Gurzadyan-Savvidy relaxation

In cosmology, Gurzadyan-Savvidy (GS) relaxation is a theory developed by Vahe Gurzadyan and George Savvidy to explain the relaxation over time of the dynamics of N-body gravitating systems such as star clusters and galaxies.^[1]^[2] Stellar systems observed in the Universe – globular clusters and elliptical galaxies – reveal their relaxed state reflected in the high degree of regularity of some of their physical characteristics such as surface luminosity, velocity dispersion, geometric shapes, etc. The basic mechanism of relaxation of stellar systems has been considered the 2-body encounters (of stars), to lead to the observed fine-grained equilibrium. The coarse-grained phase of evolution of gravitating systems is described by violent relaxation developed by Donald Lynden-Bell.^[3] The 2-body mechanism of relaxation is known in plasma physics. The difficulties with description of collective effects in N-body gravitating systems arise due to the long-range character of gravitational interaction, as distinct of plasma where due to two different signs of charges the Debye screening takes place. The 2-body relaxation mechanism e.g. for elliptical galaxies predicts around $10^{13}$ years i.e. time scales exceeding the age of the Universe. The problem of relaxation and evolution of stellar systems and the role of collective effects are studied by various techniques, see.^[4]^[5]^[6]^[7] Among the efficient methods of study of N-body gravitating systems are the numerical simulations, particularly, Sverre Aarseth's^[8] N-body codes are widely used.

Stellar system time scales

Using the geometric methods of theory of dynamical systems,^[9]^[10]^[11] Gurzadyan and Savvidy showed the exponential instability (chaos) of spherical N-body systems interacting by Newtonian gravity and derived the collective (N-body) relaxation time (see also ^[12])

\tau _{GS}=\left({\frac {15}{4}}\right)^{2/3}\left({\frac {1}{2\pi {\sqrt {2}}}}{\frac {v}{GMn^{2/3}}}\right),

where $v$ denotes the average stellar velocity, $M$ is the mean stellar mass and $n$ is the stellar density. Normalized for parameters of stellar systems like globular clusters it yields

\tau _{GS}\simeq 10^{8}{\text{year}}\left({\frac {v}{10{\text{km/sec}}}}\right)\left({\frac {n}{1{\text{pc}}^{-3}}}\right)^{-2/3}\left({\frac {M}{M_{\odot }}}\right)^{-1}.

For clusters of galaxies it yields 10-1000 Gyr. Comparing this (GS) relaxation time to the 2-body relaxation time (see ^[13]^[14])

\tau _{2b}={\frac {{\sqrt {2}}v^{3}}{\pi G^{2}M^{2}nln(N/2)}},

Gurzadyan and Savvidy obtain

{\frac {\tau _{2b}}{\tau _{GS}}}\simeq {\frac {v^{2}}{GMn^{1/3}}}{\frac {1}{lnN}}\simeq {\frac {d}{r_{*}}}{\frac {1}{lnN}},

where $r_{*}=GM/v^{2}$ is the radius of gravitational influence and d is the mean distance between stars. With increasing density, d decreases and approaches $r_{*}$ so that the 2-body encounters become the dominant in the relaxation mechanism. The times $\tau _{GS}$ and $\tau _{2b}$ are related to the dynamical time $\tau _{dyn}=D^{3/2}{GMN}^{1/2}$ by the relations

\tau _{GS}\simeq {\frac {D}{d}}\tau _{dyn},\quad \tau _{2b}\simeq {\frac {D}{r_{*}}}\tau _{dyn},

and reflect the fact of existence of 3 scales of time and length for stellar systems (see also ^[15]^[16]^[17]^[18])

D:\tau _{dyn}\quad ;\quad d:\tau _{GS}\quad ;\quad r_{*}:\tau _{2b}.

That approach (from the analysis of so-called two-dimensional curvature of the configuration space of the system) enabled to conclude^[19] that while the spherical systems are exponentially instable systems (Kolmogorov K-systems), the spiral galaxies "spend a large amount of time in regions with positive two-dimensional curvature" and hence "elliptical and spiral galaxies should have a different origin". Within the same geometric approach Gurzadyan and Armen Kocharyan had introduced the Ricci curvature criterion for relative instability (chaos) of dynamical systems.^[20]^[21]^[22]

Derivation of GS-time scale by stochastic differential equation approach

GS-time scale $\tau _{GS}=\tau _{dyn}N^{1/3}$ has been rederived by Gurzadyan and Kocharyan using stochastic differential equation approach^[23]

Observational indication and numerical simulations

Observational support to the GS-time scale is reported for globular clusters.^[24] Numerical simulations supporting GS-time scale are claimed in.^[25]^[26]^[27]

References

^ Gurzadyan, V.G.; Savvidy, G.K. (1984). "The problem of relaxation of stellar systems". Soviet Physics-Doklady. 29: 521.
^ Gurzadyan, V.G.; Savvidy, G.K. (1986). "Collective relaxation of stellar systems". Astronomy & Astrophysics. 160: 203. Bibcode:1986A&A...160..203G.
^ Lynden-Bell, D. (1967). "Statistical mechanics of violent relaxation in stellar systems". Monthly Notices of the Royal Astronomical Society. 136: 101–121. arXiv:astro-ph/0212205. Bibcode:1967MNRAS.136..101L. doi:10.1093/mnras/136.1.101.
^ Savvidy, G.K. (2020). "Maximally chaotic dynamical systems". Annals of Physics. 421: 168274. Bibcode:2020AnPhy.42168274S. doi:10.1016/j.aop.2020.168274. S2CID 224941547.
^ Gurzadyan, V.G.; Pfenniger, D. (1994). Ergodic Concepts in Stellar Dynamics. Lecture Notes in Physics, 430. Springer. ISBN 978-3-662-13986-8.
^ Binney, J.; Tremaine, S. (2008). Galactic Dynamics. Princeton University Press. ISBN 978-0-691-13027-9.
^ Heggie, D.; Hut, P. (2003). The Gravitational Million-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics. Cambridge University Press. ISBN 978-0-521-77486-4.
^ Aarseth, S. (2009). Gravitational N-Body Simulations: Tools and Algorithms. Cambridge University Press. ISBN 978-0-511-53524-6.
^ Anosov, D.V. (1967). "Geodesic flows on closed Riemannian manifolds of negative curvature". Proceedings of the Steklov Institute of Mathematics. 90: 1.
^ Arnold, V.I. (1997). Mathematical Methods of Classical Mechanics. Springer. ISBN 978-0-387-96890-2.
^ Savvidy, G.K. (2022). "Maximally chaotic dynamical systems of Anosov–Kolmogorov and fundamental interactions". International Journal of Modern Physics A. 37 (9): 2230001–2230333. arXiv:2202.09846. Bibcode:2022IJMPA..3730001S. doi:10.1142/S0217751X22300010. S2CID 247011398.
^ Lang, K. (1999). Astrophysical Formulae. Vol. 2. Springer. p. 95. ISBN 978-3-540-61267-4.
^ Lang, K. (1999). Astrophysical Formulae. Vol. 2. Springer. p. 95. ISBN 978-3-540-61267-4.
^ Binney, J.; Tremaine, S. (2008). Galactic Dynamics. Princeton University Press.
^ Gurzadyan, V. G (1994). "Ergodic methods in stellar dynamics". In V.G. Gurzadyan; D. Pfenniger (eds.). Ergodic Concepts in Stellar Dynamics. Lecture Notes in Physics. Vol. 430. Springer. pp. 43–55.
^ Allahverdyan, A.E.; Gurzadyan, V.G. (2003). "From Fermi-Pasta-Ulam problem to galaxies: The quest for relaxation". Nuovo Cimento. 117B (9–11): 947–964. arXiv:astro-ph/0210026. Bibcode:2002NCimB.117..947A.
^ Gurzadyan, V.G. (2005). "A physicist's view of stellar dynamics: dynamical instability of stellar systems". Highlights of Astronomy. 13: 354–357. arXiv:astro-ph/0310551. doi:10.1017/S1539299600015951.
^ Lang, K. (1999). Astrophysical Formulae. Vol. 2. Springer. ISBN 978-3-540-61267-4.
^ Gurzadyan, V.G.; Savvidy, G.K. (1984). "The problem of relaxation of stellar systems". Soviet Physics-Doklady. 29: 521.
^ Gurzadyan, V.G.; Kocharyan, A.A. (1987). "Relative chaos in stellar systems". Astrophysics and Space Science. 135 (2): 307. Bibcode:1987Ap&SS.135..307G. doi:10.1007/BF00641567. S2CID 120102431.
^ Gurzadyan, V.G.; Kocharyan, A.A. (1988). "Dynamical chaos and regular field". Doklady Akademii Nauk SSSR. 301: 323.
^ El-Zant, A.; Gurzadyan, V.G. (1998). "Relative chaos in stellar systems with massive centre". Physica D: Nonlinear Phenomena. 122: 241. arXiv:astro-ph/9806164. doi:10.1016/S0167-2789(98)00170-5. S2CID 15724268.
^ Gurzadyan, V.G.; Kocharyan, A.A. (2009). "Collective relaxation of stellar systems revisited". Astronomy & Astrophysics. 505 (2): 625–627. arXiv:0905.0517. Bibcode:2009A&A...505..625G. doi:10.1051/0004-6361/200912218. S2CID 8011915.
^ Vesperini, E (1992). "Possible observational indication for Gurzadyan-Savvidy relaxation for globular clusters". Astronomy & Astrophysics. 266 (1): 215. Bibcode:1992A&A...266..215V.
^ Beraldo e Silva, L.; Walter de Siqueira Pedra, Walter; Sodré, Laerte; L. D. Perico, Eder; Lima, Marcos (2017). "The Arrow of Time in the collapse of collisionless self-gravitating systems: non-validity of the Vlasov-Poisson equation during violent relaxation". Astrophysical Journal. 846 (2): 125. arXiv:1703.07363. Bibcode:2017ApJ...846..125B. doi:10.3847/1538-4357/aa876e. S2CID 119185622.
^ Di Cintio, P.; Casetti, L. (2019). "N-body chaos, phase-space transport and relaxation in numerical simulations". Proceedings of the IAU Symposium, Star Clusters: From the Milky Way to the Early Universe. 351: 426–429. arXiv:1907.12774. doi:10.1017/S1743921319006744. S2CID 198985679.
^ Di Cintio, P.; Casetti, L. (2020). "Discreteness effects, N-body chaos and the onset of radial-orbit instability". Monthly Notices of the Royal Astronomical Society. 494: 1027–1034. arXiv:1912.07406. doi:10.1093/mnras/staa741.

[1] Gurzadyan, V.G.; Savvidy, G.K. (1984). "The problem of relaxation of stellar systems". Soviet Physics-Doklady. 29: 521.

[2] Gurzadyan, V.G.; Savvidy, G.K. (1986). "Collective relaxation of stellar systems". Astronomy & Astrophysics. 160: 203. Bibcode:1986A&A...160..203G.

[3] Lynden-Bell, D. (1967). "Statistical mechanics of violent relaxation in stellar systems". Monthly Notices of the Royal Astronomical Society. 136: 101–121. arXiv:astro-ph/0212205. Bibcode:1967MNRAS.136..101L. doi:10.1093/mnras/136.1.101.

[4] Savvidy, G.K. (2020). "Maximally chaotic dynamical systems". Annals of Physics. 421: 168274. Bibcode:2020AnPhy.42168274S. doi:10.1016/j.aop.2020.168274. S2CID 224941547.

[5] Gurzadyan, V.G.; Pfenniger, D. (1994). Ergodic Concepts in Stellar Dynamics. Lecture Notes in Physics, 430. Springer. ISBN 978-3-662-13986-8.

[6] Binney, J.; Tremaine, S. (2008). Galactic Dynamics. Princeton University Press. ISBN 978-0-691-13027-9.

[7] Heggie, D.; Hut, P. (2003). The Gravitational Million-Body Problem: A Multidisciplinary Approach to Star Cluster Dynamics. Cambridge University Press. ISBN 978-0-521-77486-4.

[8] Aarseth, S. (2009). Gravitational N-Body Simulations: Tools and Algorithms. Cambridge University Press. ISBN 978-0-511-53524-6.

[9] Anosov, D.V. (1967). "Geodesic flows on closed Riemannian manifolds of negative curvature". Proceedings of the Steklov Institute of Mathematics. 90: 1.

[10] Arnold, V.I. (1997). Mathematical Methods of Classical Mechanics. Springer. ISBN 978-0-387-96890-2.

[11] Savvidy, G.K. (2022). "Maximally chaotic dynamical systems of Anosov–Kolmogorov and fundamental interactions". International Journal of Modern Physics A. 37 (9): 2230001–2230333. arXiv:2202.09846. Bibcode:2022IJMPA..3730001S. doi:10.1142/S0217751X22300010. S2CID 247011398.

[12] Lang, K. (1999). Astrophysical Formulae. Vol. 2. Springer. p. 95. ISBN 978-3-540-61267-4.

[13] Lang, K. (1999). Astrophysical Formulae. Vol. 2. Springer. p. 95. ISBN 978-3-540-61267-4.

[14] Binney, J.; Tremaine, S. (2008). Galactic Dynamics. Princeton University Press.

[15] Gurzadyan, V. G (1994). "Ergodic methods in stellar dynamics". In V.G. Gurzadyan; D. Pfenniger (eds.). Ergodic Concepts in Stellar Dynamics. Lecture Notes in Physics. Vol. 430. Springer. pp. 43–55.

[16] Allahverdyan, A.E.; Gurzadyan, V.G. (2003). "From Fermi-Pasta-Ulam problem to galaxies: The quest for relaxation". Nuovo Cimento. 117B (9–11): 947–964. arXiv:astro-ph/0210026. Bibcode:2002NCimB.117..947A.

[17] Gurzadyan, V.G. (2005). "A physicist's view of stellar dynamics: dynamical instability of stellar systems". Highlights of Astronomy. 13: 354–357. arXiv:astro-ph/0310551. doi:10.1017/S1539299600015951.

[18] Lang, K. (1999). Astrophysical Formulae. Vol. 2. Springer. ISBN 978-3-540-61267-4.

[19] Gurzadyan, V.G.; Savvidy, G.K. (1984). "The problem of relaxation of stellar systems". Soviet Physics-Doklady. 29: 521.

[20] Gurzadyan, V.G.; Kocharyan, A.A. (1987). "Relative chaos in stellar systems". Astrophysics and Space Science. 135 (2): 307. Bibcode:1987Ap&SS.135..307G. doi:10.1007/BF00641567. S2CID 120102431.

[21] Gurzadyan, V.G.; Kocharyan, A.A. (1988). "Dynamical chaos and regular field". Doklady Akademii Nauk SSSR. 301: 323.

[22] El-Zant, A.; Gurzadyan, V.G. (1998). "Relative chaos in stellar systems with massive centre". Physica D: Nonlinear Phenomena. 122: 241. arXiv:astro-ph/9806164. doi:10.1016/S0167-2789(98)00170-5. S2CID 15724268.

[23] Gurzadyan, V.G.; Kocharyan, A.A. (2009). "Collective relaxation of stellar systems revisited". Astronomy & Astrophysics. 505 (2): 625–627. arXiv:0905.0517. Bibcode:2009A&A...505..625G. doi:10.1051/0004-6361/200912218. S2CID 8011915.

[24] Vesperini, E (1992). "Possible observational indication for Gurzadyan-Savvidy relaxation for globular clusters". Astronomy & Astrophysics. 266 (1): 215. Bibcode:1992A&A...266..215V.

[25] Beraldo e Silva, L.; Walter de Siqueira Pedra, Walter; Sodré, Laerte; L. D. Perico, Eder; Lima, Marcos (2017). "The Arrow of Time in the collapse of collisionless self-gravitating systems: non-validity of the Vlasov-Poisson equation during violent relaxation". Astrophysical Journal. 846 (2): 125. arXiv:1703.07363. Bibcode:2017ApJ...846..125B. doi:10.3847/1538-4357/aa876e. S2CID 119185622.

[26] Di Cintio, P.; Casetti, L. (2019). "N-body chaos, phase-space transport and relaxation in numerical simulations". Proceedings of the IAU Symposium, Star Clusters: From the Milky Way to the Early Universe. 351: 426–429. arXiv:1907.12774. doi:10.1017/S1743921319006744. S2CID 198985679.

[27] Di Cintio, P.; Casetti, L. (2020). "Discreteness effects, N-body chaos and the onset of radial-orbit instability". Monthly Notices of the Royal Astronomical Society. 494: 1027–1034. arXiv:1912.07406. doi:10.1093/mnras/staa741.

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