Abstract
In cluster approximations for lattice systems the thermodynamic behavior of the infinite system is inferred from that of a relatively small, finite subsystem (cluster), approximations being made for the influence of the surrounding system. In this context we study, for translation-invariant classical lattice systems, the conditions under which a state for a cluster admits an extension to a global translation-invariant state. This extension problem is related to undecidable tiling problems. The implication is that restrictions of global translation-invariant states cannot be characterized purely locally in general. This means that there is an unavoidable element of uncertainty in the application of a cluster approximation.
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Schlijper, A.G. Tiling problems and undecidability in the cluster variation method. J Stat Phys 50, 689–714 (1988). https://doi.org/10.1007/BF01026496
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DOI: https://doi.org/10.1007/BF01026496