Skip to main content
SpringerLink
Log in

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

  • Reference work entry
Mathematics of Complexity and Dynamical Systems

  • 337 Accesses

Article Outline

Glossary

Definition of the Subject

Introduction

Weak‐Coupling Limit for Classical Systems

Weak‐Coupling Limit for Quantum Systems

Weak-Coupling Limit in the Bose–Einstein and Fermi–Dirac Statistics

Weak-Coupling Limit for a Single Particle: The Linear Theory

Future Directions

Bibliography

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 600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 329.99
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

Abbreviations

Scaling limits:

A scaling limit denotes a procedure to reduce the degree of complexity of a large particle system. It consists in scaling the space-time variables and, possibly, other quantities (like the interaction potential or the density), in order to obtain a more handable description of the same system. The initial space-time coordinates are called microscopic, while the new ones, those well suited for the description of kinetic or hydrodynamical systems, are called macroscopic.

Boltzmann equation:

It is an integro‐differential kinetic equation for the one particle distribution function in the classical phase space (see Eq. (64) below). It arises in some physical regimes, namely for a rarefied gas and for a weakly interacting quantum dense gas.

Uehling–Uhlenbeck equation:

It is a Boltzmann type equation taking into account corrections due to the Bose–Einstein or the Fermi–Dirac statistics (see Eq. (69) below). It holds in the weak‐coupling limit.

Fokker–Planck–Landau equation:

It is a kinetic equation diffusive in velocity (see Eq. (28) below). It arises in the context of a weakly interacting classical dense gas.

Hydrodynamical equations:

They are evolution equations for macroscopic quantities like density, mean velocity, temperature, and so on.

Low density limit:

Sometimes called Boltzmann–Grad limit, it is a scaling limit in which the density is vanishing. Applied to classical particle systems it gives the Boltzmann equation.

Weak coupling limit:

It is a scaling limit in which the density is constant but the interaction vanishes suitably. Applied to a quantum particle systems it gives a Boltzmann equation. Applied to a classical particle systems it gives the Fokker–Planck–Landau equation.

Hydrodynamic limit:

In this scaling we simply pass from micro to macro variables. We look at the behavior of suitable mean values, which are functions of the space and the time. We expect them to behave in a hydrodynamical way, namely to satisfy a set of hydrodynamical equations.

Wigner transform:

It is the description of a quantum state as a function in the classical phase space.

Bibliography

Primary Literature

  1. Arseniev AA, Buryak OE (1990) On a connection between the solution of the Boltzmann equation and the solution of the Landau–Fokker–Planck equation (Russian). Mat Sb 181(4):435–446 (translation in Math USSR-Sb 69(2):465–478 1991)

    Google Scholar 

  2. Balescu R (1975) Equilibrium and Nonequilibrium Statistical Mechanics. Wiley, New York

    MATH  Google Scholar 

  3. Boldrighini C, Bunimovich LA, Ya Sinai G (1983) On the Boltzmann equation for nthe Lorentz gas. J Stat Phys 32:477–501

    Article  MATH  Google Scholar 

  4. Benedetto D, Castella F, Esposito R, Pulvirenti M (2004) Some Considerations on the derivation of the nonlinear Quantum Boltzmann Equation. J Stat Phys 116(114):381–410

    Article  MathSciNet  MATH  Google Scholar 

  5. Benedetto D, Castella F, Esposito R, Pulvirenti M (2005) On The Weak–Coupling Limit for Bosons and Fermions. Math Mod Meth Appl Sci 15(12):1–33

    Article  MathSciNet  Google Scholar 

  6. Benedetto D, Castella F, Esposito R, Pulvirenti M (2006) Some Considerations on the derivation of the nonlinear Quantum Boltzmann Equation II: the low‐density regime. J Stat Phys 124(2–4):951–996

    Article  MathSciNet  MATH  Google Scholar 

  7. Benedetto D, Castella F, Esposito R, Pulvirenti M (2008) From the N-body Schrödinger equation to the quantum Boltzmann equation: a term-by-term convergence result in the weak coupling regime. Commun Math Phys 277(1):1–44

    Article  MathSciNet  MATH  Google Scholar 

  8. Benedetto D, Esposito R, Pulvirenti M (2004) Asymptotic analysis of quantum scattering under mesoscopic scaling. Asymptot Anal 40(2):163–187

    MathSciNet  MATH  Google Scholar 

  9. Benedetto D, Pulvirenti M (2007) The classical limit for the Uehling–Uhlenbeck operator. Bull Inst Math Acad Sinica 2(4):907–920

    MathSciNet  MATH  Google Scholar 

  10. Burgain J, Golse F, Wennberg B (1998) On the distribution of free path lenght for the periodic Lorentz gas. Comm Math Phys 190:491–508

    Article  MathSciNet  Google Scholar 

  11. Caglioti E, Golse F (2003) On the distribution of free path lengths for the periodic Lorentz gas. III Comm Math Phys 236(2):199–221

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen T (2005) Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3. J Stat Phys 120(1–2):279–337

    Article  MathSciNet  MATH  Google Scholar 

  13. Caglioti E, Pulvirenti M, Ricci V (2000) Derivation of a linear Boltzmann equation for a periodic Lorentz gas. Mark Proc Rel Fields 3:265–285

    MathSciNet  Google Scholar 

  14. Cercignani C, Illner R, Pulvirenti M (1994) The mathematical theory of dilute gases, Applied Mathematical Sciences, vol 106. Springer, New York

    Google Scholar 

  15. Degond P, Lucquin–Desreux B (1992) The Fokker–Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math Models Methods Appl Sci 2(2):167–182

    Google Scholar 

  16. Dürr D, Goldstain S, Lebowitz JL (1987) Asymptotic motion of a classical particle in a random potential in two dimension: Landau model. Comm Math Phys 113:209–230

    Google Scholar 

  17. Desvillettes L, Pulvirenti M (1999) The linear Boltzmann eqaution for long-range forces: a derivation for nparticle systems. Math Moduls Methods Appl Sci 9:1123–1145

    Article  MathSciNet  MATH  Google Scholar 

  18. Di Perna RJ, Lions PL (1989) On the Cauchy problem for the Boltzmann equatioin. Ann Math 130:321–366

    Article  Google Scholar 

  19. Esposito R, Pulvirenti M (2004) From particles to fluids. In: Friedlander S, Serre D (eds) Handbook of Mathematical Fluid Dynamics, vol 3. Elsevier, North Holland, pp 1–83

    Chapter  Google Scholar 

  20. Eng D, Erdös L (2005) The linear Boltzmann equation as the low‐density limit of a random Schrödinger equation. Rev Math Phys 17(6):669–743

    Google Scholar 

  21. Erdös L, Salmhofer M, Yau HT (2004) On the quantum Boltzmann equation. J Stat Phys 116:367–380

    Google Scholar 

  22. Erdös L, Yau HT (2000) Linear Boltzmann equation as a weak‐coupling limit of a random Schrödinger equation. Comm Pure Appl Math 53:667–735

    Google Scholar 

  23. Gallavotti G (1972) Rigorous theory of the Boltzmann equation in the Lorentz gas in Meccanica Statistica. reprint Quaderni CNR 50:191–204

    Google Scholar 

  24. Goudon T (1997) On Boltzmann equations and Fokker–Planck asymptotics: influence of grazing collisions. J Statist Phys 89(3–4):751–776

    Article  MathSciNet  MATH  Google Scholar 

  25. Grad H (1949) On the kinetic Theory of rarefied gases. Comm Pure Appl Math 2:331–407

    Article  MathSciNet  MATH  Google Scholar 

  26. Hugenholtz MN (1983) Derivation of the Boltzmann equation for a Fermi gas. J Stat Phys 32:231–254

    Article  MathSciNet  Google Scholar 

  27. Ho NT, Landau LJ (1997) Fermi gas in a lattice in the van Hove limit. J Stat Phys 87:821–845

    Article  MathSciNet  MATH  Google Scholar 

  28. Illner R, Pulvirenti M (1986) Global Validity of the Boltzmann equation for a two‐dimensional rare gas in the vacuum. Comm Math Phys 105:189–203 (Erratum and improved result, Comm Math Phys 121:143–146)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kesten H, Papanicolaou G (1981) A limit theorem for stochastic acceleration. Comm Math Phys 78:19–31

    Article  MathSciNet  Google Scholar 

  30. Landau LJ (1994) Observation of quantum particles on a large space-time scale. J Stat Phys 77:259–309

    Article  MATH  Google Scholar 

  31. Lanford III O (1975) Time evolution of large classical systems. In: Moser EJ (ed) Lecture Notes in Physics 38. Springer, pp 1–111

    Google Scholar 

  32. Lifshitz EM, Pitaevskii LP (1981) Course of theoretical physics “Landau–Lifshit”, vol 10. Pergamon Press, Oxford-Elmsford

    Google Scholar 

  33. Xuguang LU (2005) The Boltzmann equation for Bose–Einstein particles: velocity concentration and convergence to equilibrium. J Stat Phys 119(5–6):1027–1067

    MathSciNet  MATH  Google Scholar 

  34. Xuguang LU (2004) On isotropic distributional solutions to the Boltzmann equation for Bose–Einstein particles. J Stat Phys 116(5–6):1597–1649

    MATH  Google Scholar 

  35. CB Jr Morrey (1955) On the derivation of the equations of hydrodynamics from statistical mechanics. Comm Pure Appl Math 8:279–326

    Article  MathSciNet  Google Scholar 

  36. Nier F (1996) A semi‐classical picture of quantum scattering. Ann Sci Ec Norm Sup 29(4):149–183

    MathSciNet  MATH  Google Scholar 

  37. Nier F (1995) Asymptotic analysis of a scaled Wigner equation and quantum scattering. Transp Theory Statist Phys 24(4–5):591–628

    Article  MathSciNet  MATH  Google Scholar 

  38. Nordheim LW (1928) On the Kinetic Method in the New Statistics and Its Application in the Electron Theory of Conductivity. Proc Royal Soc Lond Ser A 119(783):689–698

    Article  MATH  Google Scholar 

  39. Spohn H (1991) Large scale dynamics of interacting particles, Texts and monographs in physics. Springer

    Google Scholar 

  40. Spohn H (1978) The Lorentz flight process converges to a random flight process. Comm Math Phys 60:277–290

    Article  MathSciNet  MATH  Google Scholar 

  41. Spohn H (1977) Derivation of the transport equation for electrons moving through random impurities. J Stat Phys 17:385–412

    Article  MathSciNet  MATH  Google Scholar 

  42. Uehling EA, Uhlembeck GE (1933) Transport Phenomena in Einstein–Bose and Fermi–Dirac Gases. Phys Rev 43:552–561

    Article  Google Scholar 

  43. Villani C (2002) A review of mathematical topics in collisional kinetic theory. In: Friedlander S, Serre D (eds) Handbook of Mathematical Fluid Dynamics, vol 1. Elsevier, North Holland, pp 71–307

    Chapter  Google Scholar 

  44. Villani C (1998) On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch Rational Mech Anal 143(3):273–307

    Article  MathSciNet  MATH  Google Scholar 

  45. Wigner E (1932) On the quantum correction for the thermodynamical equilibrium. Phys Rev 40:742–759

    Article  Google Scholar 

Books and Reviews

  1. Balescu R (1975) Equilibrium and Nonequilibrium Statistical Mechanics. Wiley, New York

    MATH  Google Scholar 

  2. Cercignani C, Boltzmann L (1998) The man who trusted atoms. Oxford University Press, Oxford

    MATH  Google Scholar 

  3. Cercignani C, Illner R, Pulvirenti M (1994) The mathematical theory of dilute gases. Applied Mathematical Sciences, vol 106. Springer, New York

    Google Scholar 

  4. Esposito R, Pulvirenti M (2004) From particles to fluids. In: Friedlander S, Serre D (eds) Handbook of Mathematical Fluid Dynamics, vol 3. Elsevier, North Holland, pp 1–83

    Chapter  Google Scholar 

  5. Spohn H (1991) Large scale dynamics of interacting particles, Texts and monographs in physics. Springer, Heidelberg

    Book  Google Scholar 

  6. Villani C (2002) A review of mathematical topics in collisional kinetic theory. In: Friedlander S, Serre D (eds) Handbook of Mathematical Fluid Dynamics, vol 1. Elsevier, North Holland, pp 71–307

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma ‘La Sapienza’, Roma, Italy

    D. Benedetto & M. Pulvirenti

Authors
  1. D. Benedetto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Pulvirenti
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Benedetto, D., Pulvirenti, M. (2012). Scaling Limits of Large Systems of Non-linear Partial Differential Equations. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_95

Download citation

Publish with us

Policies and ethics

Search

Navigation