opposite sign. Bound vortex–antivortex pairs have lower energies than free vortices, but have lower entropy as well. In order to minimize free energy, , the system undergoes a transition at a critical temperature, . Below , there are only bound vortex–antivortex pairs. Above , there are free vortices. Informal description There is an elegant thermodynamic argument for the Kosterlitz–Thouless transition. The energy of a single vortex is , where is a parameter that depends upon the system in which the vortex is located, is the system size, and is the radius of the vortex core. One assumes . In the 2D system, the number of possible positions of a vortex is approximately . From Boltzmann's entropy formula, (with W is the number of states), the entropy is , where is Boltzmann's constant. Thus, the Helmholtz free energy is When , the system will not have a vortex. On the other hand, when , entropic considerations favor the formation of a vortex. The critical temperature above which vortices may form can be found by setting and is given by The Kosterlitz–Thouless transition can be observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage (I-V) measurements. Above , the relation will be linear . Just below , the relation will be , as the number of free vortices will go as . This jump from linear dependence is indicative of a Kosterlitz–Thouless transition and may be used to determine . This approach was used in Resnick et al. to confirm the Kosterlitz–Thouless transition in proximity-coupled Josephson junction arrays. Field theoretic analysis The following discussion uses field theoretic methods. Assume a field φ(x) defined in the plane which takes on values in . For convenience, we work with the universal cover R of instead, but identify any two values of φ(x) that differ by an integer multiple of 2π. The energy is given by and the Boltzmann factor is . Taking a contour integral over any contractible closed path , we would expect it to be zero. However, this is not the case due to the singular nature of vortices. We can imagine that the theory is defined up to some energetic cut-off scale , so that we can puncture the plane at the points where the vortices are located, by removing regions of linear size of order . If winds counter-clockwise once around a puncture, the contour