skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow

Abstract

The nonequilibrium attractors of systems undergoing Gubser flow within relativistic kinetic theory are studied. In doing so we employ well-established methods of nonlinear dynamical systems which rely on finding the fixed points, investigating the structure of the flow diagrams of the evolution equations, and characterizing the basin of attraction using a Lyapunov function near the stable fixed points. We obtain the attractors of anisotropic hydrodynamics, Israel-Stewart (IS) and transient fluid (DNMR) theories and show that they are indeed nonplanar and the basin of attraction is essentially three dimensional. The attractors of each hydrodynamical model are compared with the one obtained from the exact Gubser solution of the Boltzmann equation within the relaxation time approximation. We observe that the anisotropic hydrodynamics is able to match up to high numerical accuracy the attractor of the exact solution while the second-order hydrodynamical theories fail to describe it. We show that the IS and DNMR asymptotic series expansions diverge and use resurgence techniques to perform the resummation of these divergences. We also comment on a possible link between the manifold of steepest descent paths in path integrals and the basin of attraction for the attractors via Lyapunov functions that opens a new horizon towardmore » an effective field theory description of hydrodynamics. Our findings indicate that the reorganization of the expansion series carried out by anisotropic hydrodynamics resums the Knudsen and inverse Reynolds numbers to all orders and thus, it can be understood as an effective theory for the far-from-equilibrium fluid dynamics« less

Authors:
; ;
Publication Date:
Research Org.:
North Carolina State Univ., Raleigh, NC (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1422657
Alternate Identifier(s):
OSTI ID: 1498962
Grant/Contract Number:  
FG02-03ER41260; SC0013036
Resource Type:
Published Article
Journal Name:
Physical Review D
Additional Journal Information:
Journal Name: Physical Review D Journal Volume: 97 Journal Issue: 4; Journal ID: ISSN 2470-0010
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS

Citation Formats

Behtash, Alireza, Cruz-Camacho, C. N., and Martinez, M. Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow. United States: N. p., 2018. Web. doi:10.1103/PhysRevD.97.044041.
Behtash, Alireza, Cruz-Camacho, C. N., & Martinez, M. Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow. United States. doi:10.1103/PhysRevD.97.044041.
Behtash, Alireza, Cruz-Camacho, C. N., and Martinez, M. Mon . "Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow". United States. doi:10.1103/PhysRevD.97.044041.
@article{osti_1422657,
title = {Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow},
author = {Behtash, Alireza and Cruz-Camacho, C. N. and Martinez, M.},
abstractNote = {The nonequilibrium attractors of systems undergoing Gubser flow within relativistic kinetic theory are studied. In doing so we employ well-established methods of nonlinear dynamical systems which rely on finding the fixed points, investigating the structure of the flow diagrams of the evolution equations, and characterizing the basin of attraction using a Lyapunov function near the stable fixed points. We obtain the attractors of anisotropic hydrodynamics, Israel-Stewart (IS) and transient fluid (DNMR) theories and show that they are indeed nonplanar and the basin of attraction is essentially three dimensional. The attractors of each hydrodynamical model are compared with the one obtained from the exact Gubser solution of the Boltzmann equation within the relaxation time approximation. We observe that the anisotropic hydrodynamics is able to match up to high numerical accuracy the attractor of the exact solution while the second-order hydrodynamical theories fail to describe it. We show that the IS and DNMR asymptotic series expansions diverge and use resurgence techniques to perform the resummation of these divergences. We also comment on a possible link between the manifold of steepest descent paths in path integrals and the basin of attraction for the attractors via Lyapunov functions that opens a new horizon toward an effective field theory description of hydrodynamics. Our findings indicate that the reorganization of the expansion series carried out by anisotropic hydrodynamics resums the Knudsen and inverse Reynolds numbers to all orders and thus, it can be understood as an effective theory for the far-from-equilibrium fluid dynamics},
doi = {10.1103/PhysRevD.97.044041},
journal = {Physical Review D},
number = 4,
volume = 97,
place = {United States},
year = {2018},
month = {2}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1103/PhysRevD.97.044041

Citation Metrics:
Cited by: 14 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Turbulent thermalization process in heavy-ion collisions at ultrarelativistic energies
journal, April 2014


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


Relative importance of second-order terms in relativistic dissipative fluid dynamics
journal, April 2014


Coupling Hydrodynamics to Nonequilibrium Degrees of Freedom in Strongly Interacting Quark-Gluon Plasma
journal, December 2014


Resurgence in extended hydrodynamics
journal, April 2016


Equilibration of the quark-gluon plasma produced in relativistic heavy ion collisions
journal, June 1989


Characteristics of Thermalization of Boost-Invariant Plasma from Holography
journal, May 2012


Relativistic hydrodynamics and thermodynamics of anisotropic plasmas
journal, August 1991

  • Gedalin, M.
  • Physics of Fluids B: Plasma Physics, Vol. 3, Issue 8
  • DOI: 10.1063/1.859656

Does the Chapman–Enskog Expansion for Sheared Granular Gases Converge?
journal, February 2008


Conformal hydrodynamics in Minkowski and de Sitter spacetimes
journal, May 2011


Analytic Solution of the Boltzmann Equation in an Expanding System
journal, January 2016


Homogeneous isotropization and equilibration of a strongly coupled plasma with a critical point
journal, December 2017

  • Critelli, Renato; Rougemont, Romulo; Noronha, Jorge
  • Journal of High Energy Physics, Vol. 2017, Issue 12
  • DOI: 10.1007/JHEP12(2017)029

Isotropization and Hydrodynamization in Weakly Coupled Heavy-Ion Collisions
journal, October 2015


<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2011-01-01">January 2011</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Ryblewski, R.; Florkowski, W.</span> </li> <li> Acta Physica Polonica B, Vol. 42, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.5506/APhysPolB.42.115" class="text-muted" target="_blank" rel="noopener noreferrer">10.5506/APhysPolB.42.115<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.89.034907" target="_blank" rel="noopener noreferrer" class="name">Projection method and new formulation of leading-order anisotropic hydrodynamics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2014-03-01">March 2014</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Tinti, Leonardo; Florkowski, Wojciech</span> </li> <li> Physical Review C, Vol. 89, Issue 3</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.89.034907" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.89.034907<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1090/S0002-9939-1960-0121542-7" target="_blank" rel="noopener noreferrer" class="name">A lemma in the theory of structural stability of differential equations<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1960-04-01">April 1960</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Hartman, Philip</span> </li> <li> Proceedings of the American Mathematical Society, Vol. 11, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1090/S0002-9939-1960-0121542-7" class="text-muted" target="_blank" rel="noopener noreferrer">10.1090/S0002-9939-1960-0121542-7<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.82.085027" target="_blank" rel="noopener noreferrer" class="name">Symmetry constraints on generalizations of Bjorken flow<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2010-10-01">October 2010</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Gubser, Steven S.</span> </li> <li> Physical Review D, Vol. 82, Issue 8</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.82.085027" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.82.085027<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1063/1.1761357" target="_blank" rel="noopener noreferrer" class="name">Generalized Orthogonal Polynomial Solutions of the Boltzmann Equation<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1965-01-01">January 1965</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Mintzer, David</span> </li> <li> Physics of Fluids, Vol. 8, Issue 6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1063/1.1761357" class="text-muted" target="_blank" rel="noopener noreferrer">10.1063/1.1761357<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.82.026006" target="_blank" rel="noopener noreferrer" class="name">Boost invariant flow, black hole formation, and far-from-equilibrium dynamics in <math display="inline"> <mi mathvariant="script">N</mi> <mo>=</mo> <mn>4</mn> </math> supersymmetric Yang-Mills theory<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2010-07-01">July 2010</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Chesler, Paul M.; Yaffe, Laurence G.</span> </li> <li> Physical Review D, Vol. 82, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.82.026006" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.82.026006<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.92.085048" target="_blank" rel="noopener noreferrer" class="name">Relativistic causal hydrodynamics derived from Boltzmann equation: A novel reduction theoretical approach<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-10-01">October 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Tsumura, Kyosuke; Kikuchi, Yuta; Kunihiro, Teiji</span> </li> <li> Physical Review D, Vol. 92, Issue 8</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.92.085048" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.92.085048<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.85.064913" target="_blank" rel="noopener noreferrer" class="name">Boost-invariant (2+1)-dimensional anisotropic hydrodynamics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-06-01">June 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Martinez, Mauricio; Ryblewski, Radoslaw; Strickland, Michael</span> </li> <li> Physical Review C, Vol. 85, Issue 6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.85.064913" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.85.064913<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1215/S0012-7094-98-09311-5" target="_blank" rel="noopener noreferrer" class="name">On Borel summation and Stokes phenomena for rank- $1$ nonlinear systems of ordinary differential equations<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1998-06-01">June 1998</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Costin, Ovidiu</span> </li> <li> Duke Mathematical Journal, Vol. 93, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1215/S0012-7094-98-09311-5" class="text-muted" target="_blank" rel="noopener noreferrer">10.1215/S0012-7094-98-09311-5<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.68.036004" target="_blank" rel="noopener noreferrer" class="name">Collective modes of an anisotropic quark-gluon plasma<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2003-08-01">August 2003</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Romatschke, Paul; Strickland, Michael</span> </li> <li> Physical Review D, Vol. 68, Issue 3</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.68.036004" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.68.036004<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/JHEP12(2017)079" target="_blank" rel="noopener noreferrer" class="name">Relativistic hydrodynamic attractors with broken symmetries: non-conformal and non-homogeneous<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-12-01">December 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Romatschke, Paul</span> </li> <li> Journal of High Energy Physics, Vol. 2017, Issue 12</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/JHEP12(2017)079" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/JHEP12(2017)079<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.na.2011.11.027" target="_blank" rel="noopener noreferrer" class="name">Numerical determination of the basin of attraction for asymptotically autonomous dynamical systems<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-03-01">March 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Giesl, Peter; Wendland, Holger</span> </li> <li> Nonlinear Analysis: Theory, Methods & Applications, Vol. 75, Issue 5</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.na.2011.11.027" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.na.2011.11.027<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.116.172301" target="_blank" rel="noopener noreferrer" class="name">Observation of Long-Range Elliptic Azimuthal Anisotropies in <math display="inline"> <msqrt> <mi>s</mi> </msqrt> <mo>=</mo> <mn>13</mn> </math> and 2.76 TeV <math display="inline"> <mi>p</mi> <mi>p</mi> </math> Collisions with the ATLAS Detector<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2016-04-01">April 2016</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Aad, G.; Abbott, B.; Abdallah, J.</span> </li> <li> Physical Review Letters, Vol. 116, Issue 17</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.116.172301" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.116.172301<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.nuclphysa.2017.06.001" target="_blank" rel="noopener noreferrer" class="name">Nonlocal wave turbulence in non-Abelian plasmas<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-10-01">October 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Mehtar-Tani, Yacine</span> </li> <li> Nuclear Physics A, Vol. 966</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.nuclphysa.2017.06.001" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.nuclphysa.2017.06.001<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/0003-4916(79)90130-1" target="_blank" rel="noopener noreferrer" class="name">Transient relativistic thermodynamics and kinetic theory<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1979-04-01">April 1979</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Israel, W.; Stewart, J. M.</span> </li> <li> Annals of Physics, Vol. 118, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/0003-4916(79)90130-1" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/0003-4916(79)90130-1<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.nuclphysa.2010.08.011" target="_blank" rel="noopener noreferrer" class="name">Dissipative dynamics of highly anisotropic systems<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2010-12-01">December 2010</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Martinez, Mauricio; Strickland, Michael</span> </li> <li> Nuclear Physics A, Vol. 848, Issue 1-2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.nuclphysa.2010.08.011" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.nuclphysa.2010.08.011<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.95.054907" target="_blank" rel="noopener noreferrer" class="name">Anisotropic fluid dynamics for Gubser flow<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-05-01">May 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Martinez, M.; McNelis, M.; Heinz, U.</span> </li> <li> Physical Review C, Vol. 95, Issue 5</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.95.054907" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.95.054907<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.91.064903" target="_blank" rel="noopener noreferrer" class="name">Nonconformal viscous anisotropic hydrodynamics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-06-01">June 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Bazow, Dennis; Heinz, Ulrich; Martinez, Mauricio</span> </li> <li> Physical Review C, Vol. 91, Issue 6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.91.064903" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.91.064903<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.94.106011" target="_blank" rel="noopener noreferrer" class="name">Entropy production, hydrodynamics, and resurgence in the primordial quark-gluon plasma from holography<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2016-11-01">November 2016</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Buchel, Alex; Heller, Michal P.; Noronha, Jorge</span> </li> <li> Physical Review D, Vol. 94, Issue 10</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.94.106011" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.94.106011<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.115.072501" target="_blank" rel="noopener noreferrer" class="name">Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-08-01">August 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Heller, Michal P.; Spaliński, Michał</span> </li> <li> Physical Review Letters, Vol. 115, Issue 7</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.115.072501" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.115.072501<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.90.014908" target="_blank" rel="noopener noreferrer" class="name">Bulk viscous evolution within anisotropic hydrodynamics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2014-07-01">July 2014</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Nopoush, Mohammad; Ryblewski, Radoslaw; Strickland, Michael</span> </li> <li> Physical Review C, Vol. 90, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.90.014908" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.90.014908<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.81.045015" target="_blank" rel="noopener noreferrer" class="name">Anisotropic hydrodynamics, bulk viscosities, and <math display="inline"> <mi>r</mi> </math> -modes of strange quark stars with strong magnetic fields<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2010-02-01">February 2010</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Huang, Xu-Guang; Huang, Mei; Rischke, Dirk H.</span> </li> <li> Physical Review D, Vol. 81, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.81.045015" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.81.045015<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevE.51.4901" target="_blank" rel="noopener noreferrer" class="name">Generally covariant relativistic anisotropic magnetohydrodynamics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1995-05-01">May 1995</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Gedalin, M.; Oiberman, I.</span> </li> <li> Physical Review E, Vol. 51, Issue 5</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevE.51.4901" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevE.51.4901<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.50.7222" target="_blank" rel="noopener noreferrer" class="name">Formalizing the slow-roll approximation in inflation<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1994-12-01">December 1994</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Liddle, Andrew R.; Parsons, Paul; Barrow, John D.</span> </li> <li> Physical Review D, Vol. 50, Issue 12</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.50.7222" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.50.7222<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.95.034906" target="_blank" rel="noopener noreferrer" class="name">Quasiparticle anisotropic hydrodynamics for central collisions<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-03-01">March 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Alqahtani, Mubarak; Nopoush, Mohammad; Strickland, Michael</span> </li> <li> Physical Review C, Vol. 95, Issue 3</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.95.034906" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.95.034906<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.94.125003" target="_blank" rel="noopener noreferrer" class="name">Closing the equations of motion of anisotropic fluid dynamics by a judicious choice of a moment of the Boltzmann equation<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2016-12-01">December 2016</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Molnár, E.; Niemi, H.; Rischke, D. H.</span> </li> <li> Physical Review D, Vol. 94, Issue 12</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.94.125003" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.94.125003<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.aop.2011.08.001" target="_blank" rel="noopener noreferrer" class="name">Kubo formulas for relativistic fluids in strong magnetic fields<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2011-12-01">December 2011</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Huang, Xu-Guang; Sedrakian, Armen; Rischke, Dirk H.</span> </li> <li> Annals of Physics, Vol. 326, Issue 12</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.aop.2011.08.001" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.aop.2011.08.001<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1140/epjc/s10052-016-4567-x" target="_blank" rel="noopener noreferrer" class="name">Do nuclear collisions create a locally equilibrated quark–gluon plasma?<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-01-01">January 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Romatschke, P.</span> </li> <li> The European Physical Journal C, Vol. 77, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1140/epjc/s10052-016-4567-x" class="text-muted" target="_blank" rel="noopener noreferrer">10.1140/epjc/s10052-016-4567-x<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.94.125006" target="_blank" rel="noopener noreferrer" class="name">Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaître-Robertson-Walker spacetime<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2016-12-01">December 2016</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Bazow, D.; Denicol, G. S.; Heinz, U.</span> </li> <li> Physical Review D, Vol. 94, Issue 12</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.94.125006" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.94.125006<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1002/cpa.3160020403" target="_blank" rel="noopener noreferrer" class="name">On the kinetic theory of rarefied gases<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1949-12-01">December 1949</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Grad, Harold</span> </li> <li> Communications on Pure and Applied Mathematics, Vol. 2, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1002/cpa.3160020403" class="text-muted" target="_blank" rel="noopener noreferrer">10.1002/cpa.3160020403<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.120.012301" target="_blank" rel="noopener noreferrer" class="name">Relativistic Fluid Dynamics Far From Local Equilibrium<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2018-01-01">January 2018</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Romatschke, Paul</span> </li> <li> Physical Review Letters, Vol. 120, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.120.012301" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.120.012301<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/S0370-2693(01)00191-5" target="_blank" rel="noopener noreferrer" class="name">“Bottom-up” thermalization in heavy ion collisions<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2001-03-01">March 2001</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Baier, R.; Mueller, A. H.; Schiff, D.</span> </li> <li> Physics Letters B, Vol. 502, Issue 1-4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/S0370-2693(01)00191-5" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/S0370-2693(01)00191-5<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.79.044903" target="_blank" rel="noopener noreferrer" class="name">Constraining relativistic viscous hydrodynamical evolution<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2009-04-01">April 2009</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Martinez, Mauricio; Strickland, Michael</span> </li> <li> Physical Review C, Vol. 79, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.79.044903" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.79.044903<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1007/s002220100153" target="_blank" rel="noopener noreferrer" class="name">On the formation of singularities of solutions of nonlinear differential systems in antistokes directions<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2001-09-01">September 2001</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Costin, O.; Costin, R. D.</span> </li> <li> Inventiones mathematicae, Vol. 145, Issue 3</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/s002220100153" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/s002220100153<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.87.061901" target="_blank" rel="noopener noreferrer" class="name">Holographic thermalization with radial flow<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2013-03-01">March 2013</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> van der Schee, Wilke</span> </li> <li> Physical Review D, Vol. 87, Issue 6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.87.061901" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.87.061901<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.89.114007" target="_blank" rel="noopener noreferrer" class="name">Universal attractor in a highly occupied non-Abelian plasma<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2014-06-01">June 2014</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Berges, J.; Boguslavski, K.; Schlichting, S.</span> </li> <li> Physical Review D, Vol. 89, Issue 11</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.89.114007" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.89.114007<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.106.122004" target="_blank" rel="noopener noreferrer" class="name">“Ridge” in Proton-Proton Scattering at 7 TeV<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2011-03-01">March 2011</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Werner, K.; Karpenko, Iu.; Pierog, T.</span> </li> <li> Physical Review Letters, Vol. 106, Issue 12</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.106.122004" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.106.122004<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.85.114047" target="_blank" rel="noopener noreferrer" class="name">Derivation of transient relativistic fluid dynamics from the Boltzmann equation<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-06-01">June 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Denicol, G. S.; Niemi, H.; Molnár, E.</span> </li> <li> Physical Review D, Vol. 85, Issue 11</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.85.114047" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.85.114047<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.113.202301" target="_blank" rel="noopener noreferrer" class="name">New Exact Solution of the Relativistic Boltzmann Equation and its Hydrodynamic Limit<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2014-11-01">November 2014</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Denicol, Gabriel S.; Heinz, Ulrich; Martinez, Mauricio</span> </li> <li> Physical Review Letters, Vol. 113, Issue 20</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.113.202301" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.113.202301<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.physletb.2017.11.059" target="_blank" rel="noopener noreferrer" class="name">On the hydrodynamic attractor of Yang–Mills plasma<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2018-01-01">January 2018</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Spaliński, Michał</span> </li> <li> Physics Letters B, Vol. 776</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.physletb.2017.11.059" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.physletb.2017.11.059<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.93.114025" target="_blank" rel="noopener noreferrer" class="name">Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2016-06-01">June 2016</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Molnár, Etele; Niemi, Harri; Rischke, Dirk H.</span> </li> <li> Physical Review D, Vol. 93, Issue 11</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.93.114025" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.93.114025<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.90.054910" target="_blank" rel="noopener noreferrer" class="name">Second-order <math> <mrow> <mo>(</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </math> -dimensional anisotropic hydrodynamics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2014-11-01">November 2014</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Bazow, Dennis; Heinz, Ulrich; Strickland, Michael</span> </li> <li> Physical Review C, Vol. 90, Issue 5</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.90.054910" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.90.054910<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1140/epjc/s10052-010-1530-0" target="_blank" rel="noopener noreferrer" class="name">Elliptic flow in proton–proton collisions at $\sqrt{s}=7$  TeV<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2011-01-01">January 2011</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Bożek, Piotr</span> </li> <li> The European Physical Journal C, Vol. 71, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1140/epjc/s10052-010-1530-0" class="text-muted" target="_blank" rel="noopener noreferrer">10.1140/epjc/s10052-010-1530-0<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.nuclphysa.2013.08.004" target="_blank" rel="noopener noreferrer" class="name">Anisotropic hydrodynamics for rapidly expanding systems<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2013-10-01">October 2013</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Florkowski, Wojciech; Ryblewski, Radoslaw; Strickland, Michael</span> </li> <li> Nuclear Physics A, Vol. 916</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.nuclphysa.2013.08.004" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.nuclphysa.2013.08.004<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.119.042301" target="_blank" rel="noopener noreferrer" class="name"><math display="inline"> <mrow> <mo stretchy="false">(</mo> <mrow> <mn>3</mn> <mo>+</mo> <mn>1</mn> </mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">D</mi> </mrow> </math> Quasiparticle Anisotropic Hydrodynamics for Ultrarelativistic Heavy-Ion Collisions<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-07-01">July 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Alqahtani, Mubarak; Nopoush, Mohammad; Ryblewski, Radoslaw</span> </li> <li> Physical Review Letters, Vol. 119, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.119.042301" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.119.042301<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.88.024903" target="_blank" rel="noopener noreferrer" class="name">Testing viscous and anisotropic hydrodynamics in an exactly solvable case<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2013-08-01">August 2013</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Florkowski, Wojciech; Ryblewski, Radoslaw; Strickland, Michael</span> </li> <li> Physical Review C, Vol. 88, Issue 2</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.88.024903" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.88.024903<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.91.014903" target="_blank" rel="noopener noreferrer" class="name">Solutions of conformal Israel-Stewart relativistic viscous fluid dynamics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-01-01">January 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Marrochio, Hugo; Noronha, Jorge; Denicol, Gabriel S.</span> </li> <li> Physical Review C, Vol. 91, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.91.014903" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.91.014903<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.92.125011" target="_blank" rel="noopener noreferrer" class="name">Hydrodynamics, resurgence, and transasymptotics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-12-01">December 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Başar, Gökçe; Dunne, Gerald V.</span> </li> <li> Physical Review D, Vol. 92, Issue 12</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.92.125011" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.92.125011<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.89.054908" target="_blank" rel="noopener noreferrer" class="name">Exact solution of the ( <math> <mrow> <mn>0</mn> <mo>+</mo> <mn>1</mn> </mrow> </math> )-dimensional Boltzmann equation for a massive gas<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2014-05-01">May 2014</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Florkowski, Wojciech; Maksymiuk, Ewa; Ryblewski, Radoslaw</span> </li> <li> Physical Review C, Vol. 89, Issue 5</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.89.054908" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.89.054908<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.nuclphysa.2015.08.009" target="_blank" rel="noopener noreferrer" class="name">Investigating the domain of validity of the Gubser solution to the Boltzmann equation<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-11-01">November 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Heinz, Ulrich; Martinez, Mauricio</span> </li> <li> Nuclear Physics A, Vol. 943</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.nuclphysa.2015.08.009" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.nuclphysa.2015.08.009<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1142/S0129183111016920" target="_blank" rel="noopener noreferrer" class="name">SHOCK WAVE COLLISIONS IN AdS <sub>5</sub> : APPROXIMATE NUMERICAL SOLUTIONS<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2011-12-01">December 2011</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Wu, Bin; Romatschke, Paul</span> </li> <li> International Journal of Modern Physics C, Vol. 22, Issue 12</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1142/S0129183111016920" class="text-muted" target="_blank" rel="noopener noreferrer">10.1142/S0129183111016920<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevA.92.043602" target="_blank" rel="noopener noreferrer" class="name">Dissipative fluid dynamics for the dilute Fermi gas at unitarity: Anisotropic fluid dynamics<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2015-10-01">October 2015</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Bluhm, M.; Schäfer, T.</span> </li> <li> Physical Review A, Vol. 92, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevA.92.043602" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevA.92.043602<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1016/j.na.2011.01.032" target="_blank" rel="noopener noreferrer" class="name">Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2011-07-01">July 2011</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Giesl, Peter; Wendland, Holger</span> </li> <li> Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, Issue 10</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.na.2011.01.032" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.na.2011.01.032<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.4310/AMSA.2017.v2.n1.a3" target="_blank" rel="noopener noreferrer" class="name">Toward Picard–Lefschetz theory of path integrals, complex saddles and resurgence<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-01-01">January 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Behtash, Alireza; Dunne, Gerald V.; Schäfer, Thomas</span> </li> <li> Annals of Mathematical Sciences and Applications, Vol. 2, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.4310/AMSA.2017.v2.n1.a3" class="text-muted" target="_blank" rel="noopener noreferrer">10.4310/AMSA.2017.v2.n1.a3<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.96.044910" target="_blank" rel="noopener noreferrer" class="name">Anisotropic hydrodynamic modeling of 2.76 TeV Pb-Pb collisions<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-10-01">October 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Alqahtani, Mubarak; Nopoush, Mohammad; Ryblewski, Radoslaw</span> </li> <li> Physical Review C, Vol. 96, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.96.044910" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.96.044910<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevC.85.064901" target="_blank" rel="noopener noreferrer" class="name">Highly anisotropic hydrodynamics in <math display="inline"> <mrow> <mn mathvariant="bold">3</mn> <mo>+</mo> <mn mathvariant="bold">1</mn> </mrow> </math> space-time dimensions<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-06-01">June 2012</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Ryblewski, Radoslaw; Florkowski, Wojciech</span> </li> <li> Physical Review C, Vol. 85, Issue 6</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevC.85.064901" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevC.85.064901<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.111.181601" target="_blank" rel="noopener noreferrer" class="name">From Full Stopping to Transparency in a Holographic Model of Heavy Ion Collisions<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2013-10-01">October 2013</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Casalderrey-Solana, Jorge; Heller, Michal P.; Mateos, David</span> </li> <li> Physical Review Letters, Vol. 111, Issue 18</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.111.181601" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.111.181601<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.56.1571" target="_blank" rel="noopener noreferrer" class="name">Divergence of the Chapman-Enskog Expansion<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="1986-04-01">April 1986</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Santos, Andres; Brey, J. Javier; Dufty, James W.</span> </li> <li> Physical Review Letters, Vol. 56, Issue 15</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.56.1571" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.56.1571<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> <div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevD.70.043538" target="_blank" rel="noopener noreferrer" class="name">Turbulent thermalization<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2004-08-01">August 2004</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Micha, Raphael; Tkachev, Igor I.</span> </li> <li> Physical Review D, Vol. 70, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.70.043538" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.70.043538<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div> </div> <div class="pagination-container small"> <a class="pure-button prev page" href="#" rel="prev"><span class="fa fa-angle-left"></span></a><ul class="pagination d-inline-block" style="padding-left:.2em;"></ul><a class="pure-button next page" href="#" rel="next"><span class="fa fa-angle-right"></span></a> </div> </div> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All References</a></li> <li class="small" style="margin-left:.75em; text-transform:capitalize;"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="journal"><span class="fa fa-angle-right"></span> journal<small class="text-muted"> (74)</small></a></li> </ul> <div style="margin-top:2em;"> <form class="pure-form small text-muted reference-search"> <label for="reference-search-text" class="sr-only">Search</label> <input class="search form-control pure-input-1" id="reference-search-text" placeholder="Search" style="margin-bottom:10px;" /> <fieldset> <div style="margin-left:1em; font-weight:normal; line-height: 1.6em;"><input type="radio" class="sort" name="references-sort" data-sort="name" style="position:relative;top:2px;" id="reference-search-sort-name"><label for="reference-search-sort-name" style="margin-left: .3em;">Sort by title</label></div> <div style="margin-left:1em; font-weight:normal; line-height: 1.6em;"><input type="radio" class="sort" name="references-sort" data-sort="date" data-order="desc" style="position:relative;top:2px;" id="reference-search-sort-date"><label for="reference-search-sort-date" style="margin-left: .3em;">Sort by date</label></div> </fieldset> <div class="text-left" style="margin-left:1em;"> <a href="" class="filter-clear clearfix" title="Clear filter / sort" style="font-weight:normal; float:none;">[ × clear filter / sort ]</a> </div> </form> </div> </div> </div> </section> <section id="biblio-related" class="tab-content tab-content-sec " data-tab="biblio"> <div class="row"> <div class="col-sm-9 order-sm-9"> <section id="biblio-similar" class="tab-content tab-content-sec active" data-tab="related"> <div class="padding"> <p class="lead text-muted" style="font-size: 18px; margin-top:0px;">Similar Records in DOE PAGES and OSTI.GOV collections:</p> <aside> <ul class="item-list" itemscope itemtype="http://schema.org/ItemList" style="padding-left:0; list-style-type: none;"> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="1" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1604297-exact-solutions-attractors-higher-order-viscous-fluid-dynamics-bjorken-flow" itemprop="url">Exact solutions and attractors of higher-order viscous fluid dynamics for Bjorken flow</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Jaiswal, Sunil</span> ; <span class="author">Chattopadhyay, Chandrodoy</span> ; <span class="author">Jaiswal, Amaresh</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review C</span> </span> </div> <div class="abstract">We consider causal higher order theories of relativistic viscous hydrodynamics in the limit of one-dimensional boost-invariant expansion and study the associated dynamical attractor. We obtain evolution equations for the inverse Reynolds number as a function of Knudsen number. The solutions of these equations exhibit attractor behavior which we analyze in terms of Lyapunov exponents using several different techniques. We compare the attractors of the second-order Müller-Israel-Stewart (MIS), transient Denicol-Niemi-Molnar-Rischke (DNMR), and third-order theories with the exact solution of the Boltzmann equation in the relaxation-time approximation. It is shown that for Bjorken flow the third-order theory provides a better approximation to<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> the exact kinetic theory attractor than both MIS and DNMR theories. For three different choices of the time dependence of the shear relaxation rate we find analytical solutions for the energy density and shear stress and use these to study the attractors analytically. By studying these analytical solutions at both small and large Knudsen numbers we characterize and uniquely determine the attractors and Lyapunov exponents. While for small Knudsen numbers the approach to the attractor is exponential, strong power-law decay of deviations from the attractor and rapid loss of initial state memory are found even for large Knudsen numbers. Implications for the applicability of hydrodynamics in far-off-equilibrium situations are discussed.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 1<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevC.100.034901" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1604297" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/PhysRevC.100.034901</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="2" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1502397-viscous-anisotropic-hydrodynamics-gubser-flow" itemprop="url">Viscous anisotropic hydrodynamics for the Gubser flow</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Martinez, M.</span> ; <span class="author">McNelis, M.</span> ; <span class="author">Heinz, Ulrich</span> <span class="text-muted pubdata"> - Nuclear Physics. A</span> </span> </div> <div class="abstract">In this work we describe the dynamics of a highly anisotropic system undergoing boost-invariant longitudinal and azimuthally symmetric radial expansion (Gubser flow) for arbitrary shear viscosity to entropy density ratio. We derive the equations of motion of dissipative anisotropic hydrodynamics by applying to this situation the moments method recently derived by Moln´ar et al. (MNR) [1, 2], based on an expansion around an arbitrary anisotropic one-particle distribution function. One requires an additional evolution equation in order to close the conservation laws. This is achieved by selecting the relaxation equation for the longitudinal pressure with a suitable Landau matching condition. As<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> a result one obtains two coupled differential equations for the energy density and the longitudinal pressure which respect the <em>SO</em>(3)q Ⓧ <em>SO</em>(1, 1) Ⓧ <em>Z</em><sub>2</sub> symmetry of the Gubser flow in the deSitter space. These equations are solved numerically and compared with the predictions of the recently found exact solution of the relaxation-time-approximation Boltzmann equation subject to the same flow. We also compare our numerical results with other fluid dynamical models. As a result we observe that the MNR description of anisotropic fluid dynamics reproduces the space-time evolution of the system than all other currently known hydrodynamical approaches.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1016/j.nuclphysa.2017.04.012" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1502397" data-product-type="Journal Article" data-product-subtype="AM" >10.1016/j.nuclphysa.2017.04.012</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1502397" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1502397" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="3" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1604333-anisotropic-fluid-dynamics-gubser-flow" itemprop="url">Anisotropic fluid dynamics for Gubser flow</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Martinez, M.</span> ; <span class="author">McNelis, M.</span> ; <span class="author">Heinz, Ulrich</span> <span class="text-muted pubdata"> - Physical Review C</span> </span> </div> <div class="abstract">Investigating a variety of closing schemes to the infinite hierarchy of momentum moments of the exactly solvable Boltzmann equation for systems undergoing Gubser flow, we study the precision with which the resulting hydrodynamic equations reproduce the exact evolution of hydrodynamic moments of the distribution function. We find that anisotropic hydrodynamics, obtained by expanding the distribution function around a dynamically evolving locally anisotropic background whose evolution is matched to exactly reproduce the macroscopic pressure anisotropy caused by the different longitudinal and transverse expansion rates in Gubser ow, provides the most accurate macroscopic description of the microscopic kinetic evolution. This confirms a<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> similar earlier finding for Bjorken ow [Molnar, Niemi and Rischke, Phys. Rev. D 94, 125003 (2016)]. Also, we explain the physics behind this optimal matching procedure and show that one can efficiently correct for a nonoptimized matching choice by adding a residual shear stress to the energy-momentum tensor whose evolution is again determined by the Boltzmann equation. Further insights to guide the optimal choice of a macroscopic anisotropic hydrodynamic framework for strongly-coupled systems that do not admit a microscopic kinetic description are reported.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 11<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevC.95.054907" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1604333" data-product-type="Journal Article" data-product-subtype="AM" >10.1103/PhysRevC.95.054907</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1604333" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1604333" data-product-type="Journal Article" data-product-subtype="AM" >Full Text Available</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="5" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1547837-thermalization-hydrodynamics-bjorken-gubser-flows" itemprop="url">Thermalization & hydrodynamics in Bjorken & Gubser flows</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Chattopadhyay, Chandrodoy</span> ; <span class="author">Heinz, Ulrich</span> ; <span class="author">Pal, Subrata</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Nuclear Physics. A</span> </span> </div> <div class="abstract">The dynamical scaling behavior of hydrodynamic and non-hydrodynamic moments of the distribution function is studied using third-order Chapman-Enskog hydrodynamics and anisotropic hydrodynamics for systems undergoing Bjorken and Gubser expansions, where exact solutions of the Boltzmann equation in Relaxation Time Approximation (RTA) are available for comparison. While dimensionless quantities like normalized shear, pressure anisotropy and normalized entropy show at late times universal scaling relations with small (large) Knudsen number for Bjorken (Gubser) flows, dimensionful quantities like the entropy per unit rapidity do not. Although the two hydrodynamic approximation schemes describe the exact attractors for normalized shear with high accuracy, their description<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> for the normalized entropy is less precise. We attribute this to non-negligible couplings to non-hydrodynamic modes in the entropy evolution.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1016/j.nuclphysa.2018.11.005" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1547837" data-product-type="Journal Article" data-product-subtype="PA" >10.1016/j.nuclphysa.2018.11.005</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="8" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1526329-dynamical-systems-nonlinear-transient-rheology-far-from-equilibrium-bjorken-flow" itemprop="url">Dynamical systems and nonlinear transient rheology of the far-from-equilibrium Bjorken flow</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Journal Article</small><span class="authors"> <span class="author">Behtash, Alireza</span> ; <span class="author">Kamata, Syo</span> ; <span class="author">Martinez, Mauricio</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">In relativistic kinetic theory, the one-particle distribution function is approximated by an asymptotic perturbative power series in the Knudsen number which is divergent. For the Bjorken flow, we expand the distribution function in terms of its moments and study their nonlinear evolution equations. The resulting coupled dynamical system can be solved for each moment consistently using a multiparameter transseries which makes the constitutive relations inherit the same structure. A new nonperturbative dynamical renormalization scheme is born out of this formalism that goes beyond the linear response theory.We show that there is a Lyapunov function, also known as dynamical potential, which<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> is, in general, a function of the moments and time satisfying Lyapunov stability conditions along renormalization group flows connected to the asymptotic hydrodynamic fixed point. As a result, the transport coefficients get dynamically renormalized at every order in the time-dependent perturbative expansion by receiving nonperturbative corrections present in the transseries. The connection between the integration constants and the UV data is discussed using the language of dynamical systems. Furthermore, we show that the first dissipative correction in the Knudsen number to the distribution function is not only determined by the known effective shear viscous term but also a new high-energy nonhydrodynamic mode. It is demonstrated that the survival of this new mode is intrinsically related to the nonlinear mode-to-mode coupling with the shear viscous term. Finally, we comment on some possible phenomenological applications of the proposed nonhydrodynamic transport theory.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 8<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.99.116012" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1526329" data-product-type="Journal Article" data-product-subtype="PA" >10.1103/PhysRevD.99.116012</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> </ul> </aside> </div> </section> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a class="tab-nav disabled" data-tab="related" style="color: #636c72 !important; opacity: 1;"><span class="fa fa-angle-right"></span> Similar Records</a></li> </ul> </div> </div> </section> </div></div> </div> </div> </section> <footer class="" style="background-color:#f9f9f9; /* padding-top: 0.5rem; */"> <div class="footer-minor"> <div class="container"> <hr class="footer-separator" /> <div class="text-center" style="margin-top:1.25rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list" id="footer-org-menu"> <li class="pure-menu-item"> <a href="https://energy.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-us-doe-min" alt="U.S. Department of Energy" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.energy.gov/science/office-science" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-office-of-science-min" alt="Office of Science" /> </a> </li> <li class="pure-menu-item"> <a href="/"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-osti-min" alt="Office of Scientific and Technical Information" /> </a> </li> </ul> </div> </div> <div class="text-center small" style="margin-top:0.5em;margin-bottom:2.0rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list"> <li class="pure-menu-item"><a href="/disclaim" class="pure-menu-link"><span class="fa fa-institution"></span> Website Policies <span class="hidden-xs">/ Important Links</span></a></li> <li class="pure-menu-item"><a href="/pages/contact" class="pure-menu-link"><span class="fa fa-comments-o"></span> Contact Us</a></li> <li class="d-block d-md-none"></li> <li class="pure-menu-item"><a href="https://www.facebook.com/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-facebook" style=""></span></a></li> <li class="pure-menu-item"><a href="https://twitter.com/OSTIgov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-twitter" style=""></span></a></li> <li class="pure-menu-item"><a href="https://www.youtube.com/user/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-youtube-play" style=""></span></a></li> </ul> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.200611.1302.css" rel="stylesheet"> <script src="/pages/js/pages.200611.1302.js"></script><noscript></noscript> <script defer src="/pages/js/pages.biblio.200611.1302.js"></script><noscript></noscript> <script defer src="/pages/js/lity.js"></script><noscript></noscript> <script async type="text/javascript" src="/pages/js/Universal-Federated-Analytics-Min.js?agency=DOE" id="_fed_an_ua_tag"></script><noscript></noscript> </body> <!-- DOE PAGES v.200611.1302 --> </html>