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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) (SC-26)
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.
Almaalol, Dekrayat, & Strickland, Michael. Anisotropic hydrodynamics with a scalar collisional kernel. United States. doi:10.1103/PhysRevC.97.044911.
Almaalol, Dekrayat, and Strickland, Michael. Fri . "Anisotropic hydrodynamics with a scalar collisional kernel". United States. doi:10.1103/PhysRevC.97.044911.
@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 = {2018},
month = {4}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1103/PhysRevC.97.044911

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Cited by: 4 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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margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Florkowski, Wojciech; Ryblewski, Radoslaw; Strickland, Michael</span> </li> <li> Nuclear Physics A, Vol. 916</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1016/j.nuclphysa.2013.08.004" class="text-muted" target="_blank" rel="noopener noreferrer">10.1016/j.nuclphysa.2013.08.004<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div><div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1103/PhysRevLett.119.042301" target="_blank" rel="noopener noreferrer" class="name"><math display="inline"> <mrow> <mo stretchy="false">(</mo> <mrow> <mn>3</mn> <mo>+</mo> <mn>1</mn> </mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">D</mi> </mrow> </math> Quasiparticle Anisotropic Hydrodynamics for Ultrarelativistic Heavy-Ion Collisions<span class="fa fa-external-link" aria-hidden="true"></span></a> <small class="text-muted" style="text-transform:uppercase; font-size:0.75rem;"><br/> <span class="type">journal</span>, <span class="date" data-date="2017-07-01">July 2017</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Alqahtani, Mubarak; Nopoush, Mohammad; Ryblewski, Radoslaw</span> </li> <li> Physical Review Letters, Vol. 119, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevLett.119.042301" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevLett.119.042301<span class="fa fa-external-link" aria-hidden="true"></span></a></span> </li> </ul> <hr/> </div><div> <h2 class="title" style="margin-bottom:0;" data-apporder=""> <a href="https://doi.org/10.1126/science.1215901" target="_blank" rel="noopener noreferrer" class="name">The Exploration of Hot Nuclear Matter<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="2012-07-19">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/10.1103/PhysRevD.97.044041" target="_blank" rel="noopener noreferrer" class="name">Far-from-equilibrium attractors and nonlinear dynamical systems approach to the 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="2018-02-01">February 2018</span></small> </h2> <ul class="small references-list" style="list-style-type:none; margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Behtash, Alireza; Cruz-Camacho, C. N.; Martinez, M.</span> </li> <li> Physical Review D, Vol. 97, Issue 4</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/PhysRevD.97.044041" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/PhysRevD.97.044041<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="fa fa-angle-left"></span></a><ul class="pagination d-inline-block" style="padding-left:.2em;"></ul><a class="pure-button next page" href="#" rel="next"><span class="fa fa-angle-right"></span></a> </div> </div> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a href="" class="reference-type-filter tab-nav" data-tab="biblio-references" data-filter="type" data-pattern="*"><span class="fa fa-angle-right"></span> All References</a></li> <li class="small" style="margin-left:.75em; 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float:none;">[ × clear filter / sort ]</a> </div> </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; 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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="/biblio/21192232-pre-equilibrium-dilepton-production-from-anisotropic-quark-gluon-plasma" itemprop="url">Pre-equilibrium dilepton production from an 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">Martinez, Mauricio</span> ; <span class="author">Strickland, Michael</span> ; <span class="author">Physics Department, Gettysburg College, Gettysburg, PA 17325</span> <span class="text-muted pubdata"> - Physical Review. C, Nuclear Physics</span> </span> </div> <div class="abstract">We calculate leading-order dilepton yields from a quark-gluon plasma that has a time-dependent anisotropy in momentum space. Such anisotropies can arise during the earliest stages of quark-gluon plasma evolution due to the rapid longitudinal expansion of the created matter. Two phenomenological models for the proper-time dependence of the parton hard momentum scale, p{sub hard}, and the plasma anisotropy parameter, {xi}, are constructed that describe the transition of the plasma from its initial nonequilibrium state to an isotropic thermalized state. The first model constructed interpolates between 1+1 dimensional free streaming at early times and 1+1 dimensional ideal hydrodynamical expansion at late<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> times. In the second model we include the effect of collisional broadening of the parton distribution functions in the early-time pre-equilibrium stage of plasma evolution. We find for both cases that for fixed initial conditions high-energy dilepton production is enhanced by pre-equilibrium emission. When the models are constrained to fixed final pion multiplicity the dependence of the resulting spectra on the assumed plasma isotropization time is reduced. Using our most realistic collisionally broadened model we find that high-transverse-momentum dilepton production would be enhanced by at most 40% at the Relativistic Heavy Ion Collider and 50% at the CERN Large Hadron Collider if one assumes an isotropization/thermalization time of 2 fm/c. Given sufficiently precise experimental data this enhancement could be used to determine the plasma isotropization time experimentally.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PHYSREVC.78.034917" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="21192232" data-product-type="Journal Article" data-product-subtype="" >10.1103/PHYSREVC.78.034917</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="/biblio/663085-adaptive-smoothed-particle-hydrodynamics-methodology-ii" itemprop="url">Adaptive Smoothed Particle Hydrodynamics: Methodology. II.</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">Owen, J M</span> ; <span class="author">Villumsen, J V</span> ; <span class="author">Shapiro, P R</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Astrophysical Journal, Supplement Series</span> </span> </div> <div class="abstract">Further development and additional details and tests of adaptive smoothed particle hydrodynamics (ASPH), the new version of smoothed particle hydrodynamics (SPH) described in the first paper in this series (Shapiro et al.), are presented. The ASPH method replaces the isotropic smoothing algorithm of standard SPH, in which interpolation is performed with spherical kernels of radius given by a scalar smoothing length, with anisotropic smoothing involving ellipsoidal kernels and tensor smoothing lengths. In standard SPH, the smoothing length for each particle represents the spatial resolution scale in the vicinity of that particle and is typically allowed to vary in space and<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> time so as to reflect the local value of the mean interparticle spacing. This isotropic approach is not optimal, however, in the presence of strongly anisotropic volume changes such as occur naturally in a wide range of astrophysical flows, including gravitational collapse, cosmological structure formation, cloud-cloud collisions, and radiative shocks. In such cases, the local mean interparticle spacing varies not only in time and space but also in {ital direction} as well. This problem is remedied in ASPH, where each axis of the ellipsoidal smoothing kernel for a given particle is adjusted so as to reflect the different mean interparticle spacings along different directions in the vicinity of that particle. By deforming and rotating these ellipsoidal kernels so as to follow the anisotropy of volume changes local to each particle, ASPH adapts its spatial resolution scale in time, space, and direction. This significantly improves the spatial resolving power of the method over that of standard SPH at fixed particle number per simulation. This paper presents an alternative formulation of the ASPH algorithm for evolving anisotropic smoothing kernels, in which the geometric approach of the first paper in this series, based upon the Lagrangian deformation of ellipsoidal fluid elements surrounding each particle, is replaced by an approach involving a local transformation of coordinates to those in which the underlying anisotropic volume changes appear to be isotropic. Using this formulation the ASPH method is presented in two and three dimensions, including a number of details not previously included in the earlier paper, some of which represent either advances or different choices with respect to the ASPH method detailed in the earlier paper. Among the advances included here are an asynchronous time-integration scheme with different time steps for different particles and the generalization of the ASPH method to three dimensions. In the category of different choices, the shock-tracking algorithm described in the earlier paper for locally adapting the artificial viscosity to restrict viscous heating just to particles encountering shocks is not included here. Instead, we adopt a different interpolation kernel for use with the artificial viscosity, which has the effect of spatially localizing effects of the artificial viscosity. This version of the ASPH method in two and three dimensions is then applied to a series of one-, two-, and three-dimensional test problems, and the results are compared to those of standard SPH applied to the same problems. These include the problem of cosmological pancake collapse, the Riemann shock tube, cylindrical and spherical Sedov blast waves, the collision of two strong shocks, and problems involving shearing disks intended to test the angular momentum conservation properties of the method. These results further support the idea that ASPH has significantly better resolving power than standard SPH for a wide range of problems, including that of cosmological structure formation. {copyright} {ital {copyright} 1998.} {ital The American Astronomical Society}</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1086/313100" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="663085" data-product-type="Journal Article" data-product-subtype="AC" >10.1086/313100</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/1422657-far-from-equilibrium-attractors-nonlinear-dynamical-systems-approach-gubser-flow" itemprop="url">Far-from-equilibrium attractors and nonlinear dynamical systems approach to 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">Behtash, Alireza</span> ; <span class="author">Cruz-Camacho, C. N.</span> ; <span class="author">Martinez, M.</span> <span class="text-muted pubdata"> - Physical Review D</span> </span> </div> <div class="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<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> from the exact Gubser solution of the Boltzmann equation within the relaxation time approximation. We observe that the anisotropic hydrodynamics is able to match up to high numerical accuracy the attractor of the exact solution while the second-order hydrodynamical theories fail to describe it. We show that the IS and DNMR asymptotic series expansions diverge and use resurgence techniques to perform the resummation of these divergences. We also comment on a possible link between the manifold of steepest descent paths in path integrals and the basin of attraction for the attractors via Lyapunov functions that opens a new horizon toward an effective field theory description of hydrodynamics. Our findings indicate that the reorganization of the expansion series carried out by anisotropic hydrodynamics resums the Knudsen and inverse Reynolds numbers to all orders and thus, it can be understood as an effective theory for the far-from-equilibrium fluid dynamics</span><a href='#' onclick='$(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 14<div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.97.044041" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1422657" data-product-type="Journal Article" data-product-subtype="PA" >10.1103/PhysRevD.97.044041</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/20774927-model-effect-collisions-qcd-plasma-instabilities" itemprop="url">Model of the effect of collisions on QCD plasma instabilities</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">Schenke, Bjoern</span> ; <span class="author">Greiner, Carsten</span> ; <span class="author">Strickland, Michael</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review. D, Particles Fields</span> </span> </div> <div class="abstract">We study the effect of including a Bhatnagar-Gross-Krook (BGK) collisional kernel on the collective modes of a QCD plasma which has a hard-particle distribution function which is anisotropic in momentum space. We calculate dispersion relations for both the stable and unstable modes and show that the addition of hard-particle collisions slows the rate of growth of QCD plasma unstable modes. We also show that for any anisotropy there is an upper limit on the collisional frequency beyond which no instabilities exist. Estimating a realistic value for the collisional frequency for {alpha}{sub s}{approx}0.2-0.4 we find that for the large-anisotropy case which<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> is relevant for the initial state of matter generated by free streaming in heavy-ion collisions that the collisional frequency is below this critical value.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevD.73.125004" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="20774927" data-product-type="Journal Article" data-product-subtype="" >10.1103/PhysRevD.73.125004</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="/biblio/5232770-collision-kernels-laser-spectroscopy" itemprop="url">Collision kernels and laser spectroscopy</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">Berman, P R</span> ; <span class="author">Mossberg, T W</span> ; <span class="author">Hartmann, S R</span> <span class="text-muted pubdata"> - Phys. Rev. A; (United States)</span> </span> </div> <div class="abstract">Collisional processes occurring within an atomic vapor can be conveniently described in terms of collision kernels. The population kernel W/sub i/i(v'..-->..v) gives the probability density per unit time that an ''active'' atom in state i undergoes a collision with a perturber that changes the active atom's velocity from v' to v. For active atoms in a linear superposition of states i and j, there is an analogous coherence kernel W/sub i/j(v'..-->..v) (inot =j) reflecting the effects of collisions on the off-diagonal density-matrix element rho/sub i/j. In this work, we discuss the general properties of the collision kernels with characterize a<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> two-level active atom which, owing to the action of a radiation field, is in a linear superposition of its two levels. Using arguments based on the uncertainty principle, we show that collisions can be divided roughly into the following two categories.: (1) collisions having impact parameters less than some characteristic radius which may be described classically and (2) collisions having impact parameters larger than this characteristics radius which give rise to diffractive scattering and must be treated using a quantum mechanical theory. For the population kernels, collisions of type (1) can lead to a large-angle scattering component, while those of type (2) lead to a small-angle (diffractive) scattering component. For the coherence kernel, however, assuming that the collisional interaction for states i and j differ appreciably, only collisions of type (2) contribute, and the coherence kernel contains a small-angle scattering component only. The absence of a large-angle scattering component in the coherence kernel is linked to a collision-induced spatial separation of the trajectories associated with states i and j. Interestingly enough, the width of the diffractive kernel, as measured in the laboratory frame, is found to be independent of the perturber to active-atom mass ratio.</span><a href='#' onclick='$(this).hide().prev().hide().prev().show();return false;' style='margin-left:10px;display:none;'>« less</a></div><div class="metadata-links small clearfix text-muted" style="margin-top:15px;"> <div class="pure-menu pure-menu-horizontal pull-right" style="width:unset;"> <ul class="pure-menu-list"> <li class="pure-menu-item"><span class="item-info-ftlink">DOI: <a class="misc doi-link " href="https://doi.org/10.1103/PhysRevA.25.2550" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="5232770" data-product-type="Journal Article" data-product-subtype="AC" >10.1103/PhysRevA.25.2550</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> </ul> </aside> </div> </section> </div> <div class="col-sm-3 order-sm-3"> <ul class="nav nav-stacked"> <li class="active"><a class="tab-nav disabled" data-tab="related" style="color: #636c72 !important; opacity: 1;"><span class="fa fa-angle-right"></span> Similar Records</a></li> </ul> </div> </div> </section> </div></div> </div> </div> </section> <footer class="" style="background-color:#f9f9f9; /* padding-top: 0.5rem; */"> <div class="footer-minor"> <div class="container"> <hr class="footer-separator" /> <div class="text-center" style="margin-top:1.25rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list" id="footer-org-menu"> <li class="pure-menu-item"> <a href="https://energy.gov" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-us-doe-min" alt="U.S. Department of Energy" /> </a> </li> <li class="pure-menu-item"> <a href="https://www.energy.gov/science/office-science" target="_blank" rel="noopener noreferrer"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-office-of-science-min" alt="Office of Science" /> </a> </li> <li class="pure-menu-item"> <a href="/"> <img src="data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==" class="sprite sprite-footer-osti-min" alt="Office of Scientific and Technical Information" /> </a> </li> </ul> </div> </div> <div class="text-center small" style="margin-top:0.5em;margin-bottom:2.0rem;"> <div class="pure-menu pure-menu-horizontal"> <ul class="pure-menu-list"> <li class="pure-menu-item"><a href="/disclaim" class="pure-menu-link"><span class="fa fa-institution"></span> Website Policies <span class="hidden-xs">/ Important Links</span></a></li> <li class="pure-menu-item"><a href="/pages/contact" class="pure-menu-link"><span class="fa fa-comments-o"></span> Contact Us</a></li> <li class="d-block d-md-none"></li> <li class="pure-menu-item"><a href="https://www.facebook.com/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-facebook" style=""></span></a></li> <li class="pure-menu-item"><a href="https://twitter.com/OSTIgov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-twitter" style=""></span></a></li> <li class="pure-menu-item"><a href="https://www.youtube.com/user/ostigov" target="_blank" rel="noopener noreferrer" class="pure-menu-link social"><span class="fa fa-youtube-play" style=""></span></a></li> </ul> </div> </div> </div> </div> </footer> <link href="/pages/css/pages.fonts.200319.1244.css" rel="stylesheet"> <script src="/pages/js/pages.200319.1244.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? mtextFontInherit: true } }); MathJax.Hub.Queue(["Typeset",MathJax.Hub,"item-list"]); MathJax.Hub.Queue(["Typeset",MathJax.Hub,"citation-pagetitle"]); MathJax.Hub.Queue(["Typeset",MathJax.Hub,"citation-abstract"]); </script><noscript></noscript><script defer src="/pages/js/pages.biblio.200319.1244.js"></script><noscript></noscript> <script defer src="/pages/js/lity.js"></script><noscript></noscript><script async type="text/javascript" src="/pages/js/Universal-Federated-Analytics-Min.js?agency=DOE" id="_fed_an_ua_tag"></script><noscript></noscript></body> <!-- DOE PAGES v.200319.1244 --> </html>