Abstract
A diffusion equation including source terms, representing randomly distributed sources and sinks is considered. For quasilinear growth rates the eigenvalue problem is equivalent to that of the quantum mechanical motion of electrons in random fields. Correspondingly there exist localized and extended density distributions dependent on the statistics of the random field and on the dimension of the space. Besides applications in physics (nonequilibrium processes in pumped disordered solid materials) a new evolution model is discussed which considers evolution as hill climbing in a random landscape.
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References
G. Nicolis and I. Prigogine,Self-organization in Nonequilibrium Systems (Wiley, New York, 1977); Yu. M. Romanovsky, N. V. Stepanova, and D. S. Chernavsky,Mathematical Models in Biophysics (in Russian) (Nauka, Moscow, 1975); H. Haken,Synergetics. An Introduction (Springer, New York, 1978).
W. Ebeling and R. Feistel,Physik der Selbstorganisation und Evolution (Akademie-Verlag, Berlin, 1982).
I. M. Lifshitz, S. A. Gredeskul, and L. A. Pastur,Introduction into the Theory of Disordered Systems (in Russian) (Nauka, Moscow, 1982).
L. A. Pasteur,Commun. Math. Phys. 75:179 (1980);Usp. Math. Nauk 28:3 (1973).
V. L. Bonch-Bruevich, R. Enderlein, B. Esser, R. Keiper, A. G. Mironov, and I. P. Zvyagin,Elektronentheorie ungeordneter Halbleiter (VEB Deutscher Verlag der Wissenschaften, Berlin, 1984).
P. W. Anderson,Phys. Rev. 109 (5):1492 (1958).
R. Feistel and W. Ebeling,BioSystems 15:291 (1982).
R. Feistel and W. Ebeling, inThermodynamics and Regulation in Biological Processes, A. I. Zotin, ed. (Nauka, Moscow, 1984).
R. A. Fisher,The Genetical Theory of Natural Selection (Clarendon Press, Oxford, 1930).
M. Eigen,Naturwiss. 58:465 (1971).
P. W. Anderson,Proc. Natl. Acad. Sci. 80:3386 (1983).
R. Landauer,Helv. Phys. Acta 56:847 (1983).
Ya. B. Zel'dovich,Dokl. Akad. Nauk USSR 270:1369 (1983).
E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan,Phys. Rev. Lett. 42:673 (1979).
D. J. Thouless,J. Phys. C 9:L603 (1976); J. L. Cardy,J. Phys. C 11:L321 (1978).
A. Engel, inProceedings of the 5th UNESCO-Meeting on System Theory (Akademie-Verlag, Berlin, 1984).
I. M. Lifshitz,Usp. Fiz. Nauk 83:617 (1964).
M. Conrad,Lect. Notes Biomath. 21:147 (1978);Adaptability (Plenum Press, New York, 1983).
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We dedicate this work to the memory of Ilya M. Lifshitz.
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Ebeling, W., Engel, A., Esser, B. et al. Diffusion and reaction in random media and models of evolution processes. J Stat Phys 37, 369–384 (1984). https://doi.org/10.1007/BF01011839
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DOI: https://doi.org/10.1007/BF01011839