### Abstract

Consider a distinguished, or tagged particle in zero-range dynamics on Z^{d} with rate g whose finite-range jump probabilities p possess a drift ∑ j p (j) ≠ 0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d ≥ 1, and at most order t in d = 1 and d ≥ 3 for a wide class of rates g. Also, in d = 1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g (k) increases, and g (k) / k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.

Original language | English (US) |
---|---|

Pages (from-to) | 215-232 |

Number of pages | 18 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 43 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2007 |

Externally published | Yes |

### Fingerprint

### Keywords

- Diffusive
- Invariance principle
- Tagged particle
- Variance
- Zero-range

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

**On diffusivity of a tagged particle in asymmetric zero-range dynamics.** / Sethuraman, Sunder.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On diffusivity of a tagged particle in asymmetric zero-range dynamics

AU - Sethuraman, Sunder

PY - 2007/3

Y1 - 2007/3

N2 - Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑ j p (j) ≠ 0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d ≥ 1, and at most order t in d = 1 and d ≥ 3 for a wide class of rates g. Also, in d = 1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g (k) increases, and g (k) / k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.

AB - Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑ j p (j) ≠ 0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d ≥ 1, and at most order t in d = 1 and d ≥ 3 for a wide class of rates g. Also, in d = 1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g (k) increases, and g (k) / k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.

KW - Diffusive

KW - Invariance principle

KW - Tagged particle

KW - Variance

KW - Zero-range

UR - http://www.scopus.com/inward/record.url?scp=33847266293&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847266293&partnerID=8YFLogxK

U2 - 10.1016/j.anihpb.2006.03.002

DO - 10.1016/j.anihpb.2006.03.002

M3 - Article

VL - 43

SP - 215

EP - 232

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 2

ER -