Non-additive properties of finite 1D Ising chains with long-range interactions

We study the statistical properties of Ising spin chains with a finite (although arbitrarily large) range of interaction among the elements. We examine mesoscopic subsystems (fragments of an Ising chain) with 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 the chain and its fragments. In particular, we study the variance of the fluctuating magnetization of fragments, magnetization of the chain in an external magnetic field, etc. Asymptotical expressions for the non-additive energy and entropy of mesoscopic fragments are derived in the limiting cases of weak and strong interactions.