Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time Approximation
The propagation speeds of excitations are a crucial input in the modeling of interacting systems of particles. In this paper, we assume the microscopic physics is described by a kinetic theory for massless particles, which is approximated by a generalized relaxation time approximation (RTA) where th...
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| author | Alejandra Kandus Esteban Calzetta |
| author_facet | Alejandra Kandus Esteban Calzetta |
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| description | The propagation speeds of excitations are a crucial input in the modeling of interacting systems of particles. In this paper, we assume the microscopic physics is described by a kinetic theory for massless particles, which is approximated by a generalized relaxation time approximation (RTA) where the relaxation time depends on the energy of the particles involved. We seek a solution of the kinetic equation by assuming a parameterized one-particle distribution function (1-pdf) which generalizes the Chapman–Enskog (Ch-En) solution to the RTA. If developed to all orders, this would yield an asymptotic solution to the kinetic equation; we restrict ourselves to an approximate solution by truncating the Ch-En series to the second order. Our generalized Ch-En solution contains undetermined space-time-dependent parameters, and we derive a set of dynamical equations for them by applying the moments method. We check that these dynamical equations lead to energy–momentum conservation and positive entropy production. Finally, we compute the propagation speeds for fluctuations away from equilibrium from the linearized form of the dynamical equations. Considering relaxation times of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>=</mo><msub><mi>τ</mi><mn>0</mn></msub><msup><mrow><mo>(</mo><mo>−</mo><msub><mi>β</mi><mi>μ</mi></msub><msup><mi>p</mi><mi>μ</mi></msup><mo>)</mo></mrow><mrow><mo>−</mo><mi>a</mi></mrow></msup></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mo>∞</mo><mo><</mo><mi>a</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>β</mi><mi>μ</mi></msub><mo>=</mo><msub><mi>u</mi><mi>μ</mi></msub><mo>/</mo><mi>T</mi></mrow></semantics></math></inline-formula> is the temperature vector in the Landau frame, we show that the Anderson–Witting prescription <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> yields the fastest speed in all scalar, vector and tensor sectors. This fact ought to be taken into consideration when choosing the best macroscopic description for a given physical system. |
| format | Article |
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| spelling | doaj-art-ce16f3c367054622842ff65aa12e143e2025-08-20T01:53:45ZengMDPI AGEntropy1099-43002024-10-01261192710.3390/e26110927Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time ApproximationAlejandra Kandus0Esteban Calzetta1Departamento de Ciências Exatas, Universidade Estadual de Santa Cruz, Rodov. J. Amado km 16, Salobrinho, Ilhéus 45662-900, BA, BrazilDepartamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Ciudad de Buenos Aires CP 1428, ArgentinaThe propagation speeds of excitations are a crucial input in the modeling of interacting systems of particles. In this paper, we assume the microscopic physics is described by a kinetic theory for massless particles, which is approximated by a generalized relaxation time approximation (RTA) where the relaxation time depends on the energy of the particles involved. We seek a solution of the kinetic equation by assuming a parameterized one-particle distribution function (1-pdf) which generalizes the Chapman–Enskog (Ch-En) solution to the RTA. If developed to all orders, this would yield an asymptotic solution to the kinetic equation; we restrict ourselves to an approximate solution by truncating the Ch-En series to the second order. Our generalized Ch-En solution contains undetermined space-time-dependent parameters, and we derive a set of dynamical equations for them by applying the moments method. We check that these dynamical equations lead to energy–momentum conservation and positive entropy production. Finally, we compute the propagation speeds for fluctuations away from equilibrium from the linearized form of the dynamical equations. Considering relaxation times of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>=</mo><msub><mi>τ</mi><mn>0</mn></msub><msup><mrow><mo>(</mo><mo>−</mo><msub><mi>β</mi><mi>μ</mi></msub><msup><mi>p</mi><mi>μ</mi></msup><mo>)</mo></mrow><mrow><mo>−</mo><mi>a</mi></mrow></msup></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mo>∞</mo><mo><</mo><mi>a</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>β</mi><mi>μ</mi></msub><mo>=</mo><msub><mi>u</mi><mi>μ</mi></msub><mo>/</mo><mi>T</mi></mrow></semantics></math></inline-formula> is the temperature vector in the Landau frame, we show that the Anderson–Witting prescription <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> yields the fastest speed in all scalar, vector and tensor sectors. This fact ought to be taken into consideration when choosing the best macroscopic description for a given physical system.https://www.mdpi.com/1099-4300/26/11/927relativistic hydrodynamicsrelativistic kinetic theorypropagation speeds |
| spellingShingle | Alejandra Kandus Esteban Calzetta Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time Approximation Entropy relativistic hydrodynamics relativistic kinetic theory propagation speeds |
| title | Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time Approximation |
| title_full | Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time Approximation |
| title_fullStr | Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time Approximation |
| title_full_unstemmed | Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time Approximation |
| title_short | Propagation Speeds of Relativistic Conformal Particles from a Generalized Relaxation Time Approximation |
| title_sort | propagation speeds of relativistic conformal particles from a generalized relaxation time approximation |
| topic | relativistic hydrodynamics relativistic kinetic theory propagation speeds |
| url | https://www.mdpi.com/1099-4300/26/11/927 |
| work_keys_str_mv | AT alejandrakandus propagationspeedsofrelativisticconformalparticlesfromageneralizedrelaxationtimeapproximation AT estebancalzetta propagationspeedsofrelativisticconformalparticlesfromageneralizedrelaxationtimeapproximation |