Skip to main content
SpringerLink
Log in

Phenomenology

—Renormalization and Asymptotic Analysis—

  • Chapter
  • First Online:
The Nonlinear World

Part of the book series: Springer Series in Synergetics ((SSSYN))

  • 2014 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Arnold VI (1997) Mathematical methods of classical mechanics. Springer

    Google Scholar 

  • Barenblatt GI (1996) Similarity, self-similarity, and intermediate asymptotics. Cambridge University Press

    Google Scholar 

  • Callen HB (1960) Thermodynamics. Interscience Publ.

    Google Scholar 

  • Cannone M, Friedlander S (2003) Navier: blow-up and collapse. Notices Amer Math Soc 50:7-13

    MathSciNet  MATH  Google Scholar 

  • Chaikin M, Lubensky TC (2000) Principles of condensed matter physics. Cambridge University Press

    Google Scholar 

  • Chen L-Y, Goldenfeld N, Oono Y, Paquette G (1993) Selection, stability and renormalization. Physica A 204:111-133

    Article  ADS  Google Scholar 

  • Chiba H (2008) C 1-approximation of vector fields based on the renormalization group method. SIAM J Applied Dynam Syst 7:895-932

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Chiba H (2009) Extension and unification of singular perturbation methods. SIAM J Applied Dynam Sys 8:1066-1115

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Cross MC, Hohenberg PC (1993) Pattern formation outside of equilibrium. Rev Mod Phys 65:851-1112

    Article  ADS  Google Scholar 

  • des Cloizeaux J (1975) The Lagrangian theory of polymer solutions at intermediate concentrations. J Phys (France) 36:281-291

    Google Scholar 

  • 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 

  • Elder KR, Katakowski M, Haataja M, Grant M (2002) Modeling elasticity in crystal growth. Phys Rev Lett 88:245701 (1-4)

    Google Scholar 

  • Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Indiana Univ Math J 21:193-226

    Article  MathSciNet  MATH  Google Scholar 

  • 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 Publishers

    Google Scholar 

  • Furukawa Y (1998) Inventing polymer science—Staudinger, Carothers, and the emergence of macromolecular chemistry—. University of Pennsylvania Press

    Google Scholar 

  • Goldenfeld N (1992) Lectures on phase transitions and renormalization group. Addison Wesley

    Google Scholar 

  • Goldenfeld ND, Martin O, Oono Y (1989) Intermediate asymptotics and renormalization group theory. J Scientific Comp 4:355-372

    Article  MathSciNet  Google Scholar 

  • Gunaratne H, Ouyang Q, Swinney HL (1994) Pattern formation in the presence of symmetries. Phys Rev E 50:2802-2820

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • 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 

  • Hall AR, Colegrave N (2008) Decay of unused characters by selection and drift. J Evol Biol 21:610-617

    Article  Google Scholar 

  • Heisenberg W (1971) Physics and beyond (translated by A J Pomerans) Harper & Row

    Google Scholar 

  • Herbut I (2007) A modern approach to critical phenomena. Cambridge University Press

    Google Scholar 

  • Hirsch M, Pugh C, Shub M (1970) Invariant manifolds. Bull Amer Math Soc 76:1015-1019

    Article  MathSciNet  MATH  Google Scholar 

  • Husserl E (1999) The idea of phenomenology (translated by L Hardy). Kluwer Academic Publishers

    Google Scholar 

  • Izutsu T (1991) Consciousness and essence—quest of the spiritual Orient. Iwanami paper back

    Google Scholar 

  • Keller CF (1999) Climate, modeling, and predictability. Physica D 133:296-308

    Article  ADS  MATH  Google Scholar 

  • Kihara T (1978) Intermolecular forces. Wiley

    Google Scholar 

  • Kubo R (1968) Thermodynamics. An advanced course with problems and solutions. North-Holland Pub Co.

    Google Scholar 

  • Ladyzhenskaya OA (1963) Mathematical theory of incompressible fluids. Gordon & Breach

    Google Scholar 

  • Ladyzhenskaya OA (2003) Sixth problem of the millennium: Navier-Stokes equations, existence and smoothness. Russ Math Surveys 58:251-286

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Le Bellac M (1991) Quantum and statistical field theory. Oxford University Press

    Google Scholar 

  • Lieb E, Yngvason J (1998) A guide to entropy and the second law of thermodynamics. Notices Amer Math Soc 45:571-581

    MathSciNet  MATH  Google Scholar 

  • Lieb E, Yngvason J (1999) The physics and mathematics of the second law of thermodynamics. Phys Rep 340:1-96

    Article  MathSciNet  Google Scholar 

  • Mandelbrot BB (1983) Fractal geometry of nature. W H Freeman

    Google Scholar 

  • Mandle F (1988) Statistical physics (2nd edition). Wiley

    Google Scholar 

  • Mañé R (1978) Persistent manifolds are normally hyperbolic. Trans Amer Math Soc 246:271-283

    Google Scholar 

  • Migdal AB (2000) Qualitative methods in quantum theory (translated by Leggett AJ). Westview Press

    Google Scholar 

  • 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 13:763-766

    Google Scholar 

  • Miyazaki K, Kitahara K, Bedeaux D (1996) Nonequilibrium thermodynamics of multicomponent systems. Physica A 230:600-630

    Article  ADS  Google Scholar 

  • Niwa N, Hiromi Y, Okabe M (2004) A conserved developmental program for sensory organ formation in Drosophila melanogaster. Nat Genet 36:293-297

    Article  Google Scholar 

  • Nozaki K, Oono Y (2001) Renormalization-group theoretical reduction. Phys Rev E 63:046101 (1-18)

    Google Scholar 

  • Ohta T, Nakanishi A (1983) Theory of semi-dilute polymer solutions: I Static properties in a good solvent. J Phys A 16:4155-4170

    Article  ADS  Google Scholar 

  • Ohta T, Oono Y (1982) Conformational space renormalization theory of semidilute polymer solutions. Phys Lett 89A:460-464

    ADS  Google Scholar 

  • Oono Y (1985) Statistical physics of polymer solutions. Conformational-space renormalization group approach. Adv Chem Phys 61:301-437

    Google Scholar 

  • Oono Y (1985) Dynamics in polymer solutions — a renormalization-group approach. AIP Conference Proceedings No 137 (edited by Y Rabin) p187-218

    Google Scholar 

  • Oono Y (1989) Large deviation and statistical physics. Prog Theor Phys Suppl 99:165-205

    Article  MathSciNet  ADS  Google Scholar 

  • 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 Press

    Google Scholar 

  • Oono Y, Paniconi M (1998) Steady state thermodynamics. Prog Theor Phys Suppl 130:29-44

    Article  MathSciNet  ADS  Google Scholar 

  • Pashko O, Oono Y (2000) The Boltzmann equation is a renormalization group equation. Int J Mod Phys B 14:555

    MathSciNet  ADS  MATH  Google Scholar 

  • Poggio T, Rifkin R, Mukherjee S, Niyoki P (2004) General conditions for predictivity in learning theory. Nature 428:419-422

    Article  ADS  Google Scholar 

  • Protas M, Conrad M, Gross JB, Tabin C, Borowsky R (2007) Regressive evolution in the Mexican cave tetra, Astyanax mexicanus. Curr Biol 17:452-454

    Article  Google Scholar 

  • Rajaram S, Taguchi Y-h, Oono Y (2005) Some implications of renormalization group theoretical ideas to statistics. Physica D 205:207-214

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ruelle D (1969) Statistical mechanics, rigorous results. Benjamin

    Google Scholar 

  • Sasa S, Tasaki H (2006) Steady state thermodynamics. J Stat Phys 125:125-224

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Shiwa Y (2000) Renormalization-group theoretical reduction of the Swift- Hohenberg model. Phys Rev E 63:016119 (1-7)

    Google Scholar 

  • Shiwa Y (2005) Comment on “renormalization-group theory for the phase-field crystal equation.” Phys Rev E 79:013601 (1-2)

    Google Scholar 

  • Shiwa Y (2011) Renormalization-group for amplitude equations in cellular pattern formation with and without conservation law. Prog Theor Phys 125: 871-878

    Article  ADS  MATH  Google Scholar 

  • Shukla J (1998) Predictability in the midst of chaos: a scientific basis for climate forecasting. Science 282:728-731

    Article  ADS  Google Scholar 

  • Stanley HE (1971) Introduction to phase transition and critical phenomena. Oxford University Press

    Google Scholar 

  • Swift J, Hohenberg PC (1977) Hydrodynamic fluctuations at the convective instability. Phys Rev A 15:319-328

    Article  ADS  Google Scholar 

  • Tieszen R (1998) Gödel’s path from the incompleteness theorems (1931) to phenomenology (1961). Bull Symbolic Logic 4:181-203

    Google Scholar 

  • Tourchette H (2009) The large deviation approach to statistical mechanics. Phys Rep 478:1-69

    Article  MathSciNet  ADS  Google Scholar 

  • Wall FT (1975) Theory of random walks with limited order of non-self-intersections used to simulate macromolecules. J Chem Phys 63:3713-3717

    Article  ADS  Google Scholar 

  • Wall FT, Seitz WA (1979) The excluded volume effect for self-avoiding random walks. J Chem Phys 70:1860-1863

    Article  ADS  Google Scholar 

  • Wang B, Zhou TG, Chen L (2007) Global topological dominance in the left hemisphere. Proc Natl Acad Sci USA 104:21014-21019

    Article  ADS  Google Scholar 

  • Yau H-T (1998) Asymptotic solutions to dynamics of many-body systems and classical continuum equations. In: Current Developments in Mathematics, 1998. International Press

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Illinois, Urbana, IL, USA

    Yoshitsugu Oono

Authors
  1. Yoshitsugu Oono
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Japan

About this chapter

Cite this chapter

Oono, Y. (2013). Phenomenology. In: The Nonlinear World. Springer Series in Synergetics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54029-8_3

Download citation

Publish with us

Policies and ethics

Search

Navigation