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• W2032799060 abstract "We present an alternative solution of the Ising chain in a field under free and periodic boundary conditions, in the microcanonical, canonical, and grand canonical ensembles, from a unified combinatorial and topological perspective. In particular, the computation of the per-site entropy as a function of the energy unveils a residual value for critical values of the magnetic field, a phenomenon for which we provide a topological interpretation and a connection with the Fibonacci sequence. We also show that, in the thermodynamic limit, the per-site microcanonical entropy is equal to the logarithm of the per-site Euler characteristic. The canonical and grand canonical partition functions are identified as combinatorial generating functions of the microcanonical problem, which allows us to evaluate them. A detailed analysis of the magnetic field-dependent thermodynamics, including positive and negative temperatures, reveals interesting features. Finally, we emphasize that our combinatorial approach to the canonical ensemble allows exact computation of the thermally averaged value < χ > of the Euler characteristic associated with the spin configurations of the chain, which is discontinuous at the critical fields, and whose thermal behavior is expected to determine the phase transition of the model. Indeed, our results show that the conjecture < χ > (T C) = 0, where T C is the critical temperature, is valid for the Ising chain." @default.
• W2032799060 created "2016-06-24" @default.
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• W2032799060 date "2012-11-15" @default.
• W2032799060 modified "2023-09-25" @default.
• W2032799060 title "Combinatorial and Topological Analysis of the Ising Chain in a Field" @default.
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• W2032799060 doi "https://doi.org/10.1007/s13538-012-0108-8" @default.
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