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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 University, 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.
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Behtash, Alireza, Cruz-Camacho, C. N., & Martinez, M. Far-from-equilibrium attractors and nonlinear dynamical systems approach to the Gubser flow. United States. https://doi.org/10.1103/PhysRevD.97.044041
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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. https://doi.org/10.1103/PhysRevD.97.044041.
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@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 = {Mon Feb 26 00:00:00 EST 2018},
month = {Mon Feb 26 00:00:00 EST 2018}
}
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https://doi.org/10.1103/PhysRevD.97.044041
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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; 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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="sr-only">Previous Page</span><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="sr-only">Next Page</span><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; 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float:none;">[ × clear filter / sort ]</a> </div> <input type="submit" id="sort_submit_references" name="submit" aria-label="submit" style="display: none;"/> </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/1800102-global-flow-structure-exact-formal-transseries-gubser-flow-kinetic-theory" itemprop="url">Global flow structure and exact formal transseries of the Gubser flow in kinetic theory</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"> - Journal of High Energy Physics (Online)</span> </span> </div> <div class="abstract">In this work we introduce the generic conditions for the existence of a non-equilibrium attractor that is an invariant manifold determined by the long-wavelength modes of the physical system. We investigate the topological properties of the global flow structure of the Gubser flow for the Israel-Stewart theory and a kinetic model for the Boltzmann equation by employing Morse-Smale theory. We present a complete classification of the invariant submanifolds of the flow and determine all the possible flow lines connecting any pair of UV/IR fixed points. The formal transseries solutions to the Gubser dynamical system around the early-time (UV) and late-time<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> (IR) fixed points are constructed and analyzed. It is proven that these solutions are purely perturbative (or power-law asymptotic) series with a finite radius of convergence. Based on these analyses, we find that Gubser-like expanding kinetic systems do not hydrodynamize owing to the failure of the hydrodynamization process which heavily relies on the classification of (non)hydrodynamic modes in the IR regime. This is in contrast to longitudinal boost-invariant plasmas where the asymptotic dynamics is described by a few terms of the hydrodynamic gradient expansion. We finally compare our results for both Bjorken and Gubser conformal kinetic models.</span><a href='#' onclick='$(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"><a class="misc doi-link " href="https://doi.org/10.1007/jhep07(2020)226" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1800102" data-product-type="Journal Article" data-product-subtype="AM" >https://doi.org/10.1007/jhep07(2020)226</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1800102" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1800102" 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="2" /><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 47<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"><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" >https://doi.org/10.1103/PhysRevC.100.034901</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1604297" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1604297" 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="4" /><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;"> <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"><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" >https://doi.org/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="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;"> <span class="fa fa-book text-muted" aria-hidden="true"></span> Cited by 2<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"><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" >https://doi.org/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="7" /><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 27<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"><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" >https://doi.org/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> </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; 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