Abstract
In this report, the kink-antikink interaction, in one spatial dimension, is revised in the framework of a paradigmatic model for bistability (real Ginzburg-Landau equation). In particular, it is pointed out that, when it is taking into account the fluctuations, drastically changes the main features of the interaction. To wit, since the long distance interaction is exponentially weak, the kink-antikink movement is ruled by the fluctuations. A simple random walk model, that incorporates the pair self-annihilation, is proposed. We discussed the implications that, consider the fluctuations, has in the coarsening dynamics. That is, the coarsening law, for the growing of the domains in each stable state, changes from being logarithmic to becoming in the power law \(\sqrt{t}\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Hubert, R. Scafer, Magnetic domains (Springer, Berlin, 1998)
A.J. Bray, Domain-growth scaling in systems with long-range interactions. Phys. Rev. E 47, 3191–3195 (1993)
C.L. Emmott, A.J. Bray, Coarsening dynamics of a one-dimensional driven Cahn-Hilliard system. Phys. Rev. E 54, 4568–4575 (1996)
A.J. Bray, Defect relaxation and coarsening exponents. Phys. Rev. E 58, 1508–1513 (1998)
S.J. Watson, F. Otto, B.Y. Rubinstein, S.H. Davis, Coarsening dynamics of the convective Cahn-Hilliard equation. Physica D 178, 127–148 (2003)
H. Calisto, M. Clerc, R. Rojas, E. Tirapegui, Bubbles Interactions in the Cahn-Hilliard Equation. Phys. Rev. Lett. 85, 3805–3808 (2000)
A.A. Golovin, A.A. Nepomnyashchy, S.H. Davis, M.A. Zaks, Convective Cahn-Hilliard models: from coarsening to roughening. Phys. Rev. Lett. 86, 1550–1553 (2001)
M. Argentina, M.G. Clerc, R. Soto, van der Waals-like transition in fluidized granular matter. Phys. Rev. Lett. 89, 044301 (2002)
J. Swift, P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319 (1977)
M. Bestehorn, H. Haken, Transient patterns of the convection instability: a model-calculation. Z. Phys. B 57, 329 (1984)
M. Tlidi, P. Mandel, R. Lefever, Kinetics of localized pattern formation in optical systems. Phys. Rev. Lett. 81, 979–982 (1998)
K. Kawasaki, Defect-phase dynamics for dissipative media with potential. Progr. Theory Phys. 80, 123–138 (1984)
T. Funaki, The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields. 102, 221–288 (1995)
D. Contreras. M.G. Clerc, Internal noise and system size effects induce nondiffusive kink dynamics. Phys. Rev. E 91, 032922 (2015)
N.G. van Kampen, Stochastic processes in physics and chemistry (Elsevier, North-Holland, 1981)
Acknowledgments
I would like to thank Prof. M.G. Clerc for fruitful discussions, and FONDECYT (Project N 1140128) for financial support.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Escaff, D. (2016). Random Walk Model for Kink-Antikink Annihilation in a Fluctuating Environment. In: Tlidi, M., Clerc, M. (eds) Nonlinear Dynamics: Materials, Theory and Experiments. Springer Proceedings in Physics, vol 173. Springer, Cham. https://doi.org/10.1007/978-3-319-24871-4_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-24871-4_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24869-1
Online ISBN: 978-3-319-24871-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)