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Hydrodynamic limit for a spin system on a multidimensional lattice
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  • Published: March 1993

Hydrodynamic limit for a spin system on a multidimensional lattice

  • Yuki suzuki1 &
  • Kôhei Uchiyama2 

Probability Theory and Related Fields volume 95, pages 47–74 (1993)Cite this article

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  • 7 Citations

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Summary

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), whereP is an increasing function which in a typical case equals const·ρ2.

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References

  1. Billingsley, P.: Convergence of probability measures. New York: Wiley 1968

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  2. Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys.118, 31–59 (1988)

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  3. Suzuki, Y., Uchiyama, K.: Hydrodynamic limit for a [0, ∞)-valued spin system. (preprint)

  4. Uchiyama, K.: Uniqueness of nonnegative solutions of the Cauchy problem foru t =‡f(u), (preprint)

  5. Varadhan, S.R.S.: Scaling limits for interacting diffusions. Commun. Math. Phys.135, 313–353 (1991)

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Technology, Keio University, 223, Hiyoshi Yokohama, Japan

    Yuki suzuki

  2. Department of Applied Physics, Tokyo Institute of Technology, Oh-Okayama, 152, Meguro Tokyo, Japan

    Kôhei Uchiyama

Authors
  1. Yuki suzuki
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  2. Kôhei Uchiyama
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Cite this article

suzuki, Y., Uchiyama, K. Hydrodynamic limit for a spin system on a multidimensional lattice. Probab. Th. Rel. Fields 95, 47–74 (1993). https://doi.org/10.1007/BF01197337

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  • Received: 27 July 1991

  • Revised: 27 May 1992

  • Issue Date: March 1993

  • DOI: https://doi.org/10.1007/BF01197337

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Mathematics Subject Classifications (1980)

  • 60K35
  • 82A50
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