Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaître-Robertson-Walker spacetime
Abstract
The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe is researched. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of nonhydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the nonhydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation. Yet, the latter probes additional high-momentum details unresolved by the relaxation time approximation. While the expansion of the FLRW spacetime is slow enough for the system to move towards (and not away from) local thermal equilibrium, it is not sufficiently slow for the system to actually ever reach complete local equilibrium. Equilibration is fastest in themore »
- Authors:
-
- The Ohio State Univ., Columbus, OH (United States)
- Fluminense Federal Univ., Niterói, RJ (Brazil); Brookhaven National Lab. (BNL), Upton, NY (United States)
- Univ. of São Paulo (Brazil)
- Publication Date:
- Research Org.:
- The Ohio State Univ., Columbus, OH (United States); Brookhaven National Laboratory (BNL), Upton, NY (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP)
- OSTI Identifier:
- 1604346
- Alternate Identifier(s):
- OSTI ID: 1334854
- Grant/Contract Number:
- SC0004286; SC0012704
- Resource Type:
- Journal Article: Accepted Manuscript
- Journal Name:
- Physical Review D
- Additional Journal Information:
- Journal Volume: 94; Journal Issue: 12; Journal ID: ISSN 2470-0010
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 79 ASTRONOMY AND ASTROPHYSICS
Citation Formats
Bazow, D., Denicol, G. S., Heinz, Ulrich, Martinez, M., and Noronha, J. Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaître-Robertson-Walker spacetime. United States: N. p., 2016.
Web. doi:10.1103/PhysRevD.94.125006.
Bazow, D., Denicol, G. S., Heinz, Ulrich, Martinez, M., & Noronha, J. Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaître-Robertson-Walker spacetime. United States. https://doi.org/10.1103/PhysRevD.94.125006
Bazow, D., Denicol, G. S., Heinz, Ulrich, Martinez, M., and Noronha, J. 2016.
"Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaître-Robertson-Walker spacetime". United States. https://doi.org/10.1103/PhysRevD.94.125006. https://www.osti.gov/servlets/purl/1604346.
@article{osti_1604346,
title = {Nonlinear dynamics from the relativistic Boltzmann equation in the Friedmann-Lemaître-Robertson-Walker spacetime},
author = {Bazow, D. and Denicol, G. S. and Heinz, Ulrich and Martinez, M. and Noronha, J.},
abstractNote = {The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe is researched. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of nonhydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the nonhydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation. Yet, the latter probes additional high-momentum details unresolved by the relaxation time approximation. While the expansion of the FLRW spacetime is slow enough for the system to move towards (and not away from) local thermal equilibrium, it is not sufficiently slow for the system to actually ever reach complete local equilibrium. Equilibration is fastest in the relaxation time approximation, followed, in turn, by kinetic evolution with a linearized and a fully nonlinear Boltzmann collision term.},
doi = {10.1103/PhysRevD.94.125006},
url = {https://www.osti.gov/biblio/1604346},
journal = {Physical Review D},
issn = {2470-0010},
number = 12,
volume = 94,
place = {United States},
year = {Thu Dec 08 00:00:00 EST 2016},
month = {Thu Dec 08 00:00:00 EST 2016}
}
Web of Science
Works referenced in this record:
Turbulent thermalization process in heavy-ion collisions at ultrarelativistic energies
journal, April 2014
- Berges, J.; Boguslavski, K.; Schlichting, S.
- Physical Review D, Vol. 89, Issue 7
Studying the validity of relativistic hydrodynamics with a new exact solution of the Boltzmann equation
journal, December 2014
- Denicol, Gabriel; Heinz, Ulrich; Martinez, Mauricio
- Physical Review D, Vol. 90, Issue 12
Relative importance of second-order terms in relativistic dissipative fluid dynamics
journal, April 2014
- Molnár, E.; Niemi, H.; Denicol, G. S.
- Physical Review D, Vol. 89, Issue 7
The quark–gluon plasma: collective dynamics and hard thermal loops
journal, March 2002
- Blaizot, Jean-Paul; Iancu, Edmond
- Physics Reports, Vol. 359, Issue 5-6
Quark-gluon transport theory II. Color response and color correlations in a quark-gluon plasma
journal, April 1986
- Heinz, Ulrich
- Annals of Physics, Vol. 168, Issue 1
Boltzmann hierarchy for interacting neutrinos I: formalism
journal, April 2015
- Oldengott, Isabel M.; Rampf, Cornelius; Wong, Yvonne Y. Y.
- Journal of Cosmology and Astroparticle Physics, Vol. 2015, Issue 04
Microscopic models for ultrarelativistic heavy ion collisions
journal, January 1998
- Bass, S.
- Progress in Particle and Nuclear Physics, Vol. 41
Asymptotic behaviour of the relativistic Boltzmann equation in the Robertson–Walker spacetime
journal, December 2013
- Lee, Ho
- Journal of Differential Equations, Vol. 255, Issue 11
Boltzmann equation in classical and quantum field theory
journal, July 2005
- Jeon, Sangyong
- Physical Review C, Vol. 72, Issue 1
Relativistic dynamics of non-ideal fluids: Viscous and heat-conducting fluids. II. Transport properties and microscopic description of relativistic nuclear matter
journal, July 2007
- Muronga, Azwinndini
- Physical Review C, Vol. 76, Issue 1
Analytic Solution of the Boltzmann Equation in an Expanding System
journal, January 2016
- Bazow, D.; Denicol, G. S.; Heinz, U.
- Physical Review Letters, Vol. 116, Issue 2
Erratum to “Saturation of elliptic flow and the transport opacity of the gluon plasma at RHIC”
journal, June 2002
- Molnar, D.
- Nuclear Physics A, Vol. 703, Issue 3-4
Formation of Maxwellian Tails
journal, May 1976
- Krook, Max; Wu, Tai Tsun
- Physical Review Letters, Vol. 36, Issue 19
Nonthermal Fixed Points: Effective Weak Coupling for Strongly Correlated Systems Far from Equilibrium
journal, July 2008
- Berges, Jürgen; Rothkopf, Alexander; Schmidt, Jonas
- Physical Review Letters, Vol. 101, Issue 4
Isotropization and Hydrodynamization in Weakly Coupled Heavy-Ion Collisions
journal, October 2015
- Kurkela, Aleksi; Zhu, Yan
- Physical Review Letters, Vol. 115, Issue 18
Reheating and thermalization in a simple scalar model
journal, September 1996
- Son, D. T.
- Physical Review D, Vol. 54, Issue 6
A relativistic relaxation-time model for the Boltzmann equation
journal, June 1974
- Anderson, J. L.; Witting, H. R.
- Physica, Vol. 74, Issue 3
General relativistic Boltzmann equation, II: Manifestly covariant treatment
journal, May 2009
- Debbasch, F.; van Leeuwen, W. A.
- Physica A: Statistical Mechanics and its Applications, Vol. 388, Issue 9
The Boltzmann equation and its applications
journal, November 1990
- Ramos, J. I.
- Applied Mathematical Modelling, Vol. 14, Issue 11
Saturation of elliptic flow and the transport opacity of the gluon plasma at RHIC
journal, January 2002
- Molnár, Dénes; Gyulassy, Miklos
- Nuclear Physics A, Vol. 697, Issue 1-2
Erratum: Derivation of transient relativistic fluid dynamics from the Boltzmann equation [Phys. Rev. D 85 , 114047 (2012)]
journal, February 2015
- Denicol, G. S.; Niemi, H.; Molnár, E.
- Physical Review D, Vol. 91, Issue 3
Symmetry constraints on generalizations of Bjorken flow
journal, October 2010
- Gubser, Steven S.
- Physical Review D, Vol. 82, Issue 8
Approach to Equilibrium in Weakly Coupled Non-Abelian Plasmas
journal, October 2014
- Kurkela, Aleksi; Lu, Egang
- Physical Review Letters, Vol. 113, Issue 18
Role of quantum fluctuations in a system with strong fields: Onset of hydrodynamical flow
journal, January 2011
- Dusling, Kevin; Epelbaum, Thomas; Gelis, François
- Nuclear Physics A, Vol. 850, Issue 1
Solutions of the Boltzmann equation for Maxwell interactions and singular angle-dependent cross sections
journal, August 1980
- Cornille, H.; Gervois, A.
- Journal of Statistical Physics, Vol. 23, Issue 2
Universality Far from Equilibrium: From Superfluid Bose Gases to Heavy-Ion Collisions
journal, February 2015
- Berges, J.; Boguslavski, K.; Schlichting, S.
- Physical Review Letters, Vol. 114, Issue 6
Properties of the Boltzmann equation in the classical approximation
journal, December 2014
- Epelbaum, Thomas; Gelis, François; Tanji, Naoto
- Physical Review D, Vol. 90, Issue 12
Collisional relaxation in simulations of heavy-ion collisions using Boltzmann-type equations
journal, December 1989
- Welke, G.; Malfliet, R.; Grégoire, C.
- Physical Review C, Vol. 40, Issue 6
Nonequilibrium fixed points in longitudinally expanding scalar theories: Infrared cascade, Bose condensation, and a challenge for kinetic theory
journal, November 2015
- Berges, J.; Boguslavski, K.; Schlichting, S.
- Physical Review D, Vol. 92, Issue 9
Basin of attraction for turbulent thermalization and the range of validity of classical-statistical simulations
journal, May 2014
- Berges, J.; Boguslavski, K.; Schlichting, S.
- Journal of High Energy Physics, Vol. 2014, Issue 5
Transient relativistic thermodynamics and kinetic theory
journal, April 1979
- Israel, W.; Stewart, J. M.
- Annals of Physics, Vol. 118, Issue 2
Thermalization of gluons in ultrarelativistic heavy ion collisions by including three-body interactions in a parton cascade
journal, June 2005
- Xu, Zhe; Greiner, Carsten
- Physical Review C, Vol. 71, Issue 6
General relativistic Boltzmann equation, I: Covariant treatment
journal, April 2009
- Debbasch, F.; van Leeuwen, W. A.
- Physica A: Statistical Mechanics and its Applications, Vol. 388, Issue 7
Semiclassicality and decoherence of cosmological perturbations
journal, March 1996
- Polarski, David; Starobinsky, Alexei A.
- Classical and Quantum Gravity, Vol. 13, Issue 3
Effective transport equations for non-Abelian plasmas
journal, November 1999
- Litim, Daniel F.; Manuel, Cristina
- Nuclear Physics B, Vol. 562, Issue 1-2
The Boltzmann equation in classical Yang–Mills theory
journal, May 2014
- Mathieu, V.; Mueller, A. H.; Triantafyllopoulos, D. N.
- The European Physical Journal C, Vol. 74, Issue 5
Kinetic theory of a longitudinally expanding system of scalar particles
journal, September 2015
- Epelbaum, Thomas; Gelis, François; Jeon, Sangyong
- Journal of High Energy Physics, Vol. 2015, Issue 9
A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems
journal, May 1954
- Bhatnagar, P. L.; Gross, E. P.; Krook, M.
- Physical Review, Vol. 94, Issue 3
Theory of core-collapse supernovae
journal, April 2007
- Janka, H.; Langanke, K.; Marek, A.
- Physics Reports, Vol. 442, Issue 1-6
Supernova Simulations with Boltzmann Neutrino Transport: A Comparison of Methods
journal, February 2005
- Liebendorfer, M.; Rampp, M.; Janka, H. ‐Th.
- The Astrophysical Journal, Vol. 620, Issue 2
Reheating in Inflationary Cosmology: Theory and Applications
journal, November 2010
- Allahverdi, Rouzbeh; Brandenberger, Robert; Cyr-Racine, Francis-Yan
- Annual Review of Nuclear and Particle Science, Vol. 60, Issue 1
Bulk viscosity of high-temperature QCD
journal, October 2006
- Arnold, Peter; Doğan, Çağlar; Moore, Guy D.
- Physical Review D, Vol. 74, Issue 8
Perfect fluidity of a dissipative system: Analytical solution for the Boltzmann equation in
journal, December 2015
- Noronha, Jorge; Denicol, Gabriel S.
- Physical Review D, Vol. 92, Issue 11
Anisotropic hydrodynamics for conformal Gubser flow
journal, February 2015
- Nopoush, Mohammad; Ryblewski, Radoslaw; Strickland, Michael
- Physical Review D, Vol. 91, Issue 4
DEFROST: a new code for simulating preheating after inflation
journal, November 2008
- Frolov, Andrei V.
- Journal of Cosmology and Astroparticle Physics, Vol. 2008, Issue 11
Classical Decay of the Inflaton
journal, July 1996
- Khlebnikov, S. Yu.; Tkachev, I. I.
- Physical Review Letters, Vol. 77, Issue 2
Highly relativistic nucleus-nucleus collisions: The central rapidity region
journal, January 1983
- Bjorken, J. D.
- Physical Review D, Vol. 27, Issue 1
Relativistic Turbulence: A Long Way from Preheating to Equilibrium
journal, March 2003
- Micha, Raphael; Tkachev, Igor I.
- Physical Review Letters, Vol. 90, Issue 12
On the kinetic theory of rarefied gases
journal, December 1949
- Grad, Harold
- Communications on Pure and Applied Mathematics, Vol. 2, Issue 4
On the equivalence between the Boltzmann equation and classical field theory at large occupation numbers
journal, March 2004
- Mueller, A. H.; Son, D. T.
- Physics Letters B, Vol. 582, Issue 3-4
Pressure Isotropization in High Energy Heavy Ion Collisions
journal, December 2013
- Epelbaum, Thomas; Gelis, François
- Physical Review Letters, Vol. 111, Issue 23
“Bottom-up” thermalization in heavy ion collisions
journal, March 2001
- Baier, R.; Mueller, A. H.; Schiff, D.
- Physics Letters B, Vol. 502, Issue 1-4
Analytic solutions of the relativistic Boltzmann equation
journal, April 2015
- Hatta, Yoshitaka; Martinez, Mauricio; Xiao, Bo-Wen
- Physical Review D, Vol. 91, Issue 8
Nonlinear model-Boltzmann equations and exact solutions
journal, November 1981
- Ernst, M. H.
- Physics Reports, Vol. 78, Issue 1
Finite-temperature field theory
journal, March 1989
- Kapusta, J. I.; Landshoff, P. V.
- Journal of Physics G: Nuclear and Particle Physics, Vol. 15, Issue 3
Universal attractor in a highly occupied non-Abelian plasma
journal, June 2014
- Berges, J.; Boguslavski, K.; Schlichting, S.
- Physical Review D, Vol. 89, Issue 11
Entropy Generation and the Survival of Protogalaxies in an Expanding Universe
journal, September 1971
- Weinberg, Steven
- The Astrophysical Journal, Vol. 168
Exact solutions of the Boltzmann equation
journal, January 1977
- Krook, Max; Wu, Tai Tsun
- Physics of Fluids, Vol. 20, Issue 10
Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges
journal, December 1995
- Ma, Chung-Pei; Bertschinger, Edmund
- The Astrophysical Journal, Vol. 455
Kinetic Theory for Plasmas with Non-Abelian Interactions
journal, August 1983
- Heinz, Ulrich
- Physical Review Letters, Vol. 51, Issue 5
Derivation of transient relativistic fluid dynamics from the Boltzmann equation
journal, June 2012
- Denicol, G. S.; Niemi, H.; Molnár, E.
- Physical Review D, Vol. 85, Issue 11
How particles emerge from decaying classical fields in heavy ion collisions: Towards a kinetic description of the Glasma
journal, February 2009
- Gelis, François; Jeon, Sangyong; Venugopalan, Raju
- Nuclear Physics A, Vol. 817, Issue 1-4
New Exact Solution of the Relativistic Boltzmann Equation and its Hydrodynamic Limit
journal, November 2014
- Denicol, Gabriel S.; Heinz, Ulrich; Martinez, Mauricio
- Physical Review Letters, Vol. 113, Issue 20
From quantum field theory to hydrodynamics: Transport coefficients and effective kinetic theory
journal, May 1996
- Jeon, Sangyong; Yaffe, Laurence G.
- Physical Review D, Vol. 53, Issue 10
Quark-Gluon transport theory
journal, November 1989
- Elze, Hans-Thomas; Heinz, Ulrich
- Physics Reports, Vol. 183, Issue 3
Origin of the relaxation time in dissipative fluid dynamics
journal, April 2011
- Denicol, Gabriel S.; Noronha, Jorge; Niemi, Harri
- Physical Review D, Vol. 83, Issue 7
A Boltzmann equation for the QCD plasma
journal, September 1999
- Blaizot, Jean-Paul; Iancu, Edmond
- Nuclear Physics B, Vol. 557, Issue 1-2
Semi-classical transport theory for non-Abelian plasmas
journal, July 2002
- Litim, Daniel F.; Manuel, Cristina
- Physics Reports, Vol. 364, Issue 6
Thermal equilibration in ultra-relativistic heavy-ion collisions
journal, April 1984
- Baym, Gordon
- Physics Letters B, Vol. 138, Issue 1-3
Testing viscous and anisotropic hydrodynamics in an exactly solvable case
journal, August 2013
- Florkowski, Wojciech; Ryblewski, Radoslaw; Strickland, Michael
- Physical Review C, Vol. 88, Issue 2
Solutions of conformal Israel-Stewart relativistic viscous fluid dynamics
journal, January 2015
- Marrochio, Hugo; Noronha, Jorge; Denicol, Gabriel S.
- Physical Review C, Vol. 91, Issue 1
Investigating the domain of validity of the Gubser solution to the Boltzmann equation
journal, November 2015
- Heinz, Ulrich; Martinez, Mauricio
- Nuclear Physics A, Vol. 943
Anisotropic hydrodynamics for conformal Gubser flow
journal, December 2016
- Strickland, Michael; Nopoush, Mohammad; Ryblewski, Radoslaw
- Nuclear Physics A, Vol. 956
Inhomogeneous Cosmological Models: Proceedings of the Spanish Relativity Meeting
conference, June 1995
- Molina, A.; Senovilla, M. M.
Lattice study of classical inflaton decay
journal, March 1997
- Prokopec, Tomislav; Roos, Thomas G.
- Physical Review D, Vol. 55, Issue 6
Turbulent thermalization
journal, August 2004
- Micha, Raphael; Tkachev, Igor I.
- Physical Review D, Vol. 70, Issue 4
Non-thermal Fixed Points: Effective Weak-coupling for Strongly Correlated Systems far from Equilibrium.
text, January 2008
- Berges, Juergen; Rothkopf, Alexander.; Schmidt, Jonas
- Deutsches Elektronen-Synchrotron, DESY, Hamburg
DEFROST: A New Code for Simulating Preheating after Inflation
text, January 2008
- Frolov, Andrei V.
- arXiv
Role of quantum fluctuations in a system with strong fields: Onset of hydrodynamical flow
text, January 2010
- Dusling, K.; Epelbaum, T.; Gelis, F.
- arXiv
Origin of the Relaxation Time in Dissipative Fluid Dynamics
text, January 2011
- Denicol, G. S.; Noronha, J.; Niemi, H.
- arXiv
Derivation of transient relativistic fluid dynamics from the Boltzmann equation
text, January 2012
- Denicol, G. S.; Niemi, H.; Molnar, E.
- arXiv
Turbulent thermalization process in heavy-ion collisions at ultrarelativistic energies
text, January 2013
- Berges, J.; Boguslavski, K.; Schlichting, S.
- arXiv
On the relative importance of second-order terms in relativistic dissipative fluid dynamics
text, January 2013
- Molnár, E.; Niemi, H.; Denicol, G. S.
- arXiv
Universal attractor in a highly occupied non-Abelian plasma
text, January 2013
- Berges, Juergen; Boguslavski, Kirill; Schlichting, Soeren
- arXiv
The Boltzmann Equation in Classical Yang-Mills Theory
text, January 2014
- Mathieu, V.; Mueller, A. H.; Triantafyllopoulos, D. N.
- arXiv
Approach to equilibrium in weakly coupled nonabelian plasmas
text, January 2014
- Kurkela, Aleksi; Lu, Egang
- arXiv
Studying the validity of relativistic hydrodynamics with a new exact solution of the Boltzmann equation
text, January 2014
- Denicol, Gabriel S.; Heinz, Ulrich W.; Martinez, Mauricio
- arXiv
Kinetic theory of a longitudinally expanding system of scalar particles
text, January 2015
- Epelbaum, Thomas; Gelis, Francois; Jeon, Sangyong
- arXiv
Isotropization and hydrodynamization in weakly coupled heavy-ion collisions
text, January 2015
- Kurkela, Aleksi; Zhu, Yan
- arXiv
Investigating the domain of validity of the Gubser solution to the Boltzmann equation
text, January 2015
- Heinz, Ulrich W.; Martinez, Mauricio
- arXiv
Semiclassicality and Decoherence of Cosmological Perturbations
text, January 1995
- Polarski, David; Starobinsky, Alexei A.
- arXiv
The Quark-Gluon Plasma: Collective Dynamics and Hard Thermal Loops
text, January 2001
- Blaizot, Jean-Paul; Iancu, Edmond
- arXiv
Semi-classical transport theory for non-Abelian plasmas
text, January 2001
- Litim, Daniel F.; Manuel, Cristina
- arXiv
Relativistic Turbulence: A Long Way from Preheating to Equilibrium
text, January 2002
- Micha, R.; Tkachev, I. I.
- arXiv
Thermalization of gluons in ultrarelativistic heavy ion collisions by including three-body interactions in a parton cascade
text, January 2004
- Xu, Zhe; Greiner, Carsten
- arXiv
Lattice study of classical inflaton decay
text, January 1996
- Prokopec, Tomislav; Roos, Thomas G.
- arXiv
Effective Transport Equations for non-Abelian Plasmas
text, January 1999
- Litim, Daniel F.; Manuel, Cristina
- arXiv
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