{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T23:40:25Z","timestamp":1680392425392},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,12,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We review the derivation of the Kac master equation model for random collisions of particles, its relationship to the Poisson process, and existing algorithms for simulating values from the marginal distribution of velocity for a single particle at any given time. We describe and implement a new algorithm that efficiently and more fully leverages properties of the Poisson process, show that it performs at least as well as existing methods, and give empirical evidence that it may perform better at capturing the tails of the single particle velocity distribution. Finally, we derive and implement a novel \u201c\u03b5-perfect sampling\u201d algorithm for the limiting marginal distribution as time goes to infinity. In this case the importance is a proof of concept that has the potential to be expanded to more interesting (DSMC) direct simulation Monte Carlo applications.<\/jats:p>","DOI":"10.1515\/mcma-2016-0114","type":"journal-article","created":{"date-parts":[[2016,10,19]],"date-time":"2016-10-19T10:02:12Z","timestamp":1476871332000},"page":"291-305","source":"Crossref","is-referenced-by-count":0,"title":["Perfect and \u03b5-perfect simulation methods for the one-dimensional Kac equation"],"prefix":"10.1515","volume":"22","author":[{"given":"Jem N.","family":"Corcoran","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics,University of Colorado, Box 526, Boulder CO 80309-0526, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dale","family":"Jennings","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics,University of Colorado, Box 526, CO 80309-0526, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paul","family":"VaughanMiller","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics,University of Colorado, Box 526, CO 80309-0526, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2016,10,19]]},"reference":[{"key":"2023040102000316888_j_mcma-2016-0114_ref_001_w2aab2b8d506b1b7b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"Babovsky H. and Illner R.,\nA convergence proof for Nanbu\u2019s simulation method for the full Boltzmann equation,\nSIAM J. 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Volume 3: Contributions to Astronomy and Physics,\nUniversity of California Press, Berkeley (1956), 171\u2013197."},{"key":"2023040102000316888_j_mcma-2016-0114_ref_008_w2aab2b8d506b1b7b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"Krook M. and Wu T. T.,\nFormation of Maxwellian tails,\nPhys. Rev. Lett. 36 (1976), no. 19, 1107\u20131109.","DOI":"10.1103\/PhysRevLett.36.1107"},{"key":"2023040102000316888_j_mcma-2016-0114_ref_009_w2aab2b8d506b1b7b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"Nanbu K.,\nDirect simulation scheme derived from the Boltzmann equation. I: Monocomponent gases,\nJ. Phys. Soc. Japan 49 (1980), no. 5, 2042\u20132049.","DOI":"10.1143\/JPSJ.49.2042"},{"key":"2023040102000316888_j_mcma-2016-0114_ref_010_w2aab2b8d506b1b7b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"Wagner W.,\nA convergence proof for Bird\u2019s direct simulation Monte Carlo method for the Boltzmann equation,\nJ. Statist. 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