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We take the initial velocity distribution of the particles to be a Borel probability measure $\\mu$ on $\\mathbb{R}^n$ that is symmetric in all its variables, has mean $\\vec{0}$ and finite second moment. Let $\\mu_t(dv)$ denote the Kac-evolved distribution at time $t$, and let $R_\\mu$ be the angular average of $\\mu$. We give an upper bound to $d_2(\\mu_t, R_\\mu)$ of the form $\\min\\{ B e^{-\\frac{4 \\lambda_1}{n+3}t}, d_2(\\mu,R_\\mu)\\}$, where $\\lambd"},"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":"1610.09601","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-10-30T03:38:05Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"ed7d5768b535aff39e49745a449fdd8f155c5da1eb5271cd7ecdc326fd15802e","abstract_canon_sha256":"ca677de8e6fe9ec36409878cb04e46b1628d64e9ccc7d3e499585e7a3cd57fbe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:24.399864Z","signature_b64":"NsExvSsBWL/cBlSrK8mnHbhI76Zh9qqDsWXIEYaZ5Ulrxe+qyrAbxqE+K6x0F3NuOR6zq4EUFT208wZJwEpABA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4945b17cb4434735d16ac6f0e3a4962199025928c7374388b6b7bc4cf2e6b3cf","last_reissued_at":"2026-05-18T00:35:24.399343Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:24.399343Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equilibration in the Kac Model using the GTW Metric $d_2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Hagop Tossounian","submitted_at":"2016-10-30T03:38:05Z","abstract_excerpt":"We use the Fourier based Gabetta-Toscani-Wennberg (GTW) metric $d_2$ to study the rate of convergence to equilibrium for the Kac model in $1$ dimension. 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