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Using ideas from dynamical systems and functional analysis, we show that the free energy $f$ in the vicinity of the second-order phase transition is given, exactly, by $$ f \\sim \\frac t{\\log t}-\\frac1{2} \\frac{h^2}t \\quad \\text{for} \\quad h^2\\ll t \\ll 1 . $$\n  Here $t=\\lambda_{G}\\log(2)(1-\\frac{\\beta}{\\beta_c})$ is a reduced temperature, so that the deviation from the critical point is scaled by the Lyapunov exponent of the Gauss map, $\\lambda_G$. It follows that $\\lambda_G$ determines the amplitude of both the specific heat a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0909.2878","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2009-09-15T20:36:41Z","cross_cats_sorted":["math-ph","math.DS","math.MP","math.NT"],"title_canon_sha256":"dc7db48fab90f24d1aed3816c8b0366aa0a44b344d18077f07ec3632fea6eb5c","abstract_canon_sha256":"4f77dbca7e74abd3ace5899fb4a104fff9f32cd3373a7df5cd1f8dde38dee6c3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:11:47.619127Z","signature_b64":"qPUlXTJ70UzyBZYleXAmjXMFGP3ZXwY8r4L8oeOtub36iV4jOBq/2z+zEIsaGMwZl7nx9YEG85/J/vEef7TdCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cc43e95d6227a16583be13f5eb69eee4d7d7452caaa6cb937cd17877d61215f","last_reissued_at":"2026-05-18T02:11:47.618559Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:11:47.618559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.DS","math.MP","math.NT"],"primary_cat":"cond-mat.stat-mech","authors_text":"J. Fiala, O.F. Bandtlow, P. Kleban, T. Prellberg","submitted_at":"2009-09-15T20:36:41Z","abstract_excerpt":"We consider the Farey fraction spin chain in an external field $h$. Using ideas from dynamical systems and functional analysis, we show that the free energy $f$ in the vicinity of the second-order phase transition is given, exactly, by $$ f \\sim \\frac t{\\log t}-\\frac1{2} \\frac{h^2}t \\quad \\text{for} \\quad h^2\\ll t \\ll 1 . $$\n  Here $t=\\lambda_{G}\\log(2)(1-\\frac{\\beta}{\\beta_c})$ is a reduced temperature, so that the deviation from the critical point is scaled by the Lyapunov exponent of the Gauss map, $\\lambda_G$. 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