{"title":"多维马尔可夫随机飞行特征函数的序列表示","authors":"Alexander D. Kolesnik","doi":"10.1007/s10955-024-03290-z","DOIUrl":null,"url":null,"abstract":"<p>We consider the symmetric Markov random flight, also called the persistent random walk, performed by a particle that moves at constant finite speed in the Euclidean space <span>\\(\\mathbb {R}^m, \\; m\\ge 2,\\)</span> and changes its direction at Poisson-distributed time instants by taking it at random according to the uniform distribution on the surface of the unit <span>\\((m-1)\\)</span>-dimensional sphere. Such stochastic motion has become a very popular object of modern statistical physics because it can serve as an appropriate model for describing the isotropic finite-velocity transport in multidimensional Euclidean spaces. In recent decade this approach was also developed in the framework of the run-and-tumble theory. In this article we study one of the most important characteristics of the multidimensional symmetric Markov random flight, namely, its characteristic function. We derive two series representations of the characteristic function of the process with respect to Bessel functions with variable indices and with respect to the powers of time variable. The coefficients of these series are given by recurrent relations, as well as in the form of special determinants. As an application of these results, an asymptotic formula for the second moment function <span>\\(\\mu _{(2,2,2)}(t), \\; t>0,\\)</span> of the three-dimensional Markov random flight, is presented. The moment function <span>\\(\\mu _{(2,0,0)}(t), \\; t>0,\\)</span> is obtained in an explicit form.</p>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Series Representations for the Characteristic Function of the Multidimensional Markov Random Flight\",\"authors\":\"Alexander D. Kolesnik\",\"doi\":\"10.1007/s10955-024-03290-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider the symmetric Markov random flight, also called the persistent random walk, performed by a particle that moves at constant finite speed in the Euclidean space <span>\\\\(\\\\mathbb {R}^m, \\\\; m\\\\ge 2,\\\\)</span> and changes its direction at Poisson-distributed time instants by taking it at random according to the uniform distribution on the surface of the unit <span>\\\\((m-1)\\\\)</span>-dimensional sphere. Such stochastic motion has become a very popular object of modern statistical physics because it can serve as an appropriate model for describing the isotropic finite-velocity transport in multidimensional Euclidean spaces. In recent decade this approach was also developed in the framework of the run-and-tumble theory. In this article we study one of the most important characteristics of the multidimensional symmetric Markov random flight, namely, its characteristic function. We derive two series representations of the characteristic function of the process with respect to Bessel functions with variable indices and with respect to the powers of time variable. The coefficients of these series are given by recurrent relations, as well as in the form of special determinants. As an application of these results, an asymptotic formula for the second moment function <span>\\\\(\\\\mu _{(2,2,2)}(t), \\\\; t>0,\\\\)</span> of the three-dimensional Markov random flight, is presented. The moment function <span>\\\\(\\\\mu _{(2,0,0)}(t), \\\\; t>0,\\\\)</span> is obtained in an explicit form.</p>\",\"PeriodicalId\":667,\"journal\":{\"name\":\"Journal of Statistical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-06-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Statistical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1007/s10955-024-03290-z\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s10955-024-03290-z","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Series Representations for the Characteristic Function of the Multidimensional Markov Random Flight
We consider the symmetric Markov random flight, also called the persistent random walk, performed by a particle that moves at constant finite speed in the Euclidean space \(\mathbb {R}^m, \; m\ge 2,\) and changes its direction at Poisson-distributed time instants by taking it at random according to the uniform distribution on the surface of the unit \((m-1)\)-dimensional sphere. Such stochastic motion has become a very popular object of modern statistical physics because it can serve as an appropriate model for describing the isotropic finite-velocity transport in multidimensional Euclidean spaces. In recent decade this approach was also developed in the framework of the run-and-tumble theory. In this article we study one of the most important characteristics of the multidimensional symmetric Markov random flight, namely, its characteristic function. We derive two series representations of the characteristic function of the process with respect to Bessel functions with variable indices and with respect to the powers of time variable. The coefficients of these series are given by recurrent relations, as well as in the form of special determinants. As an application of these results, an asymptotic formula for the second moment function \(\mu _{(2,2,2)}(t), \; t>0,\) of the three-dimensional Markov random flight, is presented. The moment function \(\mu _{(2,0,0)}(t), \; t>0,\) is obtained in an explicit form.
期刊介绍:
The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.