An easy argument for the dominating logarithmic term is, that the magnitude of the strain induced by an isolated dislocation decays according to ∝ ∝ 1 r with distance. Assuming Hooke's approximation, the associated stress is linear with the strain. Integrating the strain ~1/r gives the energy proportional to the logarithm. The logarithmic distance dependence of the energy is the reason, why KTHNY-theory is one of the few theories of phase transitions which can be solved analytically: in statistical physics one has to calculate partition functions, e.g. the probability distribution for all possible configurations of dislocation pairs given by the Boltzmann distribution e H l o c k B T . Here, k B T is the thermal energy with Boltzmann constant k B . For the majority of problems in statistical physics one can hardly solve the partition function due to the enormous amount of particles and degrees of freedoms. This is different in KTHNY theory due to the logarithmic energy functions of dislocations H l o c and the e-function from the Boltzmann factor as inverse which can be solved easily.