Introductory material:
1. Stochastic processes. Review of some elementary
results and definitions.
2. Master equation. Birth and death processes. Steady
states. Carriers of intrinsic semiconductor.
3. Branching processes, generating function. Did Eve
ever meet Adam?
4. Van Kampen’s system size expansion for systems
approaching a steady state. Mean field and Fokker Planck equation.
5. Detailed balance. Einstein relations. Monte Carlo
methods. The settling/unsettling of
dust.
6. Examples: Ludwig’s spruce budworm model. Spreading of
an epidemic.
7. The Langevin approach. Ito vs. Stratanovich.
8. Systems with extrinsic noise.
9. Limitations of system size expansion. Sex-life of male lizards.
Metastable and driven
systems:
10. Absorbing
states. Diffusion near a cliff.
11. Kimura-Weiss
model of neutral genetic drift.
12. Kramer’s escape
rate.
13.
Molecular motors
Models of growth and
growing networks:
14. Pareto-Zipf
law. Income distribution and earthquake magnitudes
15. Growth of
cities.
16. Random
networks and small worlds.
17. Scale free
networks.
18. Spreading of
infections on networks.
Time series:
19. Stable
processes. Levy flights. The Noah effect.
20. Return
distribution for random walks.
21. The Joseph
effect and the Assuan dam.
22. Colored
noise. Fractal Brownian motion.
23. Power
spectra, fft, rescaled range method.
Moving interfaces:
24. Ballistic growth and diffusion limited aggregation.
25. Self-similar
and self-affine systems. Scaling.
26. Edwards
Wilkinson model.
27. Kardar-Parisi-Zhang model in 1 dimension.
28. KPZ in 2-d.
Conservative vs. non-conservative noise.
More spatial problems:
29. Percolation.
30. Self-avoiding random walk. Polymer models.
References:
R. Albert and A.-L. Barabási, Statistical mechanics of complex
networks, arXiv:cond-mat/0106096, Rev. Mod. Phys. 74 47 (2002).
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A.-L. Barabási and H.E. Stanley, (1995), Fractal concepts in surface
growth,
Cambridge University
Press.
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A.-L. Barabási, (2002), Linked: the
new science of networks, Perseus Publishing 2002.
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D. C. Dennett, (1995), Darwin's dangerous idea, Simon & Schuster.
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T.L. Duty, (1998), Branching processes.
Lecture notes.
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J. Feder, (1988), Fractals, Plenum.
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M. Ifti and B. Bergersen, (2002), Survival and Extinction in Cyclic
and Neutral Three-species Systems, arXiv:nlin.AO/0208023
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M. Kimura and G. H. Weiss, (1964), The stepping stone model of
population
struccture and the
decrease of genetic correlation with distance, Genetics, 49, 561-76.
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P.L. Krapivsky and S.Redner, Life and death in an expanding cage and
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G. Lakatos, (1998),A thermodynamic model of insect populations,
Physics 449 thesis UBC .
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R.H. Luchsinger, (1998), Molecular motors.
Lecture notes
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D.Ludwig, D.D. Jones and C.S. Holling, (1978), Qualitative analysis of
insect outbreak
systems: the spruce
budworm and forest, J. Animal
Ecology 47 315-332 (1978)
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D.Ludwig, D.G. Aronson and H.F. Weinberger, (1979), Spatial patterning of the
spruce budworm, J.Math Biology 8
217-58.
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B. B. Mandelbrot, (1997), Fractals and scaling in finance; discontinuity,
concentration, risk, Springer.
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S. C. Manrubia and D. H. Zanette, At the boundary between biological
and cultural evolution: The origin of surname distributions,
arXiv:cond-mat/0201559 30 Jan 2002
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M. Marsili and Y-C. Zhang (1998),
Interacting individuals leading to Zipf's law, cond-mat/9801289, Phys.
Rev. Lett March 98
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E.W. Montroll and B.J. West, (1976), On an enriched collection of
stochastic
processes, Chapter 2 in E.W. Montroll and J.L. Lebowitz eds.
, Fluctuation Phenomena, North Holland.
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J.D. Murray, (1993), Mathematical Biology, Springer Biomathematics
texts {\bf 19}
2nd ed.
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R. Pastor-Satorras and A. Vespignani, (2001), Epidemic spreading in scale-free networks, Phys. Rev. Letters,86, 3200-3.
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M. Plischke and B. Bergersen, (1994), Equilibrium Statistical
Physics, 2nd Ed World Scientific.
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Sidney Redner, (2001), A guide to first-passage processes, Cambridge
University Press.
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H. Risken, (1989), The Fokker-Planck equation, 2nd ed. Springer.
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Stauffer, D. and Aharony, A.
(1992). Introduction to Percolation Theory. 2nd ed. Taylor and Francis.
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N.G. van Kampen, (1981a), Stochastic processes in physics and chemistry,
North Holland (1981).
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N.G. van Kampen, (1981b) Ito
versus Stratanovich, J. of Statistical Physics 24, 175-187 (1981).
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N.G. van Kampen, The expansion of the master equation, Adv. Chem. Phys. 34,
245-309 (1976).
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G. H. Weiss and M. Kimura, (1965), A mathematical analysis of the
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