### Abstract

The hydrodynamic limit for a Markov process of [0, ∞)-valued spin fields on a periodic multidimensional lattice is studied. In the process a positive real number, called energy, is attached to each site of the lattice and each couple of adjacent sites exchange thier energy by random amounts at random times. The law of the exchange is such that the sum of the total energy is conserved, and that the process is reversible and of gradient type for the energy distribution. We show that under diffusion type scaling of space and time, the macroscopic energy distribution converges to a deterministic limit which is characterized by a non-linear diffusion equation ∂ρ/∂t=2^{-1}ΔP(ρ), where P is an increasing function which in a typical case equals const·ρ^{2}.

Original language | English |
---|---|

Pages (from-to) | 47-74 |

Number of pages | 28 |

Journal | Probability Theory and Related Fields |

Volume | 95 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1993 Mar |

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### Keywords

- Mathematics Subject Classifications (1980): 60K35, 82A50

### ASJC Scopus subject areas

- Statistics and Probability
- Analysis
- Mathematics(all)

### Cite this

*Probability Theory and Related Fields*,

*95*(1), 47-74. https://doi.org/10.1007/BF01197337

**Hydrodynamic limit for a spin system on a multidimensional lattice.** / Suzuki, Yuki; Uchiyama, Kôhei.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 95, no. 1, pp. 47-74. https://doi.org/10.1007/BF01197337

}

TY - JOUR

T1 - Hydrodynamic limit for a spin system on a multidimensional lattice

AU - Suzuki, Yuki

AU - Uchiyama, Kôhei

PY - 1993/3

Y1 - 1993/3

N2 - The hydrodynamic limit for a Markov process of [0, ∞)-valued spin fields on a periodic multidimensional lattice is studied. In the process a positive real number, called energy, is attached to each site of the lattice and each couple of adjacent sites exchange thier energy by random amounts at random times. The law of the exchange is such that the sum of the total energy is conserved, and that the process is reversible and of gradient type for the energy distribution. We show that under diffusion type scaling of space and time, the macroscopic energy distribution converges to a deterministic limit which is characterized by a non-linear diffusion equation ∂ρ/∂t=2-1ΔP(ρ), where P is an increasing function which in a typical case equals const·ρ2.

AB - The hydrodynamic limit for a Markov process of [0, ∞)-valued spin fields on a periodic multidimensional lattice is studied. In the process a positive real number, called energy, is attached to each site of the lattice and each couple of adjacent sites exchange thier energy by random amounts at random times. The law of the exchange is such that the sum of the total energy is conserved, and that the process is reversible and of gradient type for the energy distribution. We show that under diffusion type scaling of space and time, the macroscopic energy distribution converges to a deterministic limit which is characterized by a non-linear diffusion equation ∂ρ/∂t=2-1ΔP(ρ), where P is an increasing function which in a typical case equals const·ρ2.

KW - Mathematics Subject Classifications (1980): 60K35, 82A50

UR - http://www.scopus.com/inward/record.url?scp=21144462288&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21144462288&partnerID=8YFLogxK

U2 - 10.1007/BF01197337

DO - 10.1007/BF01197337

M3 - Article

VL - 95

SP - 47

EP - 74

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1

ER -