There is an elegant thermodynamic argument for the Kosterlitz–Thouless transition. The energy of a single vortex is κ κ ln ⁡ ⁡ ( R / a ) , where κ κ is a parameter that depends upon the system in which the vortex is located, R is the system size, and a is the radius of the vortex core. One assumes R ≫ ≫ a . In the 2D system, the number of possible positions of a vortex is approximately ( R / a ) 2 . From Boltzmann's entropy formula, S = k B ln ⁡ ⁡ W (with W is the number of states), the entropy is S = 2 k B ln ⁡ ⁡ ( R / a ) , where k B is the Boltzmann constant. Thus, the Helmholtz free energy is