Abstract.
We discuss the dynamics and thermodynamics of the Hamiltonian Mean Field model (HMF) which is a prototypical system with long-range interactions. The HMF model can be seen as the one Fourier component of a one-dimensional self-gravitating system. Interestingly, it exhibits many features of real self-gravitating systems (violent relaxation, persistence of metaequilibrium states, slow collisional dynamics, phase transitions,...) while avoiding complicated problems posed by the singularity of the gravitational potential at short distances and by the absence of a large-scale confinement. We stress the deep analogy between the HMF model and self-gravitating systems by developing a complete parallel between these two systems. This allows us to apply many technics introduced in plasma physics and astrophysics to a new problem and to see how the results depend on the dimension of space and on the form of the potential of interaction. This comparative study brings new light in the statistical mechanics of self-gravitating systems. We also mention simple astrophysical applications of the HMF model in relation with the formation of bars in spiral galaxies.
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References
Dynamics and thermodynamics of systems with long range interactions, Lecture Notes in Physics, edited by T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens (Springer, 2002)
T. Padmanabhan, Phys. Rep. 188, 285 (1990)
P.H. Chavanis, Statistical mechanics of two-dimensional vortices and stellar systems, in: Dynamics and thermodynamics of systems with long range interactions, edited by T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens, Lecture Notes in Physics (Springer, 2002); e-print: cond-mat/0212223
V.A. Antonov, Vest. Leningr. Gos. Univ. 7, 135 (1962)
D. Lynden-Bell, Runaway Centers, in: Extrait du Bulletin Astronomique, Série 3, Tome III, Fascicule 2 (Éditions du CNRS, 1968)
T. Konishi, K. Kaneko, J. Phys. A 25, 6283 (1992)
S. Inagaki, T. Konishi, Publ. Astron. Soc. Jpn 45, 733 (1993)
S. Inagaki, Prog. Theor. Phys. 90, 557 (1993)
S. Inagaki, Prog. Theor. Phys. 96, 1307 (1996)
C. Pichon, Ph.D. thesis, Cambridge (1994)
D. Lynden-Bell, MNRAS 136, 101 (1967)
M. Antoni, S. Ruffo, Phys. Rev. E 52, 2361 (1995)
T. Dauxois, V. Latora, A. Rapisarda, S. Ruffo, A. Torcini, The Hamiltonian Mean Field Model: from Dynamics to Statistical Mechanics and back, in: Dynamics and thermodynamics of systems with long range interactions, edited by T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens, Lecture Notes in Physics (Springer, 2002) cond-mat/0208456
S. Tremaine, M. Hénon, D. Lynden-Bell, MNRAS 227, 543 (1986)
J. Binney, S. Tremaine, Galactic Dynamics (Princeton Series in Astrophysics, 1987)
Y.Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois, S. Ruffo, Physica A 337, 36 (2004)
P.H. Chavanis, 2004, cond-mat/0409641
P.H. Chavanis, A&A 432, 117 (2005)
P.H. Chavanis, A&A 401, 15 (2003)
P.H. Chavanis, A&A 381, 340 (2002)
P.H. Chavanis, Phys. Rev. E 65, 056123 (2002)
J. Katz, MNRAS 183, 765 (1978)
J. Katz, MNRAS 190, 497 (1980)
J. Katz, Found. Phys. 33, 223 (2003)
S. Chandrasekhar, J. von Neumann, ApJ 95, 489 (1942)
P.H. Chavanis, C. Sire, Phys. Rev. E 62, 490 (2000)
D.D. Holm, J.E. Marsden, T. Ratiu, A. Weinstein, Phys. Rep. 123, 1 (1985)
P.H. Chavanis, C. Sire, 2004, cond-mat/0409569
M. Vergassola, B. Dubrulle, U. Frisch, A. Noullez, A&A 289, 325 (1994)
J. Barré, F. Bouchet, T. Dauxois, S. Ruffo, Eur. Phys. J. B 29, 577 (2002)
P.H. Chavanis, Phys. Rev. E 68, 036108 (2003)
F. Bouchet, J. Barré, J. Stat. Phys. 118, 1073 (2005)
R. Ellis, K. Haven, B. Turkington, Nonlinearity 15, 239 (2002)
M.Y. Choi, J. Choi, Phys. Rev. Lett. 91, 124101 (2003)
S. Ichimaru, Basic Principles of Plasma Physics, edited by W.A. Benjamin (Inc. Reading, Mass., 1973)
C. Tsallis, J. Stat. Phys. 52, 479 (1988)
H. Brands, P.H. Chavanis, J. Sommeria and R. Pasmanter, Phys. Fluids 11, 3465 (1999)
P.H. Chavanis, 2004, cond-mat/0409511
P.H. Chavanis, J. Sommeria, R. Robert, ApJ 471, 385 (1996)
P.H. Chavanis, MNRAS 300, 981 (1998)
P.H. Chavanis, C. Sire, Phys. Rev. E 69, 016116 (2004)
P.H. Chavanis, Banach Center Publ. 66, 79 (2004)
S. Chandrasekhar, An Introduction to the Theory of Stellar Structure (Dover, New York, 1939)
H. Kandrup, ApJ 244, 316 (1981)
P.H. Chavanis, Phys. Rev. E 64, 026309 (2001)
P.H. Chavanis, unpublished notes (2003)
D. Dubin, T.M. O’Neil, Phys. Rev. Lett. 60, 1286 (1988)
D. Dubin, Phys. Plasmas 10, 1338 (2003)
H. Kandrup, Astr. Space. Sci. 97, 435 (1983)
P.H. Chavanis, Physica A 332, 89 (2004)
H. Risken, The Fokker-Planck equation (Springer, 1989)
F. Bouchet, Phys. Rev. E 70, 036113 (2004)
F. Bouchet, T. Dauxois 2004, cond-mat/0407703
S. Chandrasekhar, ApJ 99, 47 (1944)
Y.Y. Yamaguchi, Phys. Rev. E 68, 066210 (2003)
A. Pluchino, V. Latora, A. Rapisarda, Phys. Rev. E 69, 056113 (2004)
P.H. Chavanis, C. Rosier, C. Sire, Phys. Rev. E 66, 036105 (2002)
P.H. Chavanis, M. Ribot, C. Rosier, C. Sire, Banach Center Publ. 66, 103 (2004)
P.H. Chavanis, P. Laurençot, M. Lemou, Physica A 341, 145 (2004)
B.M. Boghosian, Phys. Rev. E 53, 4754 (1996)
V. Latora, A. Rapisarda, C. Tsallis, Physica A 305, 129 (2002)
A. Taruya, M. Sakagami, Physica A 322, 285 (2003)
G.L. Camm, MNRAS 110, 305 (1950)
M. Hénon, Astrophys. Space Sci. 14, 751 (1971)
D. Lynden-Bell, R.M. Lynden-Bell, MNRAS 181, 405 (1977)
J. Katz, I. Okamoto, MNRAS 317, 163 (2000)
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Chavanis, P., Vatteville, J. & Bouchet, F. Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model. Eur. Phys. J. B 46, 61–99 (2005). https://doi.org/10.1140/epjb/e2005-00234-0
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DOI: https://doi.org/10.1140/epjb/e2005-00234-0