{"title":"具有长程相互作用的不对称排斥过程","authors":"V. Belitsky, N. P. N. Ngoc, G. M. Schütz","doi":"arxiv-2409.05017","DOIUrl":null,"url":null,"abstract":"We consider asymmetric simple exclusion processes with $N$ particles on the\none-dimensional discrete torus with $L$ sites with following properties: (i)\nnearest-neighbor jumps on the torus, (ii) the jump rates depend only on the\ndistance to the next particle in the direction of the jump, (iii) the jump\nrates are independent of $N$ and $L$. For measures with a long-range two-body\ninteraction potential that depends only on the distance between neighboring\nparticles we prove a relation between the interaction potential and particle\njump rates that is necessary and sufficient for the measure to be invariant for\nthe process. The normalization of the measure and the stationary current are\ncomputed both for finite $L$ and $N$ and in the thermodynamic limit. For a\nfinitely many particles that evolve on $\\mathbb{Z}$ with totally asymmetric\njumps it is proved, using reverse duality, that a certain family of\nnonstationary measures with a microscopic shock and antishock evolves into a\nconvex combination of such measures with weights given by random walk\ntransition probabilities. On macroscopic scale this domain random walk is a\ntravelling wave phenomenon tantamount to phase separation with a stable shock\nand stable antishock. Various potential applications of this result and open\nquestions are outlined.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"20 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Asymmetric exclusion process with long-range interactions\",\"authors\":\"V. Belitsky, N. P. N. Ngoc, G. M. Schütz\",\"doi\":\"arxiv-2409.05017\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider asymmetric simple exclusion processes with $N$ particles on the\\none-dimensional discrete torus with $L$ sites with following properties: (i)\\nnearest-neighbor jumps on the torus, (ii) the jump rates depend only on the\\ndistance to the next particle in the direction of the jump, (iii) the jump\\nrates are independent of $N$ and $L$. For measures with a long-range two-body\\ninteraction potential that depends only on the distance between neighboring\\nparticles we prove a relation between the interaction potential and particle\\njump rates that is necessary and sufficient for the measure to be invariant for\\nthe process. The normalization of the measure and the stationary current are\\ncomputed both for finite $L$ and $N$ and in the thermodynamic limit. For a\\nfinitely many particles that evolve on $\\\\mathbb{Z}$ with totally asymmetric\\njumps it is proved, using reverse duality, that a certain family of\\nnonstationary measures with a microscopic shock and antishock evolves into a\\nconvex combination of such measures with weights given by random walk\\ntransition probabilities. On macroscopic scale this domain random walk is a\\ntravelling wave phenomenon tantamount to phase separation with a stable shock\\nand stable antishock. Various potential applications of this result and open\\nquestions are outlined.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"20 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Probability\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.05017\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05017","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Asymmetric exclusion process with long-range interactions
We consider asymmetric simple exclusion processes with $N$ particles on the
one-dimensional discrete torus with $L$ sites with following properties: (i)
nearest-neighbor jumps on the torus, (ii) the jump rates depend only on the
distance to the next particle in the direction of the jump, (iii) the jump
rates are independent of $N$ and $L$. For measures with a long-range two-body
interaction potential that depends only on the distance between neighboring
particles we prove a relation between the interaction potential and particle
jump rates that is necessary and sufficient for the measure to be invariant for
the process. The normalization of the measure and the stationary current are
computed both for finite $L$ and $N$ and in the thermodynamic limit. For a
finitely many particles that evolve on $\mathbb{Z}$ with totally asymmetric
jumps it is proved, using reverse duality, that a certain family of
nonstationary measures with a microscopic shock and antishock evolves into a
convex combination of such measures with weights given by random walk
transition probabilities. On macroscopic scale this domain random walk is a
travelling wave phenomenon tantamount to phase separation with a stable shock
and stable antishock. Various potential applications of this result and open
questions are outlined.