Consider first a one-dimensional system with short range interactions. Creating a domain wall requires a fixed energy amount ϵ ϵ . Extracting this energy from other degrees of freedom decreases entropy by Δ Δ S = − − ϵ ϵ / T . This entropy change must be compared with the entropy of the domain wall itself. In a system of length L there are L / a positions for the domain wall, leading (according to Boltzmann's principle) to an entropy gain Δ Δ S = k B log ⁡ ⁡ ( L / a ) . For nonzero temperature T and L large enough the entropy gain always dominates, and thus there is no phase transition in one-dimensional systems with short-range interactions at T > 0 . Space dimension d 1 = 1 thus is a lower bound for the lower critical dimension of such systems.