Abstract
We consider a stochastic PDE with locahzed coherent structures known as kinks. The availabihty of exact results for the steady state and increasing computer power means that precise quantitative comparison between theory and numerics is possible. One of the quantities of interest is the steady-state density of kinks, maintained by a balance between nucleation and annihilation of kink-antikink pairs. The density as measured from numerical solutions is sufficiently accurate to resolve the difference between analytical predictions based on the exact value of the correlation length, and those based on the WKB approximation to it.
http//stochastic.org.uk
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. J. Alexander and S. Habib, “Statistical Mechanics of Kinks in (l-i-l)-Dimensions,” Phys. Rev. Lett., vol. 71, p. 955,1993; F. J. Alexander, S, Habib, and A. Kovner, “Statistical Mechanics of Kinks in (l+l)-Dimensions: Numerical Simulations and Double Gaussian Approximation,” Phys. Rev. E, vol. 48, p. 4284, 1993.
S. Aubry, “A unified approach to the interpretation of displacive and order-disorder systems,” J Chem. Phys., vol. 64, p. 3392, 1976.
L. Bettencourt, S. Habib, and G. Lythe, “Controlling one-dimensional Langevin dynamics on the lattice,” Physical Review D, vol. 60, p. 105039, 1999.
A. R. Bishop, J. A. Krumhansl, and J. R. Schrieffer, “Solitons in condensed matter: a paradigm,” Physica D, vol. 1, pp. 1–4, 1980.
A. R. Bishop and T. Schneider (Eds), Solitons and Condensed Matter Physics, Springer, Berlin, 1978.
M. Büttiker and T. Christen, “Nucleation of Weakly Driven Kinks,” Phys. Rev Lett. vol. 75, p. 1895,1995; “Diffusion controlled initial recombination,” Phys. Rev. E, vol. 58, p. 1533, 1998.
M. Büttiker and R. Landauer, “Nucleation Theory of Overdamped Soliton Motion,” Phys. Rev. Lett., vol. 43, p. 1453, 1979; “Long-term behavior of the equilibrium Sine-Gordon chain,” J. Phys. C, vol. 13, p. L325, 1980.
S. Habib, K. Lindenberg, G. Lythe, and C. Mohna-Paris, “Diffusion-hmited reaction in one dimension: paired and unpaired nucleation,” J. Chem. Phys., vol. 115, pp. 73–89, 2001.
S. Habib and G. Lythe, “Dynamics of kinks: nucleation, diffusion and annihilation,” Physical Review Letters, vol. 84, p. 1070, 2000.
K. Jansons and G. Lythe, “Stochastic Calculus: apphcation to dynamic bifurcations and threshold crossings,” Journal of Statistical Physics, vol. 90, pp. 227–251, 1998.
D. J. Kaup, “Thermal corrections to overdamped soliton motion,” Phys. Rev. B, vol. 27, pp. 6787–6795, 1983.
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.
G. Lythe, “Defect formation in a dynamic transition,” Int. J. Theor Phys., vol. 40, p. 2309, 2001.
G. Lythe and S. Habib, “Stochastic PDEs: convergence to the continuum?,” Computer Physics Communications, vol. 142, p. 29, 2001.
D. W. McLaughhn and A. C. Scott, “Perturbation analysis of fluxon dynamics,” Phys. Rev. A, vol. 18, p. 1652, 1978.
M. J. Rice, A. R. Bishop, J. A. Krumhansl, and S. E. Trullinger, “Weakly Pinned Charge-Density-Wave Condensates,” Phys. Rev Lett., vol. 36, p. 432, 1976.
D. J. Scalapino, M. Sears, and R. A. Ferrell, “Statistical Mechanics of One-Dimensional Ginzburg-Landau Fields,” Phys. Rev B, vol. 6, p. 3409, 1972; J. A. Krumhansl and J. R. Schrieffer, “Dynamics and statistical mechanics of a one-dimensional model Hamiltonian for structural phase transitions,” Phys. Rev B ibid, vol. 11, p. 3535, 1975; J. F. Currie, J. A. Krumhansl, A. R. Bishop, and S. E. Trullinger, “Statistical mechanics of one-dimensional sohtary-wave-bearing scalar fields: Exact results and ideal-gas phenomenology,” Phys. Rev B ibid, vol. 22, p. 477, 1980.
W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in Polyacetylene,” Phys. Rev. Lett., vol. 42, p. 1698, 1979.
S. E. Trullinger and K. Sasaki, “Lattice-discreteness corrections in the transfer-operator method for kink-bearing chains, Physica D, vol. 28, p. 181, 1987.
J. B. Walsh, “An introduction to stochastic partial differential equations,” pp. 266–439, in Ecole d’été de probabilités de St-FlourXIV, ed. P. L. Hennequin, Springer, Berlin, 1986.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Lythe, G., Habib, S. (2003). Kinks in a Stochastic PDE. In: Namachchivaya, N.S., Lin, Y.K. (eds) IUTAM Symposium on Nonlinear Stochastic Dynamics. Solid Mechanics and Its Applications, vol 110. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0179-3_38
Download citation
DOI: https://doi.org/10.1007/978-94-010-0179-3_38
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-3985-7
Online ISBN: 978-94-010-0179-3
eBook Packages: Springer Book Archive