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We discuss the problem of anyonic statistics in one and two spatialdimensions from the point of view of statistical physics. In particular we wantto understand how the choice of the Bornvon Karman or the twisted periodicboundary conditions necessary in a Monte Carlo simulation to mimic thethermodynamic limit of the many body system influences the statistical natureof the particles. The particles can either be just bosons, when theconfiguration space is simply connected as for example for particles on a line.They can be bosons and fermions, when the configuration space is doublyconnected as for example for particles in the tridimensional space or in aRiemannian surface of genus greater or equal to one (on the torus, etc . . . ).They can be scalar anyons with arbitrary statistics, when the configurationspace is infinitely connected as for particles on the plane or in the circle.They can be scalar anyons with fractional statistics, when the configurationspace is the one of particles on a sphere. One can further have multicomponents anyons with fractional statistics when the configuration space isdoubly connected as for particles on a Riemannian surface of genus greater orequal to one. We determine an expression for the canonical partition functionof hard core particles (including anyons) on various geometries. We then showhow the choice of boundary condition (periodic or open) in one and twodimensions determine which particles can exist on the considered surface. Inthe conclusion, we mention the Laughlin wavefunction and give a few commentsabout experiments.

How should we choose the boundary conditions in a simulation which could detect anyons in one and two dimensions?