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Derivation of Cahn-Hilliard equations from Ginzburg-Landau models.(English)Zbl 0924.60065

The authors derive the generalized Cahn-Hilliard equation as a hydrodynamic scaling limit from a stochastic Ginzburg-Landau model [see alsoC. Collet,F. Dunlop andT. Gobron, J. Stat. Phys. 79, 215-229 (1995) andM. Z. Guo,G. C. Papanicolaou andS. R. S. Varadhan, Commun. Math. Phys. 118, No. 1, 31-59 (1988;Zbl 0652.60107)]. The approach is analogous to the derivation of reaction-diffusion equations as the hydrodynamic limit of a scale depending on Glauber and Kawasaki dynamics [see alsoA. de Masi,P. A. Ferrari andJ. L. Lebowitz, J. Stat. Phys. 44, 589-644 (1986;Zbl 0629.60107)]. A large deviation principle is also proved and the explicit formula for the rate function is given [see alsoM. D. Donsker andS. R. S. Varadhan, Commun. Pure Appl. Math. 42, No. 3, 243-270 (1989;Zbl 0780.60027) andC. Kipnis,S. Olla andS. R. S. Varadhan, Commun. Pure Appl. Math. 42, No. 2, 115-137 (1989;Zbl 0644.76001)].

MSC:

60J60 Diffusion processes
82C22 Interacting particle systems in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
35Q55 NLS equations (nonlinear Schrödinger equations)
60F10 Large deviations

Cite

References:

[1][BKL] O. Benois, C. Kipnis, and C. Landim, Large deviations from the hydrodynamical limit of mean zero asymmetric zero-range processes,Stoch. Proc. Applic. 55:65–89 (1995). ·Zbl 0822.60091 ·doi:10.1016/0304-4149(95)91543-A
[2][CDG] P. Collet, F. Dunlop, and T. Gobron, Conservative Langevin dynamics of solid-on-solid interface,J. Stat. Phys. 79:215–229 (1995). ·Zbl 1081.82597 ·doi:10.1007/BF02179387
[3][DFL] A. De Masi, P. Ferrari, and J. L. Lebowitz, Reaction-diffusion equations for interacting particle systems,J. Stat. Phys. 44:589–644 (1986). ·Zbl 0629.60107 ·doi:10.1007/BF01011311
[4][DOPT] A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, Glauber evolution with Kac potential I. Mesoscopic and macroscopic limits, interface dynamics,Nonlinearity 7:633–696 (1994). ·Zbl 0797.60088 ·doi:10.1088/0951-7715/7/3/001
[5][DV] M. D. Donsker and S. R. S. Varadhan, Large deviations from a hydrodynamic scaling limit,Comm. Pure Appl. Math. 42:243–270 (1989). ·Zbl 0780.60027 ·doi:10.1002/cpa.3160420303
[6][FS] T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau {\(\Delta\)} interface model, preprint (1994).
[7][JLV] D. Gabrielli, G. Jona-Lasinio, C. Landim, and M. E. Vares, Microscopic reversibility and thermodynamic fluctuations, to appear in The proceedings of the conference ”Boltzmann’s Legacy” Roma, 1994.
[8][GL] G. Giacomin and J. L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions,Phys. Rev. Let. 76:1094–1097 (1996). ·doi:10.1103/PhysRevLett.76.1094
[9][GPV] M. Z. Guo, G. C. Papanicolaou, and S. R. S. Varadhan, Nonlinear diffusion for a system with nearest neighbor interactions,Commun. Math. Phys. 118:31–59 (1988). ·Zbl 0652.60107 ·doi:10.1007/BF01218476
[10][JLV] G. Jona-Lasinio, C. Landim, and M. E. Vares, Large deviations for a reaction-diffusion model,Prob. Th. Rel. Fields 97:339–361 (1993). ·Zbl 0792.60096 ·doi:10.1007/BF01195070
[11][KOV] C. Kipnis, S. Olla, and S. R. S. Varadhan, Hydrodynamics and large deviations for simple exclusion processes,Comm. Pure Appl. Math. 42:115–137 (1989). ·Zbl 0644.76001 ·doi:10.1002/cpa.3160420202
[12][LY] C. Landim and H. T. Yau, Large deviations of interacting particle systems in infinite volume,Comm. Pure Appl. Math. 48:339–379 (1995). ·Zbl 0822.60092 ·doi:10.1002/cpa.3160480401
[13][S] H. Spohn, Interface motion in models with stochastic dynamics,J. Stat. Phys. 71:1081–1132 (1993). ·Zbl 0935.82546 ·doi:10.1007/BF01049962
[14][V] S. R. S. Varadhan,Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 46, 1984.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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