—Renormalization and Asymptotic Analysis—
  • Yoshitsugu Oono
Part of the Springer Series in Synergetics book series (SSSYN)


Despite scale interference that characterizes our nonlinear world, this world does not look so lawless as chaos suggests. There are extensive effects of noises (interference with the unknowable scales) to the phenomena we experience on our scale. However, these effects show up rather systematically at restricted places. The phenomenological way to appreciate the world is to exploit this special feature of the world. The existence of mathematical structures we can phenomenologically recognize in the world is a prerequisite of the existence of intelligent beings. The chapter begins with characterization of phenomenology. The renormalization group approach is then introduced as a means to extract phenomenological descriptions of various phenomena. An elementary introduction to renormalization group theory is followed by its application to system reduction and singular perturbation with some technical details. Technical aspects of the renormalization group theory applied to critical phenomena and polymers may be found in the accompanying webpages to the book.


Renormalization Group Singular Perturbation Invariant Manifold Renormalization Group Equation Renormalization Constant 
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  1. Arnold VI (1997) Mathematical methods of classical mechanics. SpringerGoogle Scholar
  2. Barenblatt GI (1996) Similarity, self-similarity, and intermediate asymptotics. Cambridge University PressGoogle Scholar
  3. Callen HB (1960) Thermodynamics. Interscience Publ.Google Scholar
  4. Cannone M, Friedlander S (2003) Navier: blow-up and collapse. Notices Amer Math Soc 50:7-13MathSciNetMATHGoogle Scholar
  5. Chaikin M, Lubensky TC (2000) Principles of condensed matter physics. Cambridge University PressGoogle Scholar
  6. Chen L-Y, Goldenfeld N, Oono Y, Paquette G (1993) Selection, stability and renormalization. Physica A 204:111-133ADSCrossRefGoogle Scholar
  7. Chiba H (2008) C 1-approximation of vector fields based on the renormalization group method. SIAM J Applied Dynam Syst 7:895-932MathSciNetADSMATHCrossRefGoogle Scholar
  8. Chiba H (2009) Extension and unification of singular perturbation methods. SIAM J Applied Dynam Sys 8:1066-1115MathSciNetADSMATHCrossRefGoogle Scholar
  9. Cross MC, Hohenberg PC (1993) Pattern formation outside of equilibrium. Rev Mod Phys 65:851-1112ADSCrossRefGoogle Scholar
  10. des Cloizeaux J (1975) The Lagrangian theory of polymer solutions at intermediate concentrations. J Phys (France) 36:281-291Google Scholar
  11. Elder KR, Grant M (2004) Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys Rev E 70:051605 (1-18)Google Scholar
  12. Elder KR, Katakowski M, Haataja M, Grant M (2002) Modeling elasticity in crystal growth. Phys Rev Lett 88:245701 (1-4)Google Scholar
  13. Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Indiana Univ Math J 21:193-226MathSciNetMATHCrossRefGoogle Scholar
  14. Fisher ME (1988) Condensed matter physics: does quantum mechanics matter? In: Feshbach H, Matsui T, Oleson A (ed) Niels Bohr: physics and the world (Proceedings of the Niels Bohr Centennial Symposium). Harwood Academic PublishersGoogle Scholar
  15. Furukawa Y (1998) Inventing polymer science—Staudinger, Carothers, and the emergence of macromolecular chemistry—. University of Pennsylvania PressGoogle Scholar
  16. Goldenfeld N (1992) Lectures on phase transitions and renormalization group. Addison WesleyGoogle Scholar
  17. Goldenfeld ND, Martin O, Oono Y (1989) Intermediate asymptotics and renormalization group theory. J Scientific Comp 4:355-372MathSciNetCrossRefGoogle Scholar
  18. Gunaratne H, Ouyang Q, Swinney HL (1994) Pattern formation in the presence of symmetries. Phys Rev E 50:2802-2820MathSciNetADSMATHCrossRefGoogle Scholar
  19. Haataja M, Gränäsy L, LÖwen H (2010) Classical density functional theory methods in soft and hard matter. J Phys: Cond Mat 22:360301 (1-8)Google Scholar
  20. Hall AR, Colegrave N (2008) Decay of unused characters by selection and drift. J Evol Biol 21:610-617CrossRefGoogle Scholar
  21. Heisenberg W (1971) Physics and beyond (translated by A J Pomerans) Harper & RowGoogle Scholar
  22. Herbut I (2007) A modern approach to critical phenomena. Cambridge University PressGoogle Scholar
  23. Hirsch M, Pugh C, Shub M (1970) Invariant manifolds. Bull Amer Math Soc 76:1015-1019MathSciNetMATHCrossRefGoogle Scholar
  24. Husserl E (1999) The idea of phenomenology (translated by L Hardy). Kluwer Academic PublishersGoogle Scholar
  25. Izutsu T (1991) Consciousness and essence—quest of the spiritual Orient. Iwanami paper backGoogle Scholar
  26. Keller CF (1999) Climate, modeling, and predictability. Physica D 133:296-308ADSMATHCrossRefGoogle Scholar
  27. Kihara T (1978) Intermolecular forces. WileyGoogle Scholar
  28. Kubo R (1968) Thermodynamics. An advanced course with problems and solutions. North-Holland Pub Co.Google Scholar
  29. Ladyzhenskaya OA (1963) Mathematical theory of incompressible fluids. Gordon & BreachGoogle Scholar
  30. Ladyzhenskaya OA (2003) Sixth problem of the millennium: Navier-Stokes equations, existence and smoothness. Russ Math Surveys 58:251-286MathSciNetADSMATHCrossRefGoogle Scholar
  31. Le Bellac M (1991) Quantum and statistical field theory. Oxford University PressGoogle Scholar
  32. Lieb E, Yngvason J (1998) A guide to entropy and the second law of thermodynamics. Notices Amer Math Soc 45:571-581MathSciNetMATHGoogle Scholar
  33. Lieb E, Yngvason J (1999) The physics and mathematics of the second law of thermodynamics. Phys Rep 340:1-96MathSciNetCrossRefGoogle Scholar
  34. Mandelbrot BB (1983) Fractal geometry of nature. W H FreemanGoogle Scholar
  35. Mandle F (1988) Statistical physics (2nd edition). WileyGoogle Scholar
  36. Mañé R (1978) Persistent manifolds are normally hyperbolic. Trans Amer Math Soc 246:271-283Google Scholar
  37. Migdal AB (2000) Qualitative methods in quantum theory (translated by Leggett AJ). Westview PressGoogle Scholar
  38. Miklósi A, Kubinyi E, Topál J, Gáacsi M, Virányi, Z, Csányi V (2003) A simple reason for a big difference: wolves do not look back at humans, but dogs do. Curr Biol 13:763-766Google Scholar
  39. Miyazaki K, Kitahara K, Bedeaux D (1996) Nonequilibrium thermodynamics of multicomponent systems. Physica A 230:600-630ADSCrossRefGoogle Scholar
  40. Niwa N, Hiromi Y, Okabe M (2004) A conserved developmental program for sensory organ formation in Drosophila melanogaster. Nat Genet 36:293-297CrossRefGoogle Scholar
  41. Nozaki K, Oono Y (2001) Renormalization-group theoretical reduction. Phys Rev E 63:046101 (1-18)Google Scholar
  42. Ohta T, Nakanishi A (1983) Theory of semi-dilute polymer solutions: I Static properties in a good solvent. J Phys A 16:4155-4170ADSCrossRefGoogle Scholar
  43. Ohta T, Oono Y (1982) Conformational space renormalization theory of semidilute polymer solutions. Phys Lett 89A:460-464ADSGoogle Scholar
  44. Oono Y (1985) Statistical physics of polymer solutions. Conformational-space renormalization group approach. Adv Chem Phys 61:301-437Google Scholar
  45. Oono Y (1985) Dynamics in polymer solutions — a renormalization-group approach. AIP Conference Proceedings No 137 (edited by Y Rabin) p187-218Google Scholar
  46. Oono Y (1989) Large deviation and statistical physics. Prog Theor Phys Suppl 99:165-205MathSciNetADSCrossRefGoogle Scholar
  47. Oono Y, Ohta T, In Goldbart PM, Goldenfeld N, Sherrington D (ed) Stealing the gold: a celebration of the pioneering physics of Sam Edwards. Oxford University PressGoogle Scholar
  48. Oono Y, Paniconi M (1998) Steady state thermodynamics. Prog Theor Phys Suppl 130:29-44MathSciNetADSCrossRefGoogle Scholar
  49. Pashko O, Oono Y (2000) The Boltzmann equation is a renormalization group equation. Int J Mod Phys B 14:555MathSciNetADSMATHGoogle Scholar
  50. Poggio T, Rifkin R, Mukherjee S, Niyoki P (2004) General conditions for predictivity in learning theory. Nature 428:419-422ADSCrossRefGoogle Scholar
  51. Protas M, Conrad M, Gross JB, Tabin C, Borowsky R (2007) Regressive evolution in the Mexican cave tetra, Astyanax mexicanus. Curr Biol 17:452-454CrossRefGoogle Scholar
  52. Rajaram S, Taguchi Y-h, Oono Y (2005) Some implications of renormalization group theoretical ideas to statistics. Physica D 205:207-214MathSciNetADSMATHCrossRefGoogle Scholar
  53. Ruelle D (1969) Statistical mechanics, rigorous results. BenjaminGoogle Scholar
  54. Sasa S, Tasaki H (2006) Steady state thermodynamics. J Stat Phys 125:125-224MathSciNetADSMATHCrossRefGoogle Scholar
  55. Shiwa Y (2000) Renormalization-group theoretical reduction of the Swift- Hohenberg model. Phys Rev E 63:016119 (1-7)Google Scholar
  56. Shiwa Y (2005) Comment on “renormalization-group theory for the phase-field crystal equation.” Phys Rev E 79:013601 (1-2)Google Scholar
  57. Shiwa Y (2011) Renormalization-group for amplitude equations in cellular pattern formation with and without conservation law. Prog Theor Phys 125: 871-878ADSMATHCrossRefGoogle Scholar
  58. Shukla J (1998) Predictability in the midst of chaos: a scientific basis for climate forecasting. Science 282:728-731ADSCrossRefGoogle Scholar
  59. Stanley HE (1971) Introduction to phase transition and critical phenomena. Oxford University PressGoogle Scholar
  60. Swift J, Hohenberg PC (1977) Hydrodynamic fluctuations at the convective instability. Phys Rev A 15:319-328ADSCrossRefGoogle Scholar
  61. Tieszen R (1998) Gödel’s path from the incompleteness theorems (1931) to phenomenology (1961). Bull Symbolic Logic 4:181-203Google Scholar
  62. Tourchette H (2009) The large deviation approach to statistical mechanics. Phys Rep 478:1-69MathSciNetADSCrossRefGoogle Scholar
  63. Wall FT (1975) Theory of random walks with limited order of non-self-intersections used to simulate macromolecules. J Chem Phys 63:3713-3717ADSCrossRefGoogle Scholar
  64. Wall FT, Seitz WA (1979) The excluded volume effect for self-avoiding random walks. J Chem Phys 70:1860-1863ADSCrossRefGoogle Scholar
  65. Wang B, Zhou TG, Chen L (2007) Global topological dominance in the left hemisphere. Proc Natl Acad Sci USA 104:21014-21019ADSCrossRefGoogle Scholar
  66. Yau H-T (1998) Asymptotic solutions to dynamics of many-body systems and classical continuum equations. In: Current Developments in Mathematics, 1998. International PressGoogle Scholar

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© Springer Japan 2013

Authors and Affiliations

  • Yoshitsugu Oono
    • 1
  1. 1.University of IllinoisUrbanaUSA

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