Abstract
We review a simple model of random walks on a rugged landscape, representing energy or (negative) fitness, and with many peaks and valleys. This paradigm for trapping on long timescales by metastable states in complex systems may be visualized as a terrain with lakes in the valleys whose water level depends on the observational timescale. The “broken ergodicity” structure in space and time of trapping in the valleys can be analyzed using invasion percolation, picturesquely in terms of “ponds and outlets” as water flows to a distant sea. Two main conclusions concern qualitative dependence on the spatial dimension \(d\) of the system landscape. The first is that the much-used example of a one-dimensional rugged landscape is entirely misleading for any larger \(d\). The second is that once \(d\) is realistically large (above \(6\) seems to suffice), there are many nonmerging paths to the sea; this may be relevant for the issue of contingency vs. convergence in macro-evolution.
This paper is based on a talk at the Interdisciplinary Symposium on Complex Systems, Kos, Greece, September 19–25, 2012.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
J. Jäckle, On the glass transition and the residual entropy of glasses. Phil. Mag. B 44, 533–545 (1981)
R.G. Palmer, Broken Ergodicity. Adv. Phys. 31, 669–735 (1982)
R.G. Palmer, Broken ergodicity in spin glasses. in Heidelberg Colloquium on Spin Glasses, (Springer, Berlin, 1983), pp. 234–251
A.C.D. van Enter, J.L. van Hemmen, Statistical-mechanical formalism for spin-glasses. Phys. Rev. A 29, 355–365 (1984)
R.G. Palmer, D.L. Stein, Broken Ergodicity in Glass. in Relaxations in Complex Systems, (U.S. GPO, Washington, 1985), pp. 253–259
S.A. Kauffman, Origins of Order (Oxford University Press, Oxford, 1993)
K. Binder, A.P. Young, Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58, 801–976 (1986)
P.F. Stadler, W. Schnabl, The landscape of the traveling salesman problem. Phys. Lett. A 161, 337–344 (1992)
D.L. Stein, C.M. Newman, Broken ergodicity and the geometry of rugged landscapes. Phys. Rev. E 51, 5228–5238 (1995)
D.L. Stein, C.M. Newman, Spin Glasses and Complexity (Princeton University Press, Princeton, 2013)
U. Krey, Amorphous magnetism: theoretical aspects. J. Magn. Magn. Mater. 6, 27–37 (1977)
This happened to us
C.M. Newman, D.L. Stein, Random walk in a strongly inhomogeneous environment and invasion percolation. Ann. Inst. Henri Poincaré 31, 249–261 (1995)
R. Lenormand, S. Bories, Description d’un mécanisme de connexion de liaison destiné à l’étude du drainage avec piègeage en milieu poreux. C.R. Acad. Sci. 291, 279–282 (1980)
R. Chandler, J. Koplick, K. Lerman, J.F. Willemsen, Capillary displacement and percolation in porous media. J. Fluid Mech. 119, 249–267 (1982)
D. Wilkinson, J.F. Willemsen, Invasion percolation: a new form of percolation theory. J. Phys. A 16, 3365–3376 (1983)
P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987)
D. Stauffer, A. Aharony, Introduction to Percolation Theory, 2nd edn. (Taylor and Francis, London, 1992)
C.M. Newman, D.L. Stein, Spin glass model with dimension-dependent ground state multiplicity. Phys. Rev. Lett. 72, 2286–2289 (1994)
C.M. Newman, D.L. Stein, Ground state structure in a highly disordered spin glass model. J. Stat. Phys. 82, 1113–1132 (1996)
J.S. Jackson, N. Read, Theory of minimum spanning trees. I. Mean-field theory and strongly disordered spin-glass model. Phys. Rev. E 81, 021130 (2010)
J.S. Jackson, N. Read, Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold. Phys. Rev. E 81, 021131 (2010)
G. Papanicolaou, S.R.S. Varadhan, Diffusions with random coefficients. in Statistics and Probability: Essays in Honor of C.R. Rao (North-Holland, Amsterdam, 1982), pp. 547–552
C. Kipnis, S.R.S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104, 1–19 (1986)
A. De Masi, P.A. Ferrari, S. Goldstein, D.W. Wick, An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55, 787–855 (1989)
In ordinary BE there is assumed to exist some highest barrier, \(\Delta F_{\max }\) that confines the system; after escape occurs over \(\Delta F_{\max }\), dynamical processes are ergodic. Before this occurs (that is, when the system is still in the broken ergodic regime) barriers confining the system are assumed to grow logarithmically with time, which follows from the Arrhenius relation between escape time and barrier height. In contrast, there is no clear analogue in our models to \(\Delta F_{\max }\); after initial transients, barriers are roughly constant, with expected small fluctuations owing to deviations in the diffusion process from the strict \(T\rightarrow 0\) limit.
G. Grimmett, Percolation (Springer, New York, 1989)
J.M. Hammersley, A Monte Carlo solution of percolation in a cubic lattice. in Methods in Computational Physics, vol. I (Academic Press, New York 1963), pp. 281–298
A precise formulation which is valid for any \(d\ge 2\), irrespective of whether there is one or many disjoint invasion regions, and which does not conflict with recurrence, is as follows: for any fixed \(N\) and with probability approaching one as \(\beta \) tends to \(\infty \), the RWRE will visit all sites in the second through \(N\)th ponds before it either returns to the first pond or crosses a bond of larger value than the first outlet.
M. Damron, A. Sapozhnikov, Outlets of \(2D\) invasion percolation and multiple-armed incipient infinite clusters. Prob. Theory Rel. Fields 150, 257–294 (2011)
M. Damron, A. Sapozhnikov, Relations between invasion percolation and critical percolation in two dimensions. Ann. Prob. 37, 2297–2331 (2009); Limit theorems for 2D invasion percolation. arXiv:1005.5696v3 (2012)
H. Kesten, The incipient infinite cluster in two-dimensional percolation. Prob. Theory Rel. Fields 73, 369–394 (1986)
R.V. Chamberlin, M. Hardiman, L.A. Turkevich, R. Orbach, \(H-T\) phase diagram for spin-glasses: an experimental study of Ag:Mn. Phys. Rev. B 25, 6720–6729 (1982)
J.-L. Tholence, R. Tournier, Susceptibility and remanent magnetization of a spin glass. J. Phys. (Paris) 35, C4–229-C4-236 (1974)
C.N. Guy, Gold-iron spin glasses in low DC fields. I. Susceptibility and thermoremanence. J. Phys. F 7, 1505–1519 (1977)
R.W. Knitter, J.S. Kouvel, Field-induced magnetic transition in a Cu-Mn spin-glass alloy. J. Magn. Magn. Mat. 21, L316–L319 (1980)
P. Refrigier, E. Vincent, J. Hamman, M. Ocio, Ageing phenomena in a spin glass: effect of temperature changes below \({T}_g\). J. Phys. (Paris) 48, 1533–1539 (1987)
S.J. Gould, C.M. Newman, D.L. Stein (unpublished)
Acknowledgments
This work was supported in part by NSF Grants DMS-0604869, DMS-1106316, and DMS-1207678.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Stein, D.L., Newman, C.M. (2014). Rugged Landscapes and Timescale Distributions in Complex Systems. In: Zelinka, I., Sanayei, A., Zenil, H., Rössler, O. (eds) How Nature Works. Emergence, Complexity and Computation, vol 5. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00254-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-00254-5_4
Published:
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00253-8
Online ISBN: 978-3-319-00254-5
eBook Packages: EngineeringEngineering (R0)