Abstract
Extended theories are widely used in the literature to describe relativistic fluids. The motivation for this is mostly due to the causality issues allegedly present in the first order in the gradients theories. However, the decay of fluctuations in the system is also at stake when first order theories that couple heat with acceleration are used. This paper shows that although the introduction of the Maxwell–Cattaneo equation in the description of a simple relativistic fluid formally eliminates the generic instabilities identified by Hiscock and Lindblom (Phys Rev D 31:725–733, 1985), the hypothesis on the order of magnitude of the corresponding relaxation term contradicts the basic ordering in Knudsen’s parameter present in the kinetic approach to hydrodynamics. It is shown that the time derivative, stabilizing term is of second order in such parameter and thus does not belong to the Navier–Stokes regime where the so-called instability arises.
Similar content being viewed by others
References
Cercignani, C., Medeiros Kremer, G.: The Relativistic Boltzmann Equation: Theory and Applications. Birkhauser, Basel (2002)
de Groot, S.R., van Leeuwen, W.A., van der Wert, Ch.: Relativistic Kinetic Theory. North Holland Publishing Company, Amsterdam (1980)
Liu, I.-S., Mnller, I., Ruggeri, E.: Relativistic thermodynamics of gases. Ann. Phys. 169, 191–219 (1986)
Eckart, C.: The thermodynamics of irreversible processes. I. The simple fluid. Phys. Rev. 58, 267–269 (1940)
Eckart, C.: The thermodynamics of irreversible processes. III. Relativistic theory of the simple fluid. Phys. Rev. 58, 919–924 (1940)
Hiscock, W.A., Lindblom, L.: Generic instabilities in first-order dissipative relativistic fluid theories. Phys. Rev. D 31, 725–733 (1985)
Israel, W., Stewart, J.M.: Tansient relativistic thermodynamics and kinetic theory. Ann. Phys. 118, 341–372 (1979)
Israel, W.: Thermodynamics of relativistic systems. Physica 106A, 204–214 (1981)
Israel, W., Stewart, J.M.: Progress in relativistic thermodynamics and electrodynamics of continuous media. In: Held, A. (ed.) General Relativity and Gravitation. One Hundred Years After the Birth of Albert Einstein, pp. 491–525. Plenum Press, New York (1980)
Zimdahl, W.: Bulk viscous cosmology. Phys. Rev. D 53, 5483–5493 (1996)
Jou, D., Pavon, D., Casas-Vazquez, J.: On a covariant formulation of dissipative phenomena. Ann. Inst. Henri Poincare A36, 79–88 (1982)
Garcia-Perciante, A.L., Garcia-Colin, L.S., Sandoval-Villalbazo, A.: On the nature of the so-called generic instabilities in dissipative relativistic hydrodynamics. Gen. Relat. Gravit. 41, 1645–1654 (2009)
Velasco, R.M., Uribe, F.J., Garcia-Colin, L.S.: Inconsistencies in moment methods. Phys. Rev. E 66(032103), 1–4 (2002)
Garcia-Colin, L.S., Velasco, R.M., Uribe, F.J.: Inconsistency in the Moment’s method for solving the Boltzmann equation. J. Non-Equilib. Thermodyn. 29, 257–277 (2005)
Van, P.: Generic stability of dissipative non-relativistic and relativistic fluids. J. Stat. Mech. 0902(02054), 1–23 (2009)
Mondragon Suarez, J.H., Sandoval-Villalbazo, A.: Jeans instability analysis in the presence of heat in Eckart’s frame. Gen. Relativ. Gravit. 44, 139–145 (2012)
Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics, 4th edn. Springer, New York (2010)
Garcia-Perciante, A.L., Garcia-Colin, L.S., Sandoval-Villalbazo, A.: Rayleigh-Brillouin spectrum in special relativistic hydrodynamics. Phys. Rev. E 79(066310), 1–5 (2009)
Berne, B.J., Pecora, R.: Dynamic Light Scattering. Wiley, New York (2001)
Chapman, S., Cowling, T.G.: The mathematical theory of non-uniform gases, 3rd edn. Cambridge Mathematical Library, Cambridge (1971)
Sandoval-Villalbazo, A., Garcia-Perciante, A.L., Garcia-Colin, L.S.: Relativistic transport theory for simple fluids to first order in the gradients. Phys. A 388, 3765–3770 (2009)
Medeiros Kremer, G., Marques, W.: Analysis of eigenmodes in a relativistic gas. Contin. Mech. Thermodyn. 24, 719–729 (2012)
Garcia-Colin, L.S., Velasco, R.M., Uribe, F.J.: Beyond the Navier-Stokes equations: Burnett hydrodynamics. Phys. Rep. 465, 149–189 (2008)
Uribe, F.J., Velasco, R.M., Garcia-Colin, L.S., Diaz-Herrera, E.: Shock wave profiles in the Burnett approximation. Phys. Rev. E 62, 6648–6666 (2000)
Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)
Romatschke, P.: New developments in relativistic viscous hydrodynamics. Int. J. Mod. Phys. E 19, 1–53 (2010)
Acknowledgments
The authors acknowledge support from CONACyT through Grant Number CB2011/167563.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Garcia-Perciante, A.L., Mondragon-Suarez, H., Brun-Battistini, D. et al. On the Stability Problem in Special Relativistic Thermodynamics: Implications of the Chapman–Enskog Formalism. J Stat Phys 160, 760–769 (2015). https://doi.org/10.1007/s10955-015-1272-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-015-1272-3