Statistical Mechanics of Systems with Long-Range Interactions (on first lect of this part) book

Book: Physics of Long-Range Interacting Systems

Often take them to have two-body interaction of the form:

$U(r) \sim \frac{1}{r^{d+\sigma}}$

where $d$ is the dimension.

For $\sigma \leq 0$, the systems are non-additive (or non-extensive), in the sense of Statistical physics, so that, for example, energy is no simply proportional to volume.

Examples

• self-gravitating system ($\sigma = 2$)
• dipolar magnets ($\sigma =0$)
• charged plasma
• vortices in 2 dimensions. $U(r) \sim \ln{r}$, $\sigma = -2$ (effectively)