Abstract
We study the existence and stability of stationary solutions of an integrodifferential model for phase transitions, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity. As such, this model is a nonlocal extension of the Allen–Cahn equation, which incorporates long-range interactions. We find that the set of stationary solutions for this model is much larger than that of the Allen–Cahn equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Mettal. 27:1084–1095 (1979).
P. W. Bates and A. Chmaj, On a discrete convolution model for phase transitions, Arch. Rat. Mech. Anal., to appear.
P. W. Bates, P. C. Fife, X. Ren, and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rat. Mech. Anal. 138:105–136 (1997).
P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported 2:1–38 (1988).
P. W. Bates and X. Ren, Heteroclinic orbits for a higher order phase transition problem, Euro. J. Applied Math. 8:149–163 (1997).
D. Brandon and R. C. Rogers, The coercivity paradox and nonlocal ferromagnetism, Continuum Mechanics and Thermodynamics 4:1–21 (1992).
D. Brandon and R. C. Rogers, Nonlocal regularization of L.C. Young's tacking problem, Applied Mathematics and Optimization 25:287–301 (1992).
D. Brandon and R. C. Rogers, Nonlocal superconductivity, ZAMP 45:135–152 (1994).
D. Brandon, T. Lin, and R. C. Rogers, Phase transitions and hysteresis in nonlocal and order-parameter models, Meccanica 30:541–565 (1995).
W. L. Bragg and E. J. Williams, Effect of thermal agitation on atomic arrangement in alloys, Proc. Roy. Soc. London Ser. A 145:699–730 (1934).
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28:258–267 (1958).
X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Diff. Eqs. 2:125–160 (1997).
A. Chmaj and X. Ren, Homoclinic solutions of an integral equation: existence and stability, Trans. Amer. Math. Soc., to appear.
A. de Masi, T. Gobron, and E. Presutti, Traveling fronts in non local evolution equations, Arch. Rat. Mech. Anal. 132:143–205 (1995).
A. de Masi, E. Orlandi, E. Presutti, and L. Triolo, Motion by curvature by scaling non-local evolution equations, J. Stat. Phys. 73:543–570 (1993).
A. de Masi, E. Orlandi, E. Presutti, and L. Triolo, Stability of the interface in a model of phase separation, Proc. Roy. Soc. Edin. A 124:1013–1022 (1994).
A. de Masi, E. Orlandi, E. Presutti, and L. Triolo, Uniqueness of the instanton profile and global stability in nonlocal evolution equations, Rend. Math. 14:693–723 (1994).
G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of traveling waves for a neural network, Proc. Roy. Soc. Edin. A 123:461–478 (1993).
R. L. Fosdick and D. E. Mason, Nonlocal continuum mechanics, Part I: Existence and regularity, SIAM J. Appl. Math. 58:1278–1306 (1998).
R. L. Fosdick and D. E. Mason, Nonlocal continuum mechanics, Part II: Structure, asymtotics and computations, J. Elasticity 1–50 (1997).
P. C. Fife and X. Wang, A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions, Adv. Diff. Eqs. 3:85–110 (1998).
G. Giacomin and J. L. Lebowitz, Exact macroscopic description of phase segregation in model alloys with long range interactions, Phys. Rev. Lett. 76(7):1094–1097 (1996).
G. Giacomin and J. L. Lebowitz, Phase segregation dynamics in particle systems with long range interactions I: Macroscopic limits, preprint.
M. Katsoulakis and P. E. Souganidis, Interacting particle systems and generalized mean curvature evolution, Arch. Rat. Mech. Anal. 127 (1994).
M. Katsoulakis and P. E. Souganidis, Generalized motion by mean curvature as a macroscopic limit of stochastic Ising models with long range interactions and Glauber dynamics, Comm. Math. Phys. 169:61–97 (1995).
M. Katsoulakis and P. E. Souganidis, Generalized front propagation and macroscopic limits of stochastic Ising models with long range anisotropic spin-flip dynamics, preprint.
O. Penrose, A mean-field equation of motion for the dynamic Ising model, J. Stat. Phys. 63:975–986 (1991).
R. C. Rogers. Nonlocal variational problems in nonlinear electromagneto-elastostatics, SIAM J. Math. Anal. 19:1329–1347 (1988).
R. C. Rogers, A nonlocal model for the exchange energy in ferromagnetic materials, J. Integral Equations Appl. 3:85–127 (1991).
R. C. Rogers, Some remarks on nonlocal interactions and hysteresis in phase transitions, Continuum Mechanics and Thermodynamics 8:65–73 (1996).
J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1983.
P. E. Souganidis, Front propagation: Theory and applications, C.I.M.E. Lectures (1995).
C. J. Thompson, Classical Equilibrium Statistical Mechanics (Oxford University Press, 1988).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bates, P.W., Chmaj, A. An Integrodifferential Model for Phase Transitions: Stationary Solutions in Higher Space Dimensions. Journal of Statistical Physics 95, 1119–1139 (1999). https://doi.org/10.1023/A:1004514803625
Issue Date:
DOI: https://doi.org/10.1023/A:1004514803625