  We discuss the problem of anyonic statistics in one and two spatial
dimensions from the point of view of statistical physics. In particular we want
to understand how the choice of the Bornvon Karman or the twisted periodic
boundary conditions necessary in a Monte Carlo simulation to mimic the
thermodynamic limit of the many body system influences the statistical nature
of the particles. The particles can either be just bosons, when the
configuration space is simply connected as for example for particles on a line.
They can be bosons and fermions, when the configuration space is doubly
connected as for example for particles in the tridimensional space or in a
Riemannian surface of genus greater or equal to one (on the torus, etc . . . ).
They can be scalar anyons with arbitrary statistics, when the configuration
space is infinitely connected as for particles on the plane or in the circle.
They can be scalar anyons with fractional statistics, when the configuration
space is the one of particles on a sphere. One can further have multi
components anyons with fractional statistics when the configuration space is
doubly connected as for particles on a Riemannian surface of genus greater or
equal to one. We determine an expression for the canonical partition function
of hard core particles (including anyons) on various geometries. We then show
how the choice of boundary condition (periodic or open) in one and two
dimensions determine which particles can exist on the considered surface. In
the conclusion, we mention the Laughlin wavefunction and give a few comments
about experiments.
