{"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. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.09601","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}