilibrium evolution effects on heavy-ion collision observables
journal, June 2015

Event-by-event fluctuations in a perturbative QCD + saturation + hydrodynamics model: Determining QCD matter shear viscosity in ultrarelativistic heavy-ion collisions
journal, February 2016

Anisotropic hydrodynamics with a scalar collisional kernel
journal, April 2018

Studying the validity of relativistic hydrodynamics with a new exact solution of the Boltzmann equation
journal, December 2014

Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
journal, June 2016

Bottomonium suppression using a lattice QCD vetted potential
journal, January 2018

Anisotropic nonequilibrium hydrodynamic attractor
journal, February 2018

Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow
journal, February 2018

Dynamical systems and nonlinear transient rheology of the far-from-equilibrium Bjorken flow
journal, June 2019

Prescaling and Far-from-Equilibrium Hydrodynamics in the Quark-Gluon Plasma
journal, March 2019

Heavy-flavour and quarkonium production in the LHC era: from proton–proton to heavy-ion collisions
journal, February 2016

Relativistic Hydrodynamics in Heavy-Ion Collisions: General Aspects and Recent Developments
journal, January 2016

Predictions for Bottomonia Suppression in 5.023 TeV Pb-Pb Collisions
journal, August 2016