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Wang, Chi-Jen; Guo, Xiaofang; Liu, Da-Jiang; Evans, J. W.
Schloegl’s second model for autocatalysis on a cubic lattice: mean-field-type discrete reaction-diffusion equation analysis. (English) Zbl 1269.82040
J. Stat. Phys. 144, No. 6, 1308-1328 (2011).
Summary: Schloegl’s second model for autocatalysis on a hypercubic lattice of dimension \(d\geq 2\) involves: (i) spontaneous annihilation of particles at lattice sites with rate \(p\); and (ii) autocatalytic creation of particles at vacant sites at a rate proportional to the number of diagonal pairs of particles on neighboring sites. Kinetic Monte Carlo simulations for a \(d=3\) cubic lattice reveal a discontinuous transition from a populated state to a vacuum state as \(p\) increases above \(p=p_{e }\). However, stationary points, \(p=p_{\mathrm{eq}}\) \((p\leq p_{e })\), for planar interfaces separating these states depend on interface orientation. Our focus is on analysis of interface dynamics via discrete reaction-diffusion equations (dRDE’s) obtained from mean-field type approximations to the exact master equations for spatially inhomogeneous states. These dRDE can display propagation failure absent due to fluctuations in the stochastic model. However, accounting for this anomaly, dRDE analysis elucidates exact behavior with quantitative accuracy for higher-level approximations.
MSC:
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
35K57 Reaction-diffusion equations
82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
Keywords:
Schloegl’s second model; generic two-phase coexistence; discrete reaction-diffusion equations; interface propagation
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\textit{C.-J. Wang} et al., J. Stat. Phys. 144, No. 6, 1308--1328 (2011; Zbl 1269.82040)
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References:
[1] Marro, J., Dickman, R.: Nonequilibrium Phase Transitions in Lattice Models. Cambridge University Press, Cambridge (1999) · Zbl 0931.70002
[2] Hinrichsen, H.: Adv. Phys. 49, 815 (2000) · doi:10.1080/00018730050198152
[3] Odor, G.: Rev. Mod. Phys. 76, 663 (2004) · Zbl 1205.82102 · doi:10.1103/RevModPhys.76.663
[4] Ziff, R.M., Gulari, E., Barshad, Y.: Phys. Rev. Lett. 56, 2553 (1986) · doi:10.1103/PhysRevLett.56.2553
[5] Evans, J.W., Miesch, M.S.: Phys. Rev. Lett. 66, 833 (1991) · doi:10.1103/PhysRevLett.66.833
[6] Loscar, E., Albano, E.V.: Rep. Prog. Phys. 66, 1343 (2003) · doi:10.1088/0034-4885/66/8/203
[7] Liu, D.-J., Guo, X., Evans, J.W.: Phys. Rev. Lett. 98, 050601 (2007)
[8] Evans, J.W., Ray, T.R.: Phys. Rev. E 50, 4302 (1994) · doi:10.1103/PhysRevE.50.4302
[9] Goodman, R.H., Graff, D.S., Sander, L.M., Leroux-Hugon, P., ClĂ©ment, E.: Phys. Rev. E 52, 5904 (1995) · doi:10.1103/PhysRevE.52.5904
[10] Machado, E., Buendia, G.M., Rikvold, P.A.: Phys. Rev. E 71, 031603 (2005)
[11] Guo, X., Liu, D.-J., Evans, J.W.: J. Chem. Phys. 130, 074106 (2009) · doi:10.1063/1.3074308
[12] Toom, A.L.: In: Dobrushin, D.L., Sinai, Y.G. (eds.) Multicomponent Random Systems. Advances in Probability and Related Topics, vol. 6, pp. 549–575. Dekker, New York (1980), Chap. 18
[13] Bennett, C.H., Grinstein, G.: Phys. Rev. Lett. 55, 657 (1985) · doi:10.1103/PhysRevLett.55.657
[14] Schloegl, F.: Z. Phys. 253, 147 (1972) · doi:10.1007/BF01379769
[15] Grassberger, P.: Z. Phys. B Condens. Matter 47, 365 (1982) · doi:10.1007/BF01313803
[16] Boon, J.P., Dab, D., Kapral, R., Lawniczak, A.: Rep. Mod. Phys. 273, 55 (1996) · doi:10.1016/0370-1573(95)00080-1
[17] Prakash, S., Nicolis, G.: J. Stat. Phys. 86, 1289 (1997) · Zbl 0935.82025 · doi:10.1007/BF02183624
[18] Durrett, R.: SIAM Rev. 41, 677 (1999) · Zbl 0940.60086 · doi:10.1137/S0036144599354707
[19] Guo, X., de Decker, Y., Evans, J.W.: Phys. Rev. E 82, 021121 (2010)
[20] Guo, X., Liu, D.-J., Evans, J.W.: Phys. Rev. E 75, 061129 (2007)
[21] Ziff, R.M., Brosilow, B.J.: Phys. Rev. A 46, 4630 (1992) · doi:10.1103/PhysRevA.46.4630
[22] Evans, J.W.: Rev. Mod. Phys. 65, 1281 (1993) · doi:10.1103/RevModPhys.65.1281
[23] Guo, X., Evans, J.W., Liu, D.-J.: Physica A 387, 177 (2008). Note the error in site labeling in the last two loss terms in (17): replace i with i · doi:10.1016/j.physa.2007.09.002
[24] Fischer, P., Titulaer, U.M.: Surf. Sci. 221, 409 (1989) · doi:10.1016/0039-6028(89)90589-X
[25] De Decker, Y., Tsekouras, G.A., Provata, A., Erneux, Th., Nicolis, G.: Phys. Rev. E 69, 036203 (2004)
[26] Keener, J.P.: SIAM J. Appl. Math. 47, 556 (1987) · Zbl 0649.34019 · doi:10.1137/0147038
[27] Fath, G.: Physica D 116, 176 (1998) · Zbl 0935.35070 · doi:10.1016/S0167-2789(97)00251-0
[28] Mikhailov, A.S.: Introduction to Synergetics. Springer, Berlin (1990) · Zbl 0712.92001
[29] Liu, D.-J.: J. Stat. Phys. 135, 77 (2009) · Zbl 1168.82317 · doi:10.1007/s10955-009-9708-2
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