Abstract
The thermodynamics and the dynamics of particle systems with infiniterange coupling display several unusual and new features with respect to systems with short-range interactions. The Hamiltonian Mean Field (HMF) model represents a paradigmatic example of this class of systems. The present study addresses both attractive and repulsive interactions, with a particular emphasis on the description of clustering phenomena from a thermodynamical as well as from a dynamical point of view. The observed clustering transition can be first or second order, in the usual thermodynamical sense. In the former case, ensemble inequivalence naturally arises close to the transition, i.e. canonical and microcanonical ensembles give different results. In particular, in the microcanonical ensemble negative specific heat regimes and temperature jumps are observed. Moreover, having access to dynamics one can study non-equilibrium processes. Among them, the most striking is the emergence of coherent structures in the repulsive model, whose formation and dynamics can be studied either by using the tools of statistical mechanics or as a manifestation of the solutions of an associated Vlasov equation. The chaotic character of the HMF model has been also analyzed in terms of its Lyapunov spectrum.
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References
T. Padhmanaban, Statistical Mechanics of gravitating systems in static and cosmological backgrounds, in “Dynamics and Thermodynamics of Systems with Long-Range Interactions”, T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002), (in this volume)
P.-H. Chavanis, Statistical mechanics of two-dimensional vortices and stellar systems, in “Dynamics and Thermodynamics of Systems with Long-Range Interactions”, T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002), (in this volume)
Elskens, Kinetic theory for plasmas and wave-particle hamiltonian dynamics, in “Dynamics and Thermodynamics of Systems with Long-Range Interactions”, T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002), (in this volume)
F. Tamarit and C. Anteneodo, Physical Review Letters 84, 208 (2000)
J. Barré, D. Mukamel, S. Ruffo, Ensemble inequivalence in mean-field models of magnetism, in “Dynamics and Thermodynamics of Systems with Long-Range Interactions”, T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002), (in this volume)
A. Campa, A. Giansanti, D. Moroni, Physical Review E 62, 303 (2000) and Chaos Solitons and Fractals 13, 407 (2002)
J.-P. Eckmann, D. Ruelle, Review of Modern Physics 57, 615 (1985)
S. Ruffo, in Transport and Plasma Physics, edited by S. Benkadda, Y. Elskens and F. Doveil (World Scientific, Singapore, 1994), pp. 114–119; M. Antoni, S. Ruffo, Physical Review E 52, 2361 (1995)
M. Antoni and A. Torcini, Physical Review E 57, R6233 (1998); A. Torcini and M. Antoni, Physical Review E 59, 2746 (1999)
M. Antoni, S. Ruffo, A. Torcini: Physical Review E, 66, 025103 (2002)
C. Tsallis, A. Rapisarda, V. Latora and F. Baldovin Nonextensivity: from low-dimensional maps to Hamiltonian systems, in “Dynamics and Thermodynamics of Systems with Long-Range Interactions”, T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002), (in this volume)
V. Latora, A. Rapisarda and S. Ruffo, Physical Review Letters 80, 692 (1998)
C. Anteneodo and C. Tsallis, Physical Review Letters 80, 5313 (1998)
R. Livi, A. Politi, S. Ruffo, Journal of Physics A 19, 2033 (1986)
D.H. Gross, Thermo-Statistics or Topology of the Microcanonical Entropy Surface, in “Dynamics and Thermodynamics of Systems with Long-Range Interactions”, T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002), (in this volume)
M. Kac, G. Uhlenbeck and P.C. Hemmer, Journal Mathematical Physics (N.Y.) 4, 216 (1963); 4, 229 (1963); 5, 60 (1964)
S. Tanase-Nicola and J. Kurchan, Private Communication (2002)
J. Barré and F. Bouchet, unpublished (2002)
J. Barré, PHD thesis, ENS Lyon, unpublished (2002)
D.H.E. Gross, Microcanonical thermodynamics, Phase transitions in “small” systems (World Scientific, Singapore, 2000)
H. Yoshida, Physics Letters A 150, 262 (1990); R. I. McLachlan and P. Atela, Nonlinearity 5, 541 (1992)
P. Hertel and W. Thirring, Annals of Physics 63, 520 (1971)
A. Compagner, C. Bruin, A. Roelse, Physical Review A 39, 5989 (1989); H.A. Posch, H. Narnhofer, W. Thirring, Physical Review A 42, 1880 (1990)
V. Latora, A. Rapisarda and S. Ruffo, Physical Review Letters 83, 2104 (1999)
V. Latora, A. Rapisarda and C. Tsallis, Physical Review E 64, 056124-1 (2001)
V. Latora, A. Rapisarda and C. Tsallis, Physica A 305, 129 (2002)
D. Del Castillo-Negrete, Dynamics and self-consistent chaos in a mean field Hamiltonian model, in “Dynamics and Thermodynamics of Systems with Long-Range Interactions”, T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002), (in this volume)
J. Barré, F. Bouchet, T. Dauxois, S. Ruffo, Birth and long-time stabilization of out-of-equilibrium coherent structures, European Physical Journal B, to appear (2002) and Physical Review Letters 89, 110601 (2002)
T. Dauxois, P. Holdsworth, S. Ruffo, European Physical Journal B 16, 659 (2000)
M.C. Firpo, F. Leyvraz and S. Ruffo, Journal of Physics A, 35, 4413 (2002)
W. Braun, K. Hepp, Communications in Mathematical Physics 56, 101 (1977)
P.L. Kapitza, in Collected Papers of P. L. Kapitza, edited by D. Ter Harr (Pergamon, London, 1965), pp 714, 726
J.L. Lebowitz, J. Piasecki, Ya. Sinai “Scaling dynamics of a massive piston in an ideal gas”, in Hard ball systems and the Lorentz gas, 217–227, Encycl. Math. Sci., 101, Springer, Berlin (2000)
Ya. Sinai, Theoretical and Mathematical Physics (in Russian), 121, 110 (1999)
M. Antoni, Y. Elskens, D. F. Escande, Physics of Plasmas 5, 841 (1998); M-C. Firpo, Etude dynamique et statistique de l’interaction onde-particule, PhD Thesis, Université de Marseille (1999)
P.F. Embid, A. J. Majda, Com. in Partial Differential Equations 21, 619 (1996)
L. Casetti, M. Pettini, E.G.D. Cohen, Physics Reports 337, 237 (2000)
A. Bonasera, V. Latora and A. Rapisarda, Physical Review Letters 75, 3434 (1995)
G. Benettin, L. Galgani and J.M. Strelcyn, Physical Review A 14, 2338 (1976)
S.K. Nayak, R. Ramaswamy and C. Chakravarty, Physical Review E 51, 3376 (1995); V. Mehra, R. Ramaswamy, Physical Review E 56, 2508 (1997)
Y.Y. Yamaguchi, Progress of Theoretical Physics 95, 717 (1996)
M-C. Firpo, Physical Review E 57, 6599 (1998)
V. Latora, A. Rapisarda and S. Ruffo, Physica D 131, 38 (1999); Physica A 280, 81 (2000)
L. Casetti, C. Clementi and M. Pettini, Physical Review E 54, 5969 (1996); L. Caiani, L. Casetti, C. Clementi and M. Pettini, Physical Review Letters 79, 4361, (1997)
J.L. Lebowitz, J.K. Percus and L. Verlet, Physical Review A 153, 250 (1967)
V. Latora, A. Rapisarda, S. Ruffo, Progress of Theoretical Physics Supplement 139, 204 (2000)
G. Parisi and A. Vulpiani, Journal of Physics A 19, L425 (1986)
T. Tsuchiya and N. Gouda, Physical Review E 61, 948 (2000)
L. Caiani et al: Physical Review E 57, 3886 (1998)
J. Barré and T. Dauxois, Europhysics Letters 55, 164 (2001)
C. Anteneodo and R. Vallejos, Physical Review E 65, 016210 (2002)
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Dauxois, T., Latora, V., Rapisarda, A., Ruffo, S., Torcini, A. (2002). The Hamiltonian Mean Field Model: From Dynamics to Statistical Mechanics and Back. In: Dauxois, T., Ruffo, S., Arimondo, E., Wilkens, M. (eds) Dynamics and Thermodynamics of Systems with Long-Range Interactions. Lecture Notes in Physics, vol 602. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45835-2_16
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