Abstract
We study the maximization of the Tsallis functional at fixed mass and energy in the Hamiltonian Mean Field (HMF) model. We give a thermodynamical and a dynamical interpretation of this variational principle. This leads to q-distributions known as stellar polytropes in astrophysics. We study phase transitions between spatially homogeneous and spatially inhomogeneous equilibrium states. We show that there exists a particular index q c = 3 playing the role of a canonical tricritical point separating first and second order phase transitions in the canonical ensemble and marking the occurence of a negative specific heat region in the microcanonical ensemble. We apply our results to the situation considered by Antoni and Ruffo [Phys. Rev. E 52, 2361 (1995)] and show that the anomaly displayed on their caloric curve can be explained naturally by assuming that, in this region, the QSSs are polytropes with critical index q c = 3. We qualitatively justify the occurrence of polytropic (Tsallis) distributions with compact support in terms of incomplete relaxation and inefficient mixing (non-ergodicity). Our paper provides an exhaustive study of polytropic distributions in the HMF model and the first plausible explanation of the surprising result observed numerically by Antoni and Ruffo (1995). In the course of our analysis, we also report an interesting situation where the caloric curve presents both microcanonical first and second order phase transitions.
Similar content being viewed by others
References
Dynamics and thermodynamics of systems with long range interactions, edited by T. Dauxois et al., Lecture Notes in Physics 602 (Springer, 2002)
A. Campa, T. Dauxois, S. Ruffo, Phys. Rep. 480, 57 (2009)
C. Tsallis, Introduction to Nonextensive Statistical Mechanics (Springer, 2009)
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)
C. Pichon, Ph.D. thesis, Cambridge, 1994
M. Antoni, S. Ruffo, Phys. Rev. E 52, 2361 (1995)
P.H. Chavanis, J. Vatteville, F. Bouchet, Eur. Phys. J. B 46, 61 (2005)
J. Messer, H. Spohn, J. Stat. Phys. 29, 561 (1982)
G. Battle, Commun. Math. Phys. 55, 299 (1977)
J. Barré, F. Bouchet, T. Dauxois, S. Ruffo, J. Stat. Phys. 119, 677 (2005)
V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64, 056134 (2001)
V. Latora, A. Rapisarda, C. Tsallis, Physica A 305, 129 (2002)
V. Latora, A. Rapisarda, S. Ruffo, Phys. Rev. Lett. 80, 692 (1998)
C. Tsallis, J. Stat. Phys. 52, 479 (1988)
P.H. Chavanis, Physica A 365, 102 (2006)
P.H. Chavanis, Statistical mechanics of two-dimensional vortices and stellar systems in [1]
Y.Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois, S. Ruffo, Physica A 337, 36 (2004)
P.H. Chavanis, Eur. Phys. J. B 53, 487 (2006)
A. Antoniazzi, D. Fanelli, J. Barré, P.H. Chavanis, T. Dauxois, S. Ruffo, Phys. Rev. E 75, 011112 (2007)
D. Lynden-Bell, MNRAS 136, 101 (1967)
A. Antoniazzi, D. Fanelli, S. Ruffo, Y. Yamaguchi, Phys. Rev. Lett. 99, 040601 (2007)
P.H. Chavanis, G. De Ninno, D. Fanelli, S. Ruffo, in Chaos, Complexity and Transport, edited by C. Chandre, X. Leoncini, G. Zaslavsky (World Scientific, Singapore, 2008), p. 3
F. Staniscia, P.H. Chavanis, G. De Ninno, D. Fanelli, Phys. Rev. E 80, 021138 (2009)
F. Bouchet, T. Dauxois, S. Ruffo, Europhys. News 37, 9 (2006)
P.H. Chavanis, Physica A 387, 787 (2008)
J. Binney, S. Tremaine, Galactic Dynamics (Princeton Series in Astrophysics, 1987)
X.P. Huang, C.F. Driscoll, Phys. Rev. Lett. 72, 2187 (1994)
B.M. Boghosian, Phys. Rev. E 53, 4754 (1996)
H. Brands, P.H. Chavanis, R. Pasmanter, J. Sommeria, Phys. Fluids 11, 3465 (1999)
R. Bachelard, C. Chandre, A. Ciani, D. Fanelli, Y.Y. Yamaguchi, Phys. Lett. A 373, 4239 (2009)
A. Einstein, Ann. Phys. 33, 1275 (1910)
E.G.D. Cohen, Physica A 305, 19 (2002)
A. Campa, A. Giansanti, G. Morelli, Phys. Rev. E 76, 041117 (2007)
P.H. Chavanis, A&A 401, 15 (2003)
P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)
A. Campa, P.H. Chavanis, A. Giansanti, G. Morelli, Phys. Rev. E 78, R040102 (2008)
A. Taruya, M. Sakagami, Phys. Rev. Lett. 90, 181101 (2003)
P.H. Chavanis, A&A 451, 109 (2006)
A.S. Eddington, MNRAS 76, 572 (1916)
H.C. Plummer, MNRAS 71, 460 (1911)
A. Taruya, M. Sakagami, Physica A 307, 185 (2002)
A. Taruya, M. Sakagami, Physica A 318, 387 (2003)
A. Taruya, M. Sakagami, Physica A 322, 285 (2003)
P.H. Chavanis, A&A 386, 732 (2002)
P.H. Chavanis, C. Sire, Phys. Rev. E 69, 016116 (2004)
P.H. Chavanis, C. Sire, Physica A 356, 419 (2005)
P.H. Chavanis, L. Delfini, Eur. Phys. J. B 69, 389 (2009)
P.H. Chavanis, AIP Conf. Proc. 970, 39 (2008)
R. Ellis, K. Haven, B. Turkington, J. Stat. Phys. 101, 999 (2000)
P.H. Chavanis, J. Sommeria, R. Robert, ApJ 471, 385 (1996)
J. Jeans, MNRAS 76, 70 (1915)
A.R. Plastino, A. Plastino, Phys. Lett. A 174, 384 (1993)
J. Ipser, G. Horwitz, ApJ 232, 863 (1979)
J. Ipser, ApJ 193, 463 (1974)
R. Ellis, K. Haven, B. Turkington, Nonlinearity 15, 239 (2002)
S. Tremaine, M. Hénon, D. Lynden-Bell, MNRAS 219, 285 (1986)
P.H. Chavanis,[arXiv:1002.0291]
D.D. Holm, J.E. Marsden, T. Ratiu, A. Weinstein, Phys. Rep. 123, 1 (1985)
A. Campa, P.H. Chavanis, J. Stat. Mech. P06001 (2010)
S. Chandrasekhar, Principles of Stellar Dynamics (University of Chicago Press, 1942)
J. Katz, MNRAS 183, 765 (1978)
P.H. Chavanis, Int J. Mod. Phys. B 20, 3113 (2006)
C. Sire, P.H. Chavanis, Phys. Rev. E 78, 061111 (2008)
J. Barré, D. Mukamel, S. Ruffo, Phys. Rev. Lett. 87, 030601 (2001)
P.H. Chavanis, Phys. Rev. E 65, 056123 (2002)
T. Tatekawa, F. Bouchet, T. Dauxois, S. Ruffo, Phys. Rev. E 71, 056111 (2005)
P.H. Chavanis, L. Delfini, Phys. Rev. E 81, 051103 (2010)
F. Staniscia, A. Turchi, D. Fanelli, P.H. Chavanis, G. De Ninno, Phys. Rev. Lett. 105, 010601 (2010)
P.H. Chavanis, [arXiv:1007.4916]
Dynamics and thermodynamics of systems with long range interactions: Theory and experiments, edited by A. Campa et al., AIP Conf. Proc. 970 (AIP, 2008)
A. Rapisarda, A. Pluchino, Europhys. News 36, 202 (2005)
M. Stiavelli, G. Bertin, Mon. Not. R. Astron. Soc. 229, 61 (1987)
J. Hjorth, J. Madsen, Mon. Not. R. Astron. Soc. 253, 703 (1991)
R. Bachelard, C. Chandre, D. Fanelli, X. Leoncini, S. Ruffo, Phys. Rev. Lett. 101, 260603 (2008)
A. Figueiredo, T.M. Rocha Filho, M.A. Amato, Europhys. Lett. 83, 30011 (2008)
M.C. Firpo, Europhys. Lett. 88, 30010 (2009)
X. Leoncini, T.L. van den Berg, D. Fanelli, Europhys. Lett. 86, 20002 (2009)
P.H. Chavanis, J. Stat. Mech. P05019 (2010)
C. Mouhot, C. Villani, [arXiv:0905.2167]
P.H. Chavanis, Physica A 387, 1504 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chavanis, PH., Campa, A. Inhomogeneous Tsallis distributions in the HMF model. Eur. Phys. J. B 76, 581–611 (2010). https://doi.org/10.1140/epjb/e2010-00243-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2010-00243-x