Non-extensive thermodynamics of 1D systems with long-range interaction

A new approach to non-extensive thermodynamical systems with non-additive energy and entropy is proposed. The main idea of the paper is based on the statistical matching of the thermodynamical systems with the additive multi-step Markov chains. This general approach is applied to the Ising spin chain with long-range interaction between its elements. The asymptotical expressions for the energy and entropy of the system are derived for the limiting case of weak interaction. These thermodynamical quantities are found to be non-proportional to the length of the system (number of its particle).

💡 Research Summary

The paper introduces a novel theoretical framework for treating non‑extensive thermodynamic systems—systems in which energy and entropy are not additive—by statistically matching them to additive multi‑step Markov chains. The authors begin by highlighting the inadequacy of conventional Boltzmann‑Gibbs thermodynamics for systems with long‑range interactions, where the usual assumption that thermodynamic quantities scale linearly with the number of constituents breaks down. While previous attempts, notably Tsallis’ non‑extensive statistics, have introduced a deformation parameter (q) to capture non‑additivity, they often lack a direct connection to the microscopic interaction structure of concrete models.

The core idea of the present work is to map a physical system with non‑additive energy onto a Markov process whose transition probabilities are additive. In practice, each spin (s_i) in a one‑dimensional Ising chain is assigned a conditional probability that depends on the states of the preceding (N) spins:
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