Hamiltonian and Brownian systems with longrange interactions: I Statistical equilibrium states and correlation functions
Abstract
We discuss the equilibrium statistical mechanics of systems with longrange 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 meanfield approximation is exact in a proper thermodynamic limit N→+∞. The onepoint equilibrium distribution function is solution of an integrodifferential equation obtained from a static BBGKYlike 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 T_{c} separating a homogeneous phase ( T⩾T_{c}) from a clustered phase ( T⩽T_{c}). The homogeneous phase becomes unstable for T<T_{c} and this instability is a generalization of the Jeans gravitational instability in astrophysics. We derive an expression of the twobody correlation function in the homogeneous phase and show that it diverges close to the critical point. One interest of our study is to provide general expressions valid for a wide class of potentials of interaction in various dimensions of space. Explicit results are given for selfgravitating systems, 2D vortices and for the HMF model.
 Publication:

Physica A Statistical Mechanics and its Applications
 Pub Date:
 February 2006
 DOI:
 10.1016/j.physa.2005.06.087
 Bibcode:
 2006PhyA..361...55C