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Hamiltonian and Brownian systems with long-range interactions: I Statistical equilibrium states and correlation functions

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  • Chavanis, Pierre-Henri

Abstract

We discuss the equilibrium statistical mechanics of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system from the canonical description of a stochastically forced Brownian system. We show that the mean-field approximation is exact in a proper thermodynamic limit N→+∞. The one-point equilibrium distribution function is solution of an integrodifferential equation obtained from a static BBGKY-like hierarchy. It also optimizes a thermodynamical potential (entropy or free energy) under appropriate constraints. In the case of attractive potentials of interaction, we show the existence of a critical temperature Tc separating a homogeneous phase (T⩾Tc) from a clustered phase (T⩽Tc). The homogeneous phase becomes unstable for T

Suggested Citation

  • Chavanis, Pierre-Henri, 2006. "Hamiltonian and Brownian systems with long-range interactions: I Statistical equilibrium states and correlation functions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(1), pages 55-80.
    • Handle: RePEc:eee:phsmap:v:361:y:2006:i:1:p:55-80
    DOI: 10.1016/j.physa.2005.06.087
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