Abstract
The hydrodynamics of Ginzburg-Landau dynamics has previously been proved to be a nonlinear diffusion equation. The diffusion coefficient is given by the second derivative of the free energy and hence nonnegative. We consider in this paper the Ginzburg-Landau dynamics with long-range interactions. In this case the diffusion coefficient is nonnegative only in the metastable region. We prove that if the initial condition is in the metastable region, then the hydrodynamics is governed by a nonlinear diffusion equation with the diffusion coefficient given by the metastable curve. Furthermore, the lifetime of the metastable state is proved to be exponentially large.
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References
C. C. Chang and H. T. Yau, Fluctuations of one dimensional Ginzburg-Landau models in nonequilibrium,Commun. Math. Phys. 145:209–234 (1992).
J. Fritz, On the hydrodynamical limit of a Ginzburg-Landau lattic model,Prob. Theory Related Fields 81:291–318 (1989).
J. Fritz, On the diffusive nature of the entropy flow in finite systems: Remarks to a paper by Guo-Papanicolaou-Varadhan,Commun. Math. Phys. 133:331–352 (1990).
J. Fritz, private communications.
M. Donsker and S. R. S. Varadhan, Large deviation from a hydrodynamic scaling limit,Comun. Pure Appl. Math. 42:243–270 (1989).
A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, Glauber evolution with Kac potentials II, Spinodal decomposition, preprint.
M. Z. Guo, G. C. Papanicolaou, and S. R. S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions,Commun. Math. Phys. 118:31–59 (1988).
J. Lebowitz, E. Orlandi, and E. Presutti, A particle model for spinodal decomposition,J. Stat. Phys. 63:933–974 (1991).
J. Lebowitz and O. Penrose, Rigorous treatment of the van der Waals Maxwell theory of liquid vapour transition,J. Math. Phys. 7:98 (1966).
S. L. Lu, Thesis, New York University (1991).
S. L. Lu and H. T. Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics,Commun. Math. Phys., to appear.
S. L. Lu and H. T. Yau, Logarithmic Sobolev inequality for Kawasaki dynamics, in preparation.
F. Rezakhanlou, Hydrodynamic limit for a system with finite range interactions,Commun. Math. Phys. 129:445–480 (1990).
F. Martinelli, E. Olivieri, and E. Scoppola, Metastability and exponential approach to equilibrium for low-temperature stochastic Ising models,J. Stat. Phys. 61:1105–1119 (1990).
G. Giacomin, Van der Waals limit and phase separation in a particle model with Kawasaki dynamics,J. Stat. Phys. 65:217–234 (1991).
R. H. Schonmann, Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region, UCLA preprint.
B. Simon,Functional Integration and Quantum Physics (Academic Press, New York, 1979).
H. T. Yau, Relative entropoy and the hydrodynamics of Ginzburg-Landau models,Lett. Math. Phys. 22:63–80 (1991).
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Yau, HT. Metastability of Ginzburg-Landau model with a conservation law. J Stat Phys 74, 705–742 (1994). https://doi.org/10.1007/BF02188577
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DOI: https://doi.org/10.1007/BF02188577