Abstract
We investigate an operator renormalization group method to extract and describe the relevant degrees of freedom in the evolution of partial differential equations. The proposed renormalization group approach is formulated as an analytical method providing the fundamental concepts of a numerical algorithm applicable to various dynamical systems. We examine dynamical scaling characteristics in the short-time and the long-time evolution regime providing only a reduced number of degrees of freedom to the evolution process.
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REFERENCES
L. P. Kadanoff, Physics 2:263 (1966).
K. G. Wilson, Phys. Rev. B 4:3174 (1971). Phys. Rev. B 4:3184 (1971). Phys. Rev. Lett. 28:548 (1972).
K. G. Wilson. Rev. Mod. Phys. 47:773 (1975).
S. K. Ma, Modern Theory of Critical Phenomena (Benjamin/Cummings Publishing Company, Reading, 1976).
P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49:435 (1977).
D. Forster, D. Nelson, and M. Stephen, Phys. Rev. A 16:732 (1977).
M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. 56:889 (1986).
E. Medina, T. Hwa, M. Kardar, and Y. Zhang, Phys. Rev. A 39:3053 (1989).
E. Frey and U. C. Taeuber, Phys. Rev. E 50:1024 (1994).
T. Nattermann and L. H. Tang, Phys. Rev. A 45:7156 (1992).
J. P. Bouchaud and M. E. Cates, Phys. Rev. E 47:R1455 (1993).
M. Schwartz and S. F. Edwards, Europhys. Lett. 20:301 (1992).
T. Halpin-Healy, Phys. Rev. Lett. 62:442 (1990). Phys. Rev. A 42:711 (1990).
D. E. Wolf and J. Kertész, Europhys. Lett. 4:651 (1987).
B. M. Forrest and L. Tang, J. Statist. Phys. 60:181 (1990).
M. Beccaria and G. Curci, Phys. Rev. E, 4560 (1994).
S. R. White, Phys. Rev. B 48:10345 (1993).
J. R. de Sousa, Phys. Lett. A 216:321 (1996).
M. A. Martin-Delgado, J. Rodríguez-Laguna, and G. Sierra, Nucl. Phys. B 473:685 (1996).
A. Degenhard, J. Phys. A: Math. Gen. 33:6173 (2000).
J. Gonzales, M. A. Martin-Delgado, G. Sierra, and A. H. Vozmediano, New and old real-space renormalization group methods, in Quantum Electron Liquids and High-T c Superconductivity, Lecture Notes in Physics, Vol. 38 (Springer, Berlin, 1995).
N. Goldenfeld, A. McKane, and Q. Hou, J. Statist. Phys. 93:699 (1998).
Q. Hou, N. Goldenfeld, and A. McKane. Phys. Rev. E 63:36125 (2001).
A. Degenhard and J. Rodríguez-Laguna, Phys. Rev. E, cond-mat/0106155 (2001) (in press).
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipies. The Art of Scientific Computing (Cambridge, University Press, 1992).
P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (New York, Academic Press, 1981).
J. M. Burgers, The Nonlinear Diffusion Equation (Riedel, Boston, 1974).
T. Halpin-Healy and Y.-C. Zhang. Phys. Rep. 254:215 (1995).
J. Krug, Phys. Rev. Lett. 72(18):2907 (1994).
J. Krug and H. Spohn. Phys. Rev. A 38(8):4271 (1988).
S. E. Esipov and T. J. Newman, Phys. Rev. E 48(6):1046 (1993).
J. G. Amar and F. Family, Phys. Rev. E 47(3):1595 (1993).
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984).
C. Itzykson and J. M. Drouffe, Statistical Field Theory (University Press, Cambridge, 1991).
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Degenhard, A., Rodríguez-Laguna, J. Towards the Evaluation of the Relevant Degrees of Freedom in Nonlinear Partial Differential Equations. Journal of Statistical Physics 106, 1093–1120 (2002). https://doi.org/10.1023/A:1014041904951
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DOI: https://doi.org/10.1023/A:1014041904951