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Title: Anisotropic hydrodynamics with a scalar collisional kernel

Abstract

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.

Authors:
;
Publication Date:
Research Org.:
Kent State Univ., Kent, OH (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Nuclear Physics (NP)
OSTI Identifier:
1434185
Alternate Identifier(s):
OSTI ID: 1503825
Grant/Contract Number:  
SC0013470
Resource Type:
Published Article
Journal Name:
Physical Review C
Additional Journal Information:
Journal Name: Physical Review C Journal Volume: 97 Journal Issue: 4; Journal ID: ISSN 2469-9985
Publisher:
American Physical Society
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; collective flow; hydrodynamic models; quark-gluon plasma; relativistic heavy-ion collisions; relativistic hydrodynamics

Citation Formats

Almaalol, Dekrayat, and Strickland, Michael. Anisotropic hydrodynamics with a scalar collisional kernel. United States: N. p., 2018. Web. doi:10.1103/PhysRevC.97.044911.
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Almaalol, Dekrayat, & Strickland, Michael. Anisotropic hydrodynamics with a scalar collisional kernel. United States. https://doi.org/10.1103/PhysRevC.97.044911
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Almaalol, Dekrayat, and Strickland, Michael. Fri . "Anisotropic hydrodynamics with a scalar collisional kernel". United States. https://doi.org/10.1103/PhysRevC.97.044911.
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@article{osti_1434185,
title = {Anisotropic hydrodynamics with a scalar collisional kernel},
author = {Almaalol, Dekrayat and Strickland, Michael},
abstractNote = {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.},
doi = {10.1103/PhysRevC.97.044911},
journal = {Physical Review C},
number = 4,
volume = 97,
place = {United States},
year = {Fri Apr 20 00:00:00 EDT 2018},
month = {Fri Apr 20 00:00:00 EDT 2018}
}
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https://doi.org/10.1103/PhysRevC.97.044911
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Cited by: 27 works
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Figures / Tables:

FIG. 1 FIG. 1: Generic two-to-two scattering diagram.
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Works referenced in this record:

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font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2012-07-19">July 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;"> Jacak, B. V.; Muller, B.</span> </li> <li> Science, Vol. 337, Issue 6092</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1126/science.1215901" class="text-muted" target="_blank" rel="noopener noreferrer">10.1126/science.1215901<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/10.1007/JHEP09(2013)026" target="_blank" rel="noopener noreferrer" class="name">Holographic isotropization linearized<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="2013-09-01">September 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;"> Heller, Michal P.; Mateos, David; van der Schee, Wilke</span> </li> <li> Journal of High Energy Physics, Vol. 2013, Issue 9</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/JHEP09(2013)026" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/JHEP09(2013)026<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.2016.02.014" target="_blank" rel="noopener noreferrer" class="name">Anisotropic hydrodynamics for conformal 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="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;"> Strickland, Michael; Nopoush, Mohammad; Ryblewski, Radoslaw</span> </li> <li> Nuclear Physics A, Vol. 956</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.nuclphysa.2016.02.014" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.nuclphysa.2016.02.014<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/10.1103/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/10.1007/JHEP01(2017)026" target="_blank" rel="noopener noreferrer" class="name">Holographic collisions in non-conformal theories<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;"> Attems, Maximilian; Casalderrey-Solana, Jorge; Mateos, David</span> </li> <li> Journal of High Energy Physics, Vol. 2017, Issue 1</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1007/JHEP01(2017)026" class="text-muted" target="_blank" rel="noopener noreferrer">10.1007/JHEP01(2017)026<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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margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Denicol, Gabriel S.; Noronha, Jorge</span> </li> <li> Physical Review D, Vol. 99, Issue 11</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/physrevd.99.116004" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/physrevd.99.116004<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-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All Cited By</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_citations" name="submit" aria-label="submit" style="display: none;"/> </form> </div> </div> </div> </section> <section id="biblio-images" class="tab-content tab-content-sec osti-curated" data-tab="biblio"> <div class="row"> <div class="col-sm-9 order-sm-9"> <div class="padding"> <p class="lead text-muted" style="font-size: 18px; margin-top:0px;"><span id="image-type-label">Figures / Tables</span> found in this record:</p> <div class="list clearfix"> <div class="col-sm-3 float-left biblio-image-tile" 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" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51428" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51428"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">FIG. 1 <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 2)</small><span class="d-none type">figure</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051428.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </div> <div class="col-sm-3 float-left biblio-image-tile" 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 Czz/Λ6 and panel (b) shows Cxx/Λ6. For the purposes of this figure, we took η̄ = 0.2." data-ostiid="1434185" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51433" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51433"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">FIG. 2 <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 4)</small><span class="d-none type">figure</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051433.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </div> <div class="col-sm-3 float-left biblio-image-tile" 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" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51436" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51436"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">FIG. 3 <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 5)</small><span class="d-none type">figure</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051436.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </div> <div class="col-sm-3 float-left biblio-image-tile" 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" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51427" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51427"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">FIG. 4 <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 6)</small><span class="d-none type">figure</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051427.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </div> <div class="col-sm-3 float-left biblio-image-tile" 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" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51429" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51429"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">FIG. 5 <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 7)</small><span class="d-none type">figure</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051429.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </div> <div class="col-sm-3 float-left biblio-image-tile" 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) = ∑n cnξn using 101 moment evaluations in the range −0.68 ≤ ξ ≤ 99." data-ostiid="1434185" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51430" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51430"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">Table I <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 7)</small><span class="d-none type">table</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051430.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </div> <div class="col-sm-3 float-left biblio-image-tile" 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" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51432" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51432"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">FIG. 6 <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 8)</small><span class="d-none type">figure</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051432.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </div> <div class="col-sm-3 float-left biblio-image-tile" 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 PL/PT 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" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51435" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51435"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">FIG. 7 <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 9)</small><span class="d-none type">figure</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051435.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </div> <div class="col-sm-3 float-left biblio-image-tile" 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" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51434" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51434"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">FIG. 8 <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 9)</small><span class="d-none type">figure</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051434.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </div> <div class="col-sm-3 float-left biblio-image-tile" 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" style="padding-bottom: 2em; border-bottom: 1px solid #ddd;"> <a href="#img" class="biblio-image-tile-a ga-click-event" data-imgid="1434185-img51431" data-lityx data-category="Extracted Images" data-label="biblio: image thumbnail" data-value="1434185-img51431"> <div style=" padding: .5em; border: 1px solid #eee; background-color: #fff; "> <small class="name">FIG. 9 <small class="pull-right" style="margin-right: .5em;color:#999;top: 3px;position: relative;">(p. 10)</small><span class="d-none type">figure</span></small> <div style=" background-image:url('/biblio/1434185/image/000/739/0007396/2/t0051431.png'); background-repeat:no-repeat; background-size:contain; background-position-x: center; width: 100%; height: 175px; margin-top:.5em; "> </div> </div> </a> </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-images" data-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All Images</a></li> <li class="small" style="margin-left:.75em; text-transform:capitalize;"><a href="" class="reference-type-filter tab-nav ga-click-event" data-tab="biblio-images" data-filter="type" data-pattern="figure" data-category="Extracted Images" data-label="biblio: image filter - figure"><span class="fa fa-angle-right"></span> figures<small class="text-muted"> (9)</small></a></li> <li class="small" style="margin-left:.75em; text-transform:capitalize;"><a href="" class="reference-type-filter tab-nav ga-click-event" data-tab="biblio-images" data-filter="type" data-pattern="table" data-category="Extracted Images" data-label="biblio: image filter - table"><span class="fa fa-angle-right"></span> tables<small class="text-muted"> (1)</small></a></li> </ul> <div style="margin-top:2em;"> <form class="pure-form small text-muted image-search"> <label for="image-search-text" class="sr-only">Search</label> <input class="search form-control pure-input-1" id="image-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 ga-click-event" name="images-sort" data-sort="name" style="position:relative;top:2px;" id="image-search-sort-name" data-category="Extracted Images" data-label="biblio: image sort - name"><label for="image-search-sort-name" style="margin-left: .3em;">Sort by figure / table title</label></div> <div style="margin-left:1em; font-weight:normal; line-height: 1.6em;"><input type="radio" class="sort ga-click-event" name="images-sort" data-sort="order" style="position:relative;top:2px;" id="image-search-sort-date" data-category="Extracted Images" data-label="biblio: image sort - order"><label for="image-search-sort-date" style="margin-left: .3em;">Sort by page order</label></div> </fieldset> <div class="text-left" style="margin-top:1.5em;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 class="text-muted small" style="margin-top:2em; padding:1em;"> <em>Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.</em> </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/1611994-anisotropic-hydrodynamics-number-conserving-kernels" itemprop="url">Anisotropic hydrodynamics with number-conserving kernels</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">Almaalol, Dekrayat</span> ; <span class="author">Alqahtani, Mubarak</span> ; <span class="author">Strickland, Michael</span> <span class="text-muted pubdata"> - Physical Review. C</span> </span> </div> <div class="abstract">We compare anisotropic hydrodynamics (aHydro) results obtained using the relaxation-time approximation (RTA) and leading-order (LO) scalar λΦ<sup>4</sup> collisional kernels. We extend previous work by explicitly enforcing number conservation through the incorporation of a dynamical chemical potential (fugacity) in the underlying aHydro distribution function. We focus on the case of a transversally homogenous and boost-invariant system obeying classical statistics and compare the relevant moments of the two collisional kernels. We then compare the time evolution of the aHydro microscopic parameters and components of the energy-momentum tensor. We also determine the nonequilibrium attractor using both the RTA and LO massless λΦ<sup>4</sup> number-conserving<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> kernels. Furthermore, we find that the aHydro dynamics receives quantitatively important corrections when enforcing number conservation; however, the aHydro attractor itself is not modified substantially.</span><a href='#' onclick='$(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"><a class="misc doi-link " href="https://doi.org/10.1103/physrevc.99.014903" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1611994" data-product-type="Journal Article" data-product-subtype="AM" >https://doi.org/10.1103/physrevc.99.014903</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1611994" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1611994" 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/1611989-non-equilibrium-attractor-kinetic-theory-relaxation-time-approximation" itemprop="url">The non-equilibrium attractor for kinetic theory in relaxation time approximation</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">Strickland, M.</span> <span class="text-muted pubdata"> - Journal of High Energy Physics (Online)</span> </span> </div> <div class="abstract">I demonstrate that the concept of a non-equilibrium attractor can be extended beyond the lowest-order moments typically considered in hydrodynamic treatments. Using a previously obtained exact solution to the relaxation-time approximation Boltzmann equation for a transversally homogeneous and boost-invariant system subject to Bjorken flow, I derive an equation obeyed by all moments of the one-particle distribution function. Using numerical solutions, I show that, similar to the pressure anisotropy, all moments of the distribution function exhibit attractor-like behavior wherein all initial conditions converge to a universal solution after a short time with the exception of moments which are sensitive to modes<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> with zero longitudinal momentum and high transverse momentum. In addition, I compute the exact solution for the distribution function itself on very fine lattices in momentum space and demonstrate that (a) an attractor for the full distribution function exists and (b) solutions with generic initial conditions relax to this solution, first at low momentum and later at high momentum.</span><a href='#' onclick='$(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 54<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/jhep12(2018)128" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1611989" data-product-type="Journal Article" data-product-subtype="AM" >https://doi.org/10.1007/jhep12(2018)128</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/pages/servlets/purl/1611989" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1611989" 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/1664482-nonequilibrium-attractor-high-temperature-qcd-plasmas" itemprop="url">Nonequilibrium Attractor in High-Temperature QCD Plasmas</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">Almaalol, Dekrayat</span> ; <span class="author">Kurkela, Aleksi</span> ; <span class="author">Strickland, Michael</span> <span class="text-muted pubdata"> - Physical Review Letters</span> </span> </div> <div class="abstract">We establish the existence of a far-from-equilibrium attractor in weakly coupled gauge theory undergoing one-dimensional Bjorken expansion. We demonstrate that the resulting far-from-equilibrium evolution is insensitive to certain features of the initial condition, including both the initial momentum-space anisotropy and initial occupancy. We find that this insensitivity extends beyond the energy-momentum tensor to the detailed form of the one-particle distribution function. Based on our results, we assess different procedures for reconstructing the full one-particle distribution function from the energy-momentum tensor along the attractor and discuss implications for the freeze-out procedure used in the phenomenological analysis of ultrarelativistic nuclear collisions.</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/10.1103/PhysRevLett.125.122302" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1664482" data-product-type="Journal Article" data-product-subtype="PA" >https://doi.org/10.1103/PhysRevLett.125.122302</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="/biblio/20974987-fluid-theory-magnetized-plasma-dynamics-low-collisionality" itemprop="url">Fluid theory of magnetized plasma dynamics at low collisionality</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">Ramos, J J</span> <span class="text-muted pubdata"> - Physics of Plasmas</span> </span> </div> <div class="abstract">Finite Larmor radius (FLR) fluid equations for magnetized plasmas evolving on either sonic or diamagnetic drift time scales are derived consistent with a broad low-collisionality hypothesis. The fundamental expansion parameter is the ratio {delta} between the ion Larmor radius and the shortest macroscopic length scale (including fluctuation wavelengths in the absence of small scale turbulence). The low-collisionality regime of interest is specified by assuming that the other two basic small parameters--namely, the ratio between the electron and ion masses and the ratio between the ion collision and cyclotron frequencies--are comparable to or smaller than {delta}{sup 2}. First significant order FLR<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> equations for the stress tensors and the heat fluxes are given, including a detailed discussion of the collisional terms that need be retained under the assumed orderings and of the closure terms that need be determined kinetically. This analysis is valid for any magnetic geometry and for fully electromagnetic nonlinear dynamics with arbitrarily large fluctuation amplitudes. It is also valid for strong anisotropies and does not require the distribution functions to be close to Maxwellians. With a subsidiary small-parallel-gradient ordering for large-aspect-ratio toroidal plasmas in a strong but weakly inhomogeneous magnetic field, a new system of reduced two-fluid equations is derived, rigorously taking into account all the diamagnetic effects associated with arbitrary density and anisotropic temperature gradients.</span><a href='#' onclick='$(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/10.1063/1.2717595" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="20974987" data-product-type="Journal Article" data-product-subtype="" >https://doi.org/10.1063/1.2717595</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="6" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/pages/biblio/1641864-photon-production-elliptic-flow-from-momentum-anisotropic-quark-gluon-plasma" itemprop="url">Photon production and elliptic flow from a momentum-anisotropic quark-gluon plasma</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">Kasmaei, Babak S.</span> ; <span class="author">Strickland, Michael</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="abstract">The emission of real photons from a momentum-anisotropic quark-gluon plasma (QGP) is affected by both the collective flow of the radiating medium and the modification of the local rest frame emission rate due to the anisotropic momentum distribution of partonic degrees of freedom. In this paper, we first calculate the photon production rate from an ellipsoidally momentum-anisotropic QGP including hard contributions from Compton scattering and quark pair annihilation and soft contribution calculated using the hard thermal loop approximation. We introduce a parametrization of the nonequilibrium rate in order to facilitate its further application in yield and flow calculations. We convolve the anisotropic photon rate with the space-time evolution of QGP provided by 3 + 1d anisotropic hydrodynamics to obtain the yield and the elliptic flow coefficient<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> $$v_2$$ of photons from QGP generated at Pb-Pb collisions at the LHC at 2.76 TeV and Au-Au collisions at the RHIC at 200 GeV. We investigate the effects of various parameters on the results. In particular, we analyze the sensitivity of the results to initial momentum anisotropy.</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 9<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/PhysRevD.102.014037" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1641864" data-product-type="Journal Article" data-product-subtype="PA" >https://doi.org/10.1103/PhysRevD.102.014037</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;"> <div class="footer-minor"> <div class="container"> <hr class="footer-separator"/> <br/> <div class="col text-center mt-3"> <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="https://www.osti.gov" target="_blank" rel="noopener noreferrer"> <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="col text-center small" style="margin-top: 0.5em;margin-bottom:2.0rem;"> <div class="row justify-content-center" style="color:white"> <div class="pure-menu pure-menu-horizontal" style='white-space:normal'> <ul class="pure-menu-list"> <li class="pure-menu-item"><a href="https://www.osti.gov/disclaim" class="pure-menu-link" target="_blank" ref="noopener noreferrer"><span class="fa fa-institution"></span> Website Policies <span class="d-none d-sm-inline d-print-none" style="color:#737373;">/ Important Links</span></a></li> <li class="pure-menu-item" style='float:none;'><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 mb-1"></li> <li class="pure-menu-item" style='float:none;'><a target="_blank" title="Vulnerability Disclosure Program" class="pure-menu-link" href="https://doe.responsibledisclosure.com/hc/en-us" rel="noopener noreferrer">Vulnerability Disclosure Program</a></li> <li class="d-block d-lg-none mb-1"></li> <li class="pure-menu-item" style="float:none;"><a href="https://www.facebook.com/ostigov" target="_blank" class="pure-menu-link social ext fa fa-facebook" rel="noopener noreferrer"><span class="sr-only" style="background-color: #fff; color: #333;">Facebook</span></a></li> <li class="pure-menu-item" style="float:none;"><a href="https://twitter.com/OSTIgov" target="_blank" class="pure-menu-link social ext fa fa-twitter" rel="noopener noreferrer"><span class="sr-only" style="background-color: #fff; color: #333;">Twitter</span></a></li> <li class="pure-menu-item" style="float:none;"><a href="https://www.youtube.com/user/ostigov" target="_blank" class="pure-menu-link social ext fa fa-youtube-play" rel="noopener noreferrer"><span class="sr-only" style="background-color: #fff; color: #333;">Youtube</span></a></li> </ul> </div> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.240327.0205.css" rel="stylesheet"> <script src="/pages/js/pages.240327.0205.js"></script><noscript></noscript> <script src='https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-MML-AM_CHTML' async> MathJax.Hub.Config({ tex2jax: {displayMath: [], inlineMath: [['$$','$$'], ['\\(','\\)']]}, asciimath2jax: {delimiters: []}, skipStartupTypeset: true, webFont: null, CommonHTML: { scale: 84, // mathjx wants to scale everything to 119% by default? 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