Notes on the Statistical Mechanics of Systems with Long-Range Interactions

In these Notes we discuss some thermodynamic and dynamical properties of models of systems in which the two-body interaction potential between particles decays algebraically with their relative distance r at large distances. Typically the potential decays as 1/r d+σ in d dimensions and it may either be isotropic or non-isotropic (as in the case of magnetic or electric dipolar interactions). In general one can distinguish between two broad classes long range interactions (LRI). Those with −d ≤ σ ≤ 0 which we term " strong " LRI, and those with 0 < σ ≤ σ c (d) which we term " weak " LRI. The parameter σ c (d) satisfies σ c (d) = 2 for d ≥ 4, and σ c (d) = d/2 for d < 4, as will be discussed in more detail in the following sections. In systems with strong LRI the potential decays slowly with the distance, and it results in rather pronounced ther-modynamic and dynamical effects. On the other hand in systems with weak LRI the potential decays faster at large distances, resulting in less pronounced effects. Systems with σ > σ c behave thermodynamically as the more commonly studied systems with short range interactions. For recent reviews on systems with long range interactions see, for example, (Dauxois et al, 2002; Campa et al, 2007a). Long range interactions are rather common in nature. Examples include self gravitating systems (σ = −2) (Padmanabhan, 1990; Chavanis, 2002), dipolar ferroelectrics and ferromagnets in which the interactions are anisotropic with (σ = 0) (Landau and Lifshitz, 1960), non-neutral plasmas (σ = −2) (Nicholson, 1992), two dimensional geophysical vortices which interact via a weak, logarithmically decaying, potential (σ = −2) (Chavanis, 2002), charged particles interacting via their mutual electromagnetic fields, such as in free electron laser (Barré et al, 2004) and many others. Let us first consider strong LRI. Such systems are non-additive, and the energy of homogeneously distributed particles in a volume V scales super-linearly with the volume , as V 1−σ/d. The lack of additivity leads to many unusual properties, both thermal ¾ Notes on the Statistical Mechanics of Systems with Long-Range Interactions and dynamical, which are not present in systems with weak LRI or with short range interactions. For example, as has first been pointed out by Antonov (Antonov, 1962) and later elaborated by Lynden-and others, the entropy S needs not be a concave function of …