An entropic argument against perfect long range order in crystals with D < 3 is as follows (see figure): consider a chain of atoms/particles with an average particle distance of ⟨ ⟨ a ⟩ ⟩ . Thermal fluctuations between particle 0 and particle 1 will lead to fluctuations of the average particle distance of the order of ξ ξ 0 , 1 , thus the distance is given by a = ⟨ ⟨ a ⟩ ⟩ ± ± ξ ξ 0 , 1 . The fluctuations between particle − − 1 and 0 will be of the same size: | ξ ξ − − 1 , 0 | = | ξ ξ 0 , 1 | . We assume that the thermal fluctuations are statistically independent (which is evident if we consider only nearest neighbour interaction) and the fluctuations between − − 1 and particle + 1 (with double the distance) has to be summed statistically independent (or incoherent): ξ ξ − − 1 , 1 = 2 ⋅ ⋅ ξ ξ 0 , 1 . For particles N-times the average distance, the fluctuations will increase with the square root ξ ξ 0 , N = N ⋅ ⋅ ξ ξ 0 , 1 if neighbouring fluctuations are summed independently. Although the average distance ⟨ ⟨ a ⟩ ⟩ is well defined, the deviations from a perfect periodic chain increase with the square root of the system size. In three dimensions, one has to walk along three linearly independent directions to cover the whole space; in a cubic crystal, this is effectively along the space diagonal, to get from particle 0 to particle 3 . As one can easily see in the figure, there are six different possibilities to do this. This implies, that the fluctuations on the six different pathways cannot be statistically independent, since they pass the same particles at position 0 and 3 . Now, the fluctuations of the six different ways have to be summed in a coherent way and will be of the order of ξ ξ – independent of the size of the cube. The fluctuations stay finite and lattice sites are well defined. For the case of two dimensions, Herbert Wagner and David Mermin have proved rigorously, that fluctuations distances increase logarithmically with systems size ξ ξ ∝ ∝ ln ⁡ ⁡ ( L ) . This is frequently called the logarithmic divergence of displacements.