Non-additive properties of finite 1D Ising chains with long-range   interactions
  We study the statistical properties of Ising spin chains with finite (although arbitrary large) range of interaction between the elements. We examine mesoscopic subsystems (fragments of an Ising chain) with the lengths comparable with the interaction range. The equivalence of the Ising chains and the multi-step Markov sequences is used for calculating different non-additive statistical quantities of a chain and its fragments. In particular, we study the variance of fluctuating magnetization of fragments, magnetization of the chain in the external magnetic field, etc. Asymptotical expressions for the non-additive energy and entropy of the mesoscopic fragments are derived in the limiting cases of weak and strong interactions. 
S. S. Apostolov, Z. A. Mayzelis, O. V. Usatenko, V. A. Yampol'skii