Dynamics and thermodynamics of a simple model similar to self-gravitating systems : the HMF model

• Published in: The European physical journal. B, Condensed matter physics Vol. 46; no. 1; pp. 61 - 99
• Main Authors: CHAVANIS, P. H, VATTEVILLE, J, BOUCHET, F
• Format: Journal Article
• Language: English
• Published: Les Ulis Springer 01-07-2005; Berlin EDP sciences; Springer-Verlag
• Subjects: Classical statistical mechanics; Condensed Matter; Exact sciences and technology; Physics; Statistical Mechanics; Statistical physics, thermodynamics, and nonlinear dynamical systems; Autocorrelation function; Boltzmann statistics; Mean field approximation; Maxwell Boltzmann distribution; Critical points; Thermodynamic stability; Fokker-Planck equation; Density; Long range interaction; Hamiltonians; Self-gravitating systems; Non linear effect; Statistical mechanics
• ISSN: 1434-6028; 1434-6036
• DOI: 10.1140/epjb/e2005-00234-0

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.