Abstract
We discuss the interbasin kinetics approximation for random walk on a complex (rugged) landscape of energy. In this approximation the random walk is described by the system of kinetic equations corresponding to transitions between the local minima of energy. If we approximate the transition rates between the local minima by the Arrhenius formula then the system of kinetic equations will be hierarchical. We discuss for a generic landscape of energy the anzats of interbasin kinetics which is equivalent to the ultrametric diffusion generated by an ultrametric pseudodifferential operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. H. Hoffmann and P. Sibani, “Diffusion in hierarchies,” Phys. Rev. A. 38, 4261–4270 (1988).
F. H. Stillinger, “Relaxation behavior in atomic and molecular glasses,” Phys. Rev. B. 41, 2409–2416 (1990).
H. Yoshino, “Hierarchical diffusion, aging and multifractality,” J. Phys. A. 30, 1143 (1997); arXiv:cond-mat/9604033.
A. T. Ogielski and D. L. Stein, “Dynamics on ultrametric spaces,” Phys. Rev. Lett. 55(15), 1634–1637 (1985).
L. Brekke and M. Olson, “p-Adic diffusion and relaxation in glasses,” Preprint UTTG-16-89, EFI-89-23.
L. Brekke and P. G. O. Freund, “p-Adic numbers in physics,” Phys. Rept. 233, 1–66 (1993).
H. Frauenfelder, S. G. Sligar and P. G. Wolynes, “The energy landscape and motions of proteins,” Science 254, 1598–1603 (1991).
V. I. Goldansky, Yu. F. Krupiansky, K. V. Shaitan and A. B. Rubin, “Investigation of proteins using the Mossbauer spectroscopy,” Biofizika 3, 761–774 (1987) [in Russian].
F. H. Stillinger and T. A. Weber, “Hidden structure in liquids,” Phys. Rev. A 25, 978–989 (1982).
F. H. Stillinger and T. A. Weber, “Packing structures and transitions in liquids and solids,” Science 225, 983–989 (1984).
O. M. Becker and M. Karplus, “The topology of multidimensional protein energy surfaces: theory and application to peptide structure and kinetics,” J. Chem. Phys. 106, 1495–1517 (1997).
V. A. Avetisov, A. H. Bikulov and S. V. Kozyrev, “Application of p-adic analysis to models of spontaneous breaking of replica symmetry,” J. Phys. A: Math. Gen. 32(50), 8785–8791 (1999); arXiv:cond-mat/9904360.
V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev and V. A. Osipov, “p-Adic models of ultrametric diffusion constrained by hierarchical energy landscapes,” J. Phys. A: Math. Gen. 35(2), 177–189 (2002); arXiv:cond-mat/0106506
V. A. Avetisov, A. H. Bikulov and V. A. Osipov, “p-Adic description of characteristic relaxation in complex systems,” J. Phys. A: Math. Gen. 36(15), 4239–4246 (2003).
V. A. Avetisov, A. Kh. Bikulov and A. P. Zubarev, “First passage time distribution and the number of returns for ultrametric random walks,” J. Phys. A: Math. Theor. 42(8), 18 (2009).
V. A. Avetisov and A. Kh. Bikulov, “Protein ultrametricity and spectral diffusion in deeply frozen proteins,” Biophys. Rev. Lett. 3(3), 387–396 (2008).
V. A. Avetisov, A. Kh. Bikulov, O. A. Vasilyev, S. K. Nechaev and A. V. Chertovich, “Some physical applications of random hierarchical matrices,” JETP 109(3), 485–504 (2009).
V. A. Avetisov, A. V. Chertovich, S. K. Nechaev and O. A. Vasilyev, “On scale-free and poly-scale behaviors of random hierarchical networks,” J. Stat. Mech. P07008, 15 (2009).
V. A. Avetisov, A. Kh. Bikulov and S. K. Nechaev, “Random hierarchical matrices: spectral properties and relation to polymers on disordered trees,” J. Phys. A: Math. Theor. 42(7), 9 (2009).
Y. V. Fyodorov and J.-Ph. Bouchaud, “Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite-dimensional Euclidean spaces,” J. Phys. A: Math. Theor. 41(32), 25 (2008).
A. Yu. Khrennikov and S. V. Kozyrev, “Localization in space for a free particle in ultrametric quantum mechanics,” Doklady Math. 74(3), 906 (2006).
S.V. Kozyrev, “Ultrametric dynamics as a model of interbasin kinetics,” Izvestia NAN Armenia: Mathematics 41(5), 28–38 (2006); arXiv:0711.1453.
M. Mezard,G. Parisi and M. Virasoro, Spin-Glass Theory and Beyond (World Scientific, Singapore 1987).
G. Parisi and N. Sourlas, “p-Adic numbers and replica symmetry breaking,” Eur. Phys. J. B. 14, 535–542 (2000); arXiv:cond-mat/9906095.
H. Frauenfelder, B. H. McMahon and P. W. Fenimore, “Myoglobin: the hydrogen atom of biology and paradigm of complexity,” PNAS 100(15), 8615–8617 (2003).
D. J. Wales, M. A. Miller and T. R. Walsh, “Archetypal energy landscapes,” Nature 394, 758–760 (1998).
P. G. Wolynes, J. Onuchic and D. Thirumalai, “Navigating the folding routes,” Science 267, 1619–1620 (1995).
L. A. Blumenfeld, Problems of Biological Physics (Springer Verlag, Berlin, 1981).
V. S. Vladimirov, I. V. Volovich and Ye. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore 1994).
A. Yu. Khrennikov and S. V. Kozyrev, “Pseudodifferential operators on ultrametric spaces and ultrametric wavelets,” Izvestiya: Math. 69(5), 989–1003 (2005); arXiv:math-ph/0412062.
A. Yu. Khrennikov and S. V. Kozyrev, “Wavelets on ultrametric spaces,” Appl. Comput. Harm. Anal. 19, 61–76 (2005).
S. V. Kozyrev, “Wavelets and spectral analysis of ultrametric pseudodifferenial operators,” Sbornik Math. 198(1), 103–126 (2007): arXiv:math-ph/0412082.
S. V. Kozyrev, Methods and Applications of Ultrametrix and p-Adic Analysis, Modern Problems of Mathematics 12 (Steklov Math. Institute, Moscow, 2008); http://www.mi.ras.ru/spm/pdf/012.pdf. [in Russian].
B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On p-adicmathematical physics,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 1–17 (2009); arXiv:0904.4205.
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the author in English.
Rights and permissions
About this article
Cite this article
Kozyrev, S.V. Dynamics on rugged landscapes of energy and ultrametric diffusion. P-Adic Num Ultrametr Anal Appl 2, 122–132 (2010). https://doi.org/10.1134/S2070046610020044
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046610020044