Statistical Physics

F.2 Periodic Boundary Condition 199we are dealing with the properties of a bulk sample, the result should beindependent of the boundary condition. This is because the contribution fromregions near the wall is negligible compared with that from the interior of thesample, which represents most of the volume. In fact, the number of statesthat satisfy |k| ≤k M does not depend on the boundary condition. Since theenergy of a wave is determined by the frequency of the wave, which depends onthe absolute value of the wave vector, the density of states is independent ofthe boundary condition. Therefore, the properties of a bulk sample calculatedby use of statistical mechanics do not depend on the boundary condition.