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Title: Anisotropic hydrodynamics with a scalar collisional kernel

Journal Article · · Physical Review C

Prior studies of nonequilibrium dynamics using anisotropic hydrodynamics have used the relativistic Anderson-Witting scattering kernel or some variant thereof. In this paper, we make the first study of the impact of using a more realistic scattering kernel. For this purpose, we consider a conformal system undergoing transversally homogenous and boost-invariant Bjorken expansion and take the collisional kernel to be given by the leading order $$2{\leftrightarrow}2$$ scattering kernel in scalar $${\lambda}{{\phi}}^{4}$$. We consider both classical and quantum statistics to assess the impact of Bose enhancement on the dynamics. We also determine the anisotropic nonequilibrium attractor of a system subject to this collisional kernel. We find that, when the near-equilibrium relaxation-times in the Anderson-Witting and scalar collisional kernels are matched, the scalar kernel results in a higher degree of momentum-space anisotropy during the system's evolution, given the same initial conditions. Additionally, we find that taking into account Bose enhancement further increases the dynamically generated momentum-space anisotropy.

Research Organization:
Kent State Univ., Kent, OH (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Nuclear Physics (NP)
Grant/Contract Number:
SC0013470
OSTI ID:
1434185
Alternate ID(s):
OSTI ID: 1503825
Journal Information:
Physical Review C, Journal Name: Physical Review C Vol. 97 Journal Issue: 4; ISSN 2469-9985
Publisher:
American Physical SocietyCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 27 works
Citation information provided by
Web of Science

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padding-left: 0; line-height:1.8em;"> <li class="authors">Florkowski, Wojciech; Ryblewski, Radoslaw; Strickland, Michael</li> <li class="mb-0-5 source"> <span class="mr-1">Physical Review C, Vol. 88, Issue 2</span> <span class="text-muted"><a href="https://doi.org/10.1103/PhysRevC.88.024903" class="text-muted" target="_blank" rel="noopener noreferrer">https://doi.org/10.1103/PhysRevC.88.024903<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> </td> <td class="small text-muted biblio-refs-td-secondary"> <span class="biblio-refs-type-badge ml-1 type" data-type="journal">journal</span> </td> <td class="small text-muted biblio-refs-td-secondary biblio-refs-td-secondary-date"><span class="date" data-date="2013-08-01">August 2013</span></td> </tr><tr> <td> <a href="https://doi.org/10.1007/JHEP09(2013)026" target="_blank" rel="noopener noreferrer" class="title" data-title="Holographic isotropization linearized">Holographic isotropization linearized<span class="fa fa-external-link" aria-hidden="true"></span></a> <ul class="small references-list mb-0 mt-0" style="list-style-type:none; padding-left: 0; line-height:1.8em;"> <li class="authors">Heller, Michal P.; Mateos, David; van der Schee, Wilke</li> <li class="mb-0-5 source"> <span class="mr-1">Journal of High Energy Physics, Vol. 2013, Issue 9</span> <span class="text-muted"><a href="https://doi.org/10.1007/JHEP09(2013)026" class="text-muted" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/JHEP09(2013)026<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> </td> <td class="small text-muted biblio-refs-td-secondary"> <span class="biblio-refs-type-badge ml-1 type" data-type="journal">journal</span> </td> <td class="small text-muted biblio-refs-td-secondary biblio-refs-td-secondary-date"><span class="date" data-date="2013-09-01">September 2013</span></td> </tr><tr> <td> <a href="https://doi.org/10.1016/j.nuclphysa.2016.02.014" target="_blank" rel="noopener noreferrer" class="title" data-title="Anisotropic hydrodynamics for conformal Gubser flow">Anisotropic hydrodynamics for conformal Gubser flow<span class="fa fa-external-link" aria-hidden="true"></span></a> <ul class="small references-list mb-0 mt-0" style="list-style-type:none; padding-left: 0; line-height:1.8em;"> <li class="authors">Strickland, Michael; Nopoush, Mohammad; Ryblewski, Radoslaw</li> <li class="mb-0-5 source"> <span class="mr-1">Nuclear Physics A, Vol. 956</span> <span class="text-muted"><a href="https://doi.org/10.1016/j.nuclphysa.2016.02.014" class="text-muted" target="_blank" rel="noopener noreferrer">https://doi.org/10.1016/j.nuclphysa.2016.02.014<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> </td> <td class="small text-muted biblio-refs-td-secondary"> <span class="biblio-refs-type-badge ml-1 type" data-type="journal">journal</span> </td> <td class="small text-muted biblio-refs-td-secondary biblio-refs-td-secondary-date"><span class="date" data-date="2016-12-01">December 2016</span></td> </tr><tr> <td> <a href="https://doi.org/10.1103/PhysRevA.92.043602" target="_blank" rel="noopener noreferrer" class="title" data-title="Dissipative fluid dynamics for the dilute Fermi gas at unitarity Anisotropic fluid dynamics">Dissipative fluid dynamics for the dilute Fermi gas at unitarity: Anisotropic fluid dynamics<span class="fa fa-external-link" aria-hidden="true"></span></a> <ul class="small references-list mb-0 mt-0" style="list-style-type:none; padding-left: 0; line-height:1.8em;"> <li class="authors">Bluhm, M.; Schäfer, T.</li> <li class="mb-0-5 source"> <span class="mr-1">Physical Review A, Vol. 92, Issue 4</span> <span class="text-muted"><a href="https://doi.org/10.1103/PhysRevA.92.043602" class="text-muted" target="_blank" rel="noopener noreferrer">https://doi.org/10.1103/PhysRevA.92.043602<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> </td> <td class="small text-muted biblio-refs-td-secondary"> <span class="biblio-refs-type-badge ml-1 type" data-type="journal">journal</span> </td> <td class="small text-muted biblio-refs-td-secondary biblio-refs-td-secondary-date"><span class="date" data-date="2015-10-01">October 2015</span></td> </tr><tr> <td> <a href="https://doi.org/10.1103/PhysRevC.96.044910" target="_blank" rel="noopener noreferrer" class="title" data-title="Anisotropic hydrodynamic modeling of 276 TeV PbPb collisions">Anisotropic hydrodynamic modeling of 2.76 TeV Pb-Pb collisions<span class="fa fa-external-link" aria-hidden="true"></span></a> <ul class="small references-list mb-0 mt-0" style="list-style-type:none; padding-left: 0; line-height:1.8em;"> <li class="authors">Alqahtani, Mubarak; Nopoush, Mohammad; Ryblewski, Radoslaw</li> <li class="mb-0-5 source"> <span class="mr-1">Physical Review C, Vol. 96, Issue 4</span> <span class="text-muted"><a href="https://doi.org/10.1103/PhysRevC.96.044910" class="text-muted" target="_blank" rel="noopener noreferrer">https://doi.org/10.1103/PhysRevC.96.044910<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> </td> <td class="small text-muted biblio-refs-td-secondary"> <span class="biblio-refs-type-badge ml-1 type" data-type="journal">journal</span> </td> <td class="small text-muted biblio-refs-td-secondary biblio-refs-td-secondary-date"><span class="date" data-date="2017-10-01">October 2017</span></td> </tr><tr> <td> <a href="https://doi.org/10.1103/PhysRevC.85.064901" target="_blank" rel="noopener noreferrer" class="title" data-title="Highly anisotropic hydrodynamics in math displayinline mrow mn mathvariantbold3mn momo mn mathvariantbold1mn mrow math spacetime dimensions">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> <ul class="small references-list mb-0 mt-0" style="list-style-type:none; padding-left: 0; line-height:1.8em;"> <li class="authors">Ryblewski, Radoslaw; Florkowski, Wojciech</li> <li class="mb-0-5 source"> <span class="mr-1">Physical Review C, Vol. 85, Issue 6</span> <span class="text-muted"><a href="https://doi.org/10.1103/PhysRevC.85.064901" class="text-muted" target="_blank" rel="noopener noreferrer">https://doi.org/10.1103/PhysRevC.85.064901<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> </td> <td class="small text-muted biblio-refs-td-secondary"> <span class="biblio-refs-type-badge ml-1 type" data-type="journal">journal</span> </td> <td class="small text-muted biblio-refs-td-secondary biblio-refs-td-secondary-date"><span class="date" data-date="2012-06-01">June 2012</span></td> </tr><tr> <td> <a href="https://doi.org/10.1103/PhysRevLett.111.181601" target="_blank" rel="noopener noreferrer" class="title" data-title="From Full Stopping to Transparency in a Holographic Model of Heavy Ion Collisions">From Full Stopping to Transparency in a Holographic Model of Heavy Ion Collisions<span class="fa fa-external-link" aria-hidden="true"></span></a> <ul class="small references-list mb-0 mt-0" style="list-style-type:none; padding-left: 0; line-height:1.8em;"> <li class="authors">Casalderrey-Solana, Jorge; Heller, Michal P.; Mateos, David</li> <li class="mb-0-5 source"> <span class="mr-1">Physical Review Letters, Vol. 111, Issue 18</span> <span class="text-muted"><a href="https://doi.org/10.1103/PhysRevLett.111.181601" class="text-muted" target="_blank" rel="noopener noreferrer">https://doi.org/10.1103/PhysRevLett.111.181601<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> </td> <td class="small text-muted biblio-refs-td-secondary"> <span class="biblio-refs-type-badge ml-1 type" data-type="journal">journal</span> </td> <td class="small text-muted biblio-refs-td-secondary biblio-refs-td-secondary-date"><span class="date" data-date="2013-10-01">October 2013</span></td> </tr><tr> <td> <a href="https://doi.org/10.1007/JHEP01(2017)026" target="_blank" rel="noopener noreferrer" class="title" data-title="Holographic collisions in nonconformal theories">Holographic collisions in non-conformal theories<span class="fa fa-external-link" aria-hidden="true"></span></a> <ul class="small references-list mb-0 mt-0" style="list-style-type:none; padding-left: 0; line-height:1.8em;"> <li class="authors">Attems, Maximilian; Casalderrey-Solana, Jorge; Mateos, David</li> <li class="mb-0-5 source"> <span class="mr-1">Journal of High Energy Physics, Vol. 2017, Issue 1</span> <span class="text-muted"><a href="https://doi.org/10.1007/JHEP01(2017)026" class="text-muted" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/JHEP01(2017)026<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> </td> <td class="small text-muted biblio-refs-td-secondary"> <span class="biblio-refs-type-badge ml-1 type" data-type="journal">journal</span> </td> <td class="small text-muted biblio-refs-td-secondary biblio-refs-td-secondary-date"><span class="date" data-date="2017-01-01">January 2017</span></td> </tr></tbody></table> </div> </div> </section> <hr class="mt-2 mb-2"/> <section id="section-citations" data-list="list_cite" class="tab-content-sec osti-curated biblio-section section-scroll" style="scroll-margin-top:110px;"> <div class="biblio-section-navbar active row"> <div class="col"> <h2 class="abstract biblio-section-title mt-0 mb-0-5" style="cursor:pointer;"><span class="fa indicator"></span> Cited By (1) <span class="text-muted list-table-filternum"></span></h2> </div> <div class="col-md-12 col-lg-auto"> <form class="pure-form reference-filter d-print-none"> <div class="small biblio-section-navbar-tool"> <label for="list-table-search-citations" class="sr-only">Search citations: </label> <input type="text" class="list-table-search mt-0 mb-0-5 ml-1" id="list-table-search-citations" placeholder="Search citations ..."> </div> <div class="small biblio-section-navbar-tool"> <select class="list-table-sort mt-0 mb-0-5"> <option value="date">Sort by Date</option> <option value="title">Sort by Title</option> </select> </div> <div class="small biblio-section-navbar-tool"> <button type="reset" class="pure-button list-table-reset mt-0 mb-0-5">↺</button> </div> </form> </div> </div> <div class="row"> <div class="col-sm-12"> <table id="list-table-citations" class="truncate-rows list-table table-striped" data-rows="5" style="width:100%;"><tbody class="list"><tr> <td> <a href="https://doi.org/10.1103/physrevd.99.116004" target="_blank" rel="noopener noreferrer" class="title" data-title="Hydrodynamic attractor and the fate of perturbative expansions in Gubser flow">Hydrodynamic attractor and the fate of perturbative expansions in Gubser flow<span class="fa fa-external-link" aria-hidden="true"></span></a> <ul class="small references-list mb-0 mt-0" style="list-style-type:none; padding-left: 0; line-height:1.8em;"> <li class="authors">Denicol, Gabriel S.; Noronha, Jorge</li> <li class="mb-0-5 source"> <span class="mr-1">Physical Review D, Vol. 99, Issue 11</span> <span class="text-muted"><a href="https://doi.org/10.1103/physrevd.99.116004" class="text-muted" target="_blank" rel="noopener noreferrer">https://doi.org/10.1103/physrevd.99.116004<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> </td> <td class="small text-muted biblio-refs-td-secondary"> <span class="biblio-refs-type-badge ml-1 type" data-type="journal">journal</span> </td> <td class="small text-muted biblio-refs-td-secondary biblio-refs-td-secondary-date"><span class="date" data-date="2019-06-01">June 2019</span></td> </tr></tbody></table> </div> </div> </section> <hr class="mt-2 mb-2"/> <section id="section-images" data-list="list_img" class="tab-content-sec osti-curated biblio-section section-scroll" style="scroll-margin-top:110px;"> <div class="biblio-section-navbar active row"> <div class="col"> <h2 class="abstract biblio-section-title mt-0 mb-0-5" style="cursor:pointer;"><span class="fa indicator"></span> Figures / Tables (10) <span class="text-muted list-table-filternum"></span></h2> </div> <div class="col-md-12 col-lg-auto"> <form class="pure-form reference-filter d-print-none"> <div class="small biblio-section-navbar-tool"> <label for="list-table-search-figures" class="sr-only">Search figures / tables: </label> <input type="text" class="list-table-search mt-0 mb-0-5 ml-1" id="list-table-search-figures" placeholder="Search figures / tables ..."> </div> <div class="small biblio-section-navbar-tool"> <select class="list-table-sort mt-0 mb-0-5"> <option value="date">Sort by Date</option> <option value="title">Sort by Title</option> </select> </div> <div class="small biblio-section-navbar-tool"> <button type="reset" class="pure-button list-table-reset mt-0 mb-0-5">↺</button> </div> </form> </div> </div> <div class="row"> <div class="col-sm-12"> <table id="list-table-images" class="truncate-rows list-table list-table-images table-to-tiles" data-rows="1" style="width:100%;"><tbody class="list"> <tr class="biblio-image-tile "><td class="col-sm-3"> <a id="img-1434185-img51428" data-pos="first" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51428" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51428" data-apporder="p. 2" data-order="1" data-imgid="1434185-img51428" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051428.png" data-title="FIG. 1" data-desc="Generic two-to-two scattering diagram." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">FIG. 1<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 2)</small><span class="d-none type">figure</span></small> <img class="img-responsive mt-0-5" alt="FIG. 1" src="/biblio/1434185/image/000/739/0007396/2/t0051428.png"/> </div> </a> </td></tr> <tr class="biblio-image-tile "><td class="col-sm-3"> <a id="img-1434185-img51433" data-pos="" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51433" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51433" data-apporder="p. 4" data-order="2" data-imgid="1434185-img51433" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051433.png" data-title="FIG. 2" data-desc="Comparison of the LO scalar scattering kernel moments for both the classical and quantum cases with those obtained in RTA as a function of ξ . Panel (a) shows C<sup>zz</sup>/Λ<sup>6</sup> and panel (b) shows C<sup>xx</sup>/Λ<sup>6</sup>. For the purposes of this figure, we took η̄ = 0.2." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">FIG. 2<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 4)</small><span class="d-none type">figure</span></small> <img class="img-responsive mt-0-5" alt="FIG. 2" src="/biblio/1434185/image/000/739/0007396/2/t0051433.png"/> </div> </a> </td></tr> <tr class="biblio-image-tile "><td class="col-sm-3"> <a id="img-1434185-img51436" data-pos="" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51436" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51436" data-apporder="p. 5" data-order="3" data-imgid="1434185-img51436" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051436.png" data-title="FIG. 3" data-desc="Ratio of the moments of the LO scalar collisional kernel in the classical and quantum and quantum cases as a function of ξ . Panel (a) shows C$$^{zz}_{classical}$$/C$$^{zz}_{quantum}$$ and panel (b) shows C$$^{xx}_{classical}$$/C$$^{xx}_{quantum}$$. The red points show the ratio of the two Monte-Carlo results and the solid black line is a fifth-order polynomial fit to the numerical data." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">FIG. 3<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 5)</small><span class="d-none type">figure</span></small> <img class="img-responsive mt-0-5" alt="FIG. 3" src="/biblio/1434185/image/000/739/0007396/2/t0051436.png"/> </div> </a> </td></tr> <tr class="biblio-image-tile "><td class="col-sm-3"> <a id="img-1434185-img51427" data-pos="" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51427" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51427" data-apporder="p. 6" data-order="4" data-imgid="1434185-img51427" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051427.png" data-title="FIG. 4" data-desc="Comparison of W using the LO scalar and RTA kernels. Panels (a) and (b) in the top row show the classical case (a = 0) while (c) and (d) from the bottom row show the quantum case (a = 1). The left column shows the result for small values of ξ and the right column shows the result for a larger range of values of ξ ." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">FIG. 4<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 6)</small><span class="d-none type">figure</span></small> <img class="img-responsive mt-0-5" alt="FIG. 4" src="/biblio/1434185/image/000/739/0007396/2/t0051427.png"/> </div> </a> </td></tr> <tr class="biblio-image-tile tr-striped"><td class="col-sm-3"> <a id="img-1434185-img51429" data-pos="" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51429" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51429" data-apporder="p. 7" data-order="5" data-imgid="1434185-img51429" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051429.png" data-title="FIG. 5" data-desc="Comparison of the evolution of the scaled temperature (a) and pressure anisotropy (b) for an isotropic initial condition. The RTA results are indicated by a solid black line, the LO classical scalar results by a short-dashed red line, and the LO quantum scalar results by a long-dashed blue line." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">FIG. 5<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 7)</small><span class="d-none type">figure</span></small> <img class="img-responsive mt-0-5" alt="FIG. 5" src="/biblio/1434185/image/000/739/0007396/2/t0051429.png"/> </div> </a> </td></tr> <tr class="biblio-image-tile tr-striped"><td class="col-sm-3"> <a id="img-1434185-img51430" data-pos="" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51430" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51430" data-apporder="p. 7" data-order="6" data-imgid="1434185-img51430" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051430.png" data-title="Table I" data-desc="Polynomial fit coefficients for the classical and quantum LO scalar W(ξ) function defined in Eq. (51). The fit was made assuming W(ξ) = ∑<sub>n</sub> c<sub>n</sub>ξ<sup>n</sup> using 101 moment evaluations in the range −0.68 ≤ ξ ≤ 99." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">Table I<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 7)</small><span class="d-none type">table</span></small> <img class="img-responsive mt-0-5" alt="Table I" src="/biblio/1434185/image/000/739/0007396/2/t0051430.png"/> </div> </a> </td></tr> <tr class="biblio-image-tile tr-striped"><td class="col-sm-3"> <a id="img-1434185-img51432" data-pos="" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51432" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51432" data-apporder="p. 8" data-order="7" data-imgid="1434185-img51432" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051432.png" data-title="FIG. 6" data-desc="Comparison of the evolution of the scaled temperature (a) and pressure anisotropy (b) for a highly oblate initial condition. Line styles are the same is in Fig. 5." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">FIG. 6<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 8)</small><span class="d-none type">figure</span></small> <img class="img-responsive mt-0-5" alt="FIG. 6" src="/biblio/1434185/image/000/739/0007396/2/t0051432.png"/> </div> </a> </td></tr> <tr class="biblio-image-tile tr-striped"><td class="col-sm-3"> <a id="img-1434185-img51435" data-pos="" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51435" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51435" data-apporder="p. 9" data-order="8" data-imgid="1434185-img51435" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051435.png" data-title="FIG. 7" data-desc="Panel (a) shows the attractor amplitude ϕ as a function of $$\bar{w}$$ and panel (b) shows the pressure anisotropy P<sub>L</sub>/P<sub>T</sub> as a function of $$\bar{w}$$. In both panels, the scalar quantum result is indicated by a solid black line, the scalar classical result is indicated by a red short-dashed line, and the RTA result is indicated by a blue long-dashed line. In panel (a), we additionally show the Navier-Stokes (NS) result as a green dot-dashed line and the asymptotic bounds ϕ = 2/3 and ϕ = 3/4 as gray and orange dotted lines, respectively." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">FIG. 7<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 9)</small><span class="d-none type">figure</span></small> <img class="img-responsive mt-0-5" alt="FIG. 7" src="/biblio/1434185/image/000/739/0007396/2/t0051435.png"/> </div> </a> </td></tr> <tr class="biblio-image-tile "><td class="col-sm-3"> <a id="img-1434185-img51434" data-pos="" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51434" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51434" data-apporder="p. 9" data-order="9" data-imgid="1434185-img51434" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051434.png" data-title="FIG. 8" data-desc="Comparison of the classical LO scalar attractor with a set of numerical solutions to the equations of motion for a variety of initial conditions. Panel (a) shows ϕ and panel (b) shows the resulting pressure anisotropy." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">FIG. 8<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 9)</small><span class="d-none type">figure</span></small> <img class="img-responsive mt-0-5" alt="FIG. 8" src="/biblio/1434185/image/000/739/0007396/2/t0051434.png"/> </div> </a> </td></tr> <tr class="biblio-image-tile "><td class="col-sm-3"> <a id="img-1434185-img51431" data-pos="last" href="#modal-image" data-modal-toggle="true" class="biblio-image-tile-a ga-click-event apporder order title desc" data-imgid="1434185-img51431" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51431" data-apporder="p. 10" data-order="10" data-imgid="1434185-img51431" data-imgsrc="/biblio/1434185/image/000/739/0007396/2/0051431.png" data-title="FIG. 9" data-desc="Comparison of the quantum LO scalar attractor with a set of numerical solutions to the equations of motion for a variety of initial conditions. Panel (a) shows ϕ and panel (b) shows the resulting pressure anisotropy." data-ostiid="1434185"> <div class="biblio-image-tile-inner"> <small class="name">FIG. 9<small class="pull-right text-muted" style="margin-right: .5em;top: 3px;position: relative;">(p. 10)</small><span class="d-none type">figure</span></small> <img class="img-responsive mt-0-5" alt="FIG. 9" src="/biblio/1434185/image/000/739/0007396/2/t0051431.png"/> </div> </a> </td></tr> </tbody></table> </div> </div> </section> <div id="modal-image" class="modal"> <div class="modal-content"> <div class="modal-header"> <span class="close" data-modal-close="true">×</span> <header id="modal-image_label" style="margin:0px;">Image title</header> </div> <div id="modal-image_body" class="modal-body"> </div> </div> </div> <hr class="mt-2 mb-2"/> <section id="section-similar" class="tab-content-sec osti-curated biblio-section section-scroll" style="scroll-margin-top:110px;"> <div class="row"> <div class="col-8" style="padding-right:40px;"> <h2 class="abstract">Similar Records</h2> <div class="" style="line-height:1.6rem;"> <a href="/biblio/1611994" target="_blank">Anisotropic hydrodynamics with number-conserving kernels</a> <div class="small" style="line-height:1.4rem;"> Journal Article <span style="padding-left: 0.75rem;padding-right: 0.75rem;">·</span> Tue Jan 15 00:00:00 EST 2019 <span style="padding-left: 0.75rem;padding-right: 0.75rem;">·</span> Physical Review. 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