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Title: Longitudinal hydrodynamics from event-by-event Landau initial conditions

Abstract

Here we investigate three-dimensional ideal hydrodynamic evolution, with Landau initial conditions, incorporating event-by-event variation with many events and transverse density inhomogeneities. We show that the transition to boost-invariant flow occurs too late for realistic setups, with corrections of θ (20%-30%) expected at freeze-out for most scenarios. Moreover, the deviation from boost invariance is correlated with both transverse flow and elliptic flow, with the more highly transversely flowing regions also showing the most violation of boost invariance. Therefore, if longitudinal flow is not fully developed at the early stages of heavy ion collisions, hydrodynamics where boost invariance holds at midrapidity is inadequate to extract transport coefficients of the quark-gluon plasma. We conclude by arguing that developing experimental probes of boost invariance is necessary, and suggest some promising directions in this regard.

Authors:
 [1];  [2];  [3];  [4];  [5]
+ Show Author Affiliations
  1. Univ. of Tennessee, Knoxville, TN (United States)
  2. Frankfurt Inst. for Advanced Studies (FIAS), Frankfurt (Germany)
  3. Univ. of Campinas, Sao Paulo (Brazil)
  4. Univ. of Tennessee, Knoxville, TN (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
  5. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Publication Date:
Research Org.:
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1261410
Alternate Identifier(s):
OSTI ID: 1181590
Grant/Contract Number:  
AC05-00OR22725
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review C, Nuclear Physics
Additional Journal Information:
Journal Volume: 91; Journal Issue: 2; Journal ID: ISSN 0556-2813
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
73 NUCLEAR PHYSICS AND RADIATION PHYSICS

Citation Formats

Sen, Abhisek, Gerhard, Jochen, Torrieri, Giorgio, Read, Kenneth, and Wong, Cheuk-Yin. Longitudinal hydrodynamics from event-by-event Landau initial conditions. United States: N. p., 2015. Web. doi:10.1103/PhysRevC.91.024901.
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Sen, Abhisek, Gerhard, Jochen, Torrieri, Giorgio, Read, Kenneth, & Wong, Cheuk-Yin. Longitudinal hydrodynamics from event-by-event Landau initial conditions. United States. https://doi.org/10.1103/PhysRevC.91.024901
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Sen, Abhisek, Gerhard, Jochen, Torrieri, Giorgio, Read, Kenneth, and Wong, Cheuk-Yin. Mon . "Longitudinal hydrodynamics from event-by-event Landau initial conditions". United States. https://doi.org/10.1103/PhysRevC.91.024901. https://www.osti.gov/servlets/purl/1261410.
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@article{osti_1261410,
title = {Longitudinal hydrodynamics from event-by-event Landau initial conditions},
author = {Sen, Abhisek and Gerhard, Jochen and Torrieri, Giorgio and Read, Kenneth and Wong, Cheuk-Yin},
abstractNote = {Here we investigate three-dimensional ideal hydrodynamic evolution, with Landau initial conditions, incorporating event-by-event variation with many events and transverse density inhomogeneities. We show that the transition to boost-invariant flow occurs too late for realistic setups, with corrections of θ (20%-30%) expected at freeze-out for most scenarios. Moreover, the deviation from boost invariance is correlated with both transverse flow and elliptic flow, with the more highly transversely flowing regions also showing the most violation of boost invariance. Therefore, if longitudinal flow is not fully developed at the early stages of heavy ion collisions, hydrodynamics where boost invariance holds at midrapidity is inadequate to extract transport coefficients of the quark-gluon plasma. We conclude by arguing that developing experimental probes of boost invariance is necessary, and suggest some promising directions in this regard.},
doi = {10.1103/PhysRevC.91.024901},
journal = {Physical Review C, Nuclear Physics},
number = 2,
volume = 91,
place = {United States},
year = {Mon Feb 02 00:00:00 EST 2015},
month = {Mon Feb 02 00:00:00 EST 2015}
}
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https://doi.org/10.1103/PhysRevC.91.024901
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margin-top: 0.5em; padding-left: 0; line-height:1.8em;"> <li> <span style="color:#5C7B2D;"> Sarkisyan, Edward K. G.; Mishra, Aditya Nath; Sahoo, Raghunath</span> </li> <li> Physical Review D, Vol. 93, Issue 5</li> <li> <span class="text-muted related-url">DOI: <a href="https://doi.org/10.1103/physrevd.93.054046" class="text-muted" target="_blank" rel="noopener noreferrer">10.1103/physrevd.93.054046<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-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="0" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/1261463-hydrodynamics-from-landau-initial-conditions" itemprop="url">Hydrodynamics from Landau initial conditions</a></h2> <div class="metadata"> <small class="text-muted" style="text-transform:uppercase;display:block;line-height:2.5em;">Conference</small><span class="authors"> <span class="author">Sen, Abhisek</span> ; <span class="author">Gerhard, Jochen</span> ; <span class="author">Torrieri, Giorgio</span> ; <span class="author">...</span> <span class="text-muted pubdata"></span> </span> </div> <div class="abstract">We investigate ideal hydrodynamic evolution, with Landau initial conditions, both in a semi-analytical 1+1D approach and in a numerical code incorporating event-by-event variation with many events and transverse density inhomogeneities. The object of the calculation is to test how fast would a Landau initial condition transition to a commonly used boost-invariant expansion. We show that the transition to boost-invariant flow occurs too late for realistic setups, with corrections of O (20 - 30%) expected at freezeout for most scenarios. Moreover, the deviation from boost-invariance is correlated with both transverse flow and elliptic flow, with the more highly transversely flowing regions<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> also showing the most violation of boost invariance. Therefore, if longitudinal flow is not fully developed at the early stages of heavy ion collisions, 2+1 dimensional hydrodynamics is inadequate to extract transport coefficients of the quark-gluon plasma. Based on [1, 2]</span><a href='#' onclick='$(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/10.1088/1742-6596/630/1/012042" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="1261463" data-product-type="Conference" data-product-subtype="" >https://doi.org/10.1088/1742-6596/630/1/012042</a></span></li> <li class="pure-menu-item"><span class="item-info-ftlink"><a class="misc fulltext-link " href="/servlets/purl/1261463" title="Link to document media" target="_blank" rel="noopener" data-ostiid="1261463" data-product-type="Conference" data-product-subtype="" >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="1" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/21192050-soft-heavy-ion-physics-from-hydrodynamics-statistical-hadronization-predictions-collisions-radical-sub-nn-tev" itemprop="url">Soft heavy-ion physics from hydrodynamics with statistical hadronization: Predictions for collisions at {radical}(S{sub NN})=5.5 TeV</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">Chojnacki, Mikolaj</span> ; <span class="author">Florkowski, Wojciech</span> ; <span class="author">Broniowski, Wojciech</span> ; <span class="author">...</span> <span class="text-muted pubdata"> - Physical Review. C, Nuclear Physics</span> </span> </div> <div class="abstract">Hydrodynamics supplied with single-freeze-out statistical hadronization is used to describe the midrapidity hadron production in relativistic heavy-ion collisions at the highest RHIC energies ({radical}(s{sub NN})=200 GeV) and, by means of a simple extrapolation to higher multiplicities, to make predictions for the LHC ({radical}(s{sub NN})=5.5 TeV). Thermodynamic properties of the high-temperature strongly-interacting quark-gluon plasma are taken from lattice simulations, at low temperatures the hadron-gas equation of state is used, while in the cross-over region an interpolation between the two equations of state is constructed. Boost invariance is assumed for the midrapidity calculations. The initial condition for hydrodynamics is obtained from a<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> Glauber profile for the entropy, with the initial central temperature T{sub i}. The conditions obtained from the hydrodynamic expansion at the freeze-out temperature T{sub f} are used as input for the thermal event generator THERMINATOR, which accounts for a complete treatment of hadronic resonances. Basic physical observables are obtained: the transverse-momentum spectra, the elliptic flow coefficient v{sub 2}, and the HBT radii. The femtoscopic observables are evaluated with the help of the two-particle method which accounts for the resonance decays and Coulomb final-state interactions. The problem of a simultaneous description of all discussed observables is addressed, with the conclusion that at the highest RHIC energies our approach gives a quite satisfactory global description of soft hadronic observables, which agree with the data at the level of 10-15%. Some discrepancies may be attributed to uncertainties in the components of the model or to the absence of the final-state elastic interactions among hadrons. Extrapolating T{sub i} to higher values allows for the consideration of the results characterized by higher multiplicities. We interpret these results as predictions for possible behavior of soft hadronic data at the LHC. We test T{sub i}=400,450, and 500 MeV, and observe the expected increase of the flow, resulting in smaller slopes of the p{sub T}-spectra. The elliptic flow of pions exhibits saturation, with v{sub 2} remaining practically constant, while the HBT radii increase moderately with T{sub i}.</span><a href='#' onclick='$(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/10.1103/PHYSREVC.78.014905" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="21192050" data-product-type="Journal Article" data-product-subtype="" >https://doi.org/10.1103/PHYSREVC.78.014905</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/21189854-simulating-elliptic-flow-viscous-hydrodynamics" itemprop="url">Simulating elliptic flow with viscous hydrodynamics</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">Dusling, K</span> ; <span class="author">Teaney, D</span> <span class="text-muted pubdata"> - Physical Review. C, Nuclear Physics</span> </span> </div> <div class="abstract">In this work we simulate a viscous hydrodynamical model of noncentral Au-Au collisions in 2+1 dimensions, assuming longitudinal boost invariance. The model fluid equations were proposed by Oettinger and Grmela [Grmela, M., and Oettinger, H. C., Phys. Rev. E, 56, 6620 (1997)]. Freeze-out is signaled when the viscous corrections become large relative to the ideal terms. Then viscous corrections to the transverse momentum and differential elliptic flow spectra are calculated. When viscous corrections to the thermal distribution function are not included, the effects of viscosity on elliptic flow are modest. However, when these corrections are included, the elliptic flow is<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> strongly modified at large p{sub T}. We also investigate the stability of the viscous results by comparing the nonideal components of the stress tensor ({pi}{sup ij}) and their influence on the v{sub 2} spectrum to the expectation of the Navier-Stokes equations ({pi}{sup ij}=-{eta}<{partial_derivative}{sub i}u{sub j}>). We argue that when the stress tensor deviates from the Navier-Stokes form the dissipative corrections to spectra are too large for a hydrodynamic description to be reliable. For typical Relativistic Heavy Ion Colloder initial conditions this happens for {eta}/s > or approx. 0.3.</span><a href='#' onclick='$(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/10.1103/PHYSREVC.77.034905" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="21189854" data-product-type="Journal Article" data-product-subtype="" >https://doi.org/10.1103/PHYSREVC.77.034905</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="/biblio/21191963-causal-viscous-hydrodynamics-dimensions-relativistic-heavy-ion-collisions" itemprop="url">Causal viscous hydrodynamics in 2 + 1 dimensions for relativistic heavy-ion collisions</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">Huichao, Song</span> ; <span class="author">Heinz, Ulrich</span> ; <span class="author">CERN, Physics Department, Theory Division, CH-1211 Geneva 23</span> <span class="text-muted pubdata"> - Physical Review. C, Nuclear Physics</span> </span> </div> <div class="abstract">We explore the effects of shear viscosity on the hydrodynamic evolution and final hadron spectra of Cu + Cu collisions at ultrarelativistic collision energies, using the newly developed (2 + 1)-dimensional viscous hydrodynamic code VISH2+1. Based on the causal Israel-Stewart formalism, this code describes the transverse evolution of longitudinally boost-invariant systems without azimuthal symmetry around the beam direction. Shear viscosity is shown to decelerate the longitudinal and accelerate the transverse hydrodynamic expansion. For fixed initial conditions, this leads to a longer quark-gluon plasma (QGP) lifetime, larger radial flow in the final state, and flatter transverse momentum spectra for the emitted<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> hadrons compared to ideal fluid dynamic simulations. We find that the elliptic flow coefficient v{sub 2} is particularly sensitive to shear viscosity: even the lowest value allowed by the AdS/CFT conjecture {eta}/s{>=}1/4{pi} suppresses v{sub 2} enough to have significant consequences for the phenomenology of heavy-ion collisions at the BNL Relativistic Heavy Ion Collider (RHIC). A comparison between our numerical results and earlier analytic estimates of viscous effects within a blast-wave model parametrization of the expanding fireball at freeze-out reveals that the full dynamical theory leads to much tighter constraints for the specific shear viscosity {eta}/s, thereby supporting the notion that the quark-gluon plasma created at RHIC exhibits almost 'perfect fluidity.'.</span><a href='#' onclick='$(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/10.1103/PHYSREVC.77.064901" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="21191963" data-product-type="Journal Article" data-product-subtype="" >https://doi.org/10.1103/PHYSREVC.77.064901</a></span></li> </ul> </div> </div> </div> <div class="clearfix"></div> </div> </li> <li> <div class="article item document" itemprop="itemListElement" itemscope itemtype="http://schema.org/WebPage"><meta itemprop="position" content="4" /><div class="item-info"> <h2 class="title" itemprop="name headline"><a href="/biblio/21286968-ideal-hydrodynamics-elliptic-flow-cern-super-proton-synchrotron-sps-energies-importance-initial-conditions" itemprop="url">Ideal hydrodynamics and elliptic flow at CERN Super Proton Synchrotron (SPS) energies: Importance of the initial conditions</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">Petersen, Hannah</span> ; <span class="author">Institut fuer Theoretische Physik, Johann Wolfgang Goethe-Universitaet, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main</span> ; <span class="author">Bleicher, Marcus</span> <span class="text-muted pubdata"> - Physical Review. C, Nuclear Physics</span> </span> </div> <div class="abstract">The elliptic flow excitation function calculated in a full (3+1) dimensional hybrid Boltzmann approach with an intermediate hydrodynamic stage for heavy ion reactions from GSI Schwerionen Synchrotron to the highest CERN Super Proton Synchrotron (SPS) energies is discussed in the context of the experimental data. In this study, we employ a hadron gas equation of state to investigate the differences in the dynamics and viscosity effects. The specific event-by-event setup with initial conditions and freeze-out from a nonequilibrium transport model allows for a direct comparison between ideal fluid dynamics and transport simulations. At higher SPS energies, where the pure transport<a href='#' onclick='$(this).hide().next().show().next().show();return false;' style='margin-left:10px;'>more »</a><span style='display:none;'> calculation cannot account for the high elliptic flow values, the smaller mean free path in the hydrodynamic evolution leads to higher elliptic flow values. In contrast to previous studies within pure hydrodynamics, the more realistic initial conditions employed here and the inclusion of a sequential final state hadronic decoupling provides results that are in line with the experimental data almost over the whole energy range from E{sub lab}=2-160A GeV. Thus, this new approach leads to a substantially different shape of the v{sub 2}/{epsilon} scaling curve as a function of (1/SdN{sub ch}/dy) in line with the experimental data compared to previous ideal hydrodynamic calculations. This hints at a strong influence of the initial conditions for the hydrodynamic evolution on the finally observed v{sub 2} values, thus questioning the standard interpretation that the hydrodynamic limit is only reached at BNL Relativistic Heavy Ion Collider energies.</span><a href='#' onclick='$(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/10.1103/PHYSREVC.79.054904" target="_blank" rel="noopener" title="Link to document DOI" data-ostiid="21286968" data-product-type="Journal Article" data-product-subtype="" >https://doi.org/10.1103/PHYSREVC.79.054904</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 defer src="/pages/js/pages.biblio.240327.0205.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.240327.0205 --> </html>