Generalized Langevin Equations

  • Alexandre J. Chorin
  • Ole H. Hald
Part of the Texts in Applied Mathematics book series (TAM, volume 58)


We now turn to problems in statistical mechanics where the assumption of thermal equilibrium does not apply. In nonequilibrium problems, one should in principle solve the full Liouville equation, at least approximately. There are many situations in which one attempts to do that under different assumptions and conditions, giving rise to the Euler and Navier–Stokes equations, the Boltzmann equation, the Vlasov equation, etc.


Hamiltonian System Conditional Expectation Langevin Equation Noise Term Full System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

9.9 Bibliography

  1. [1].
    B. Alder and T. Wainwright, Decay of the velocity correlation function, Phys. Rev. A 1 (1970), pp. 1–12.Google Scholar
  2. [2].
    R. Balescu, Statistical Dynamics, Matter out of Equilibrium, Imperial College Press, London, 1997.Google Scholar
  3. [3].
    D. Bernstein, Optimal prediction of the Burgers equation, Mult. Mod. Sim. 6 (2007), pp. 27–52.Google Scholar
  4. [4].
    S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), pp. 1–88; reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes, Dover, New York, 1954.Google Scholar
  5. [5].
    A.J. Chorin, O.H. Hald, and R. Kupferman, Optimal prediction and the Mori–Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA 97 (2000), pp. 2968–2973.Google Scholar
  6. [6].
    A.J. Chorin, O.H. Hald, and R. Kupferman, Optimal prediction with memory, Physica D, 166 (2002), pp. 239–257.Google Scholar
  7. [7].
    A.J. Chorin and P. Stinis, Problem reduction, renormalization, and memory, Comm. Appl. Math. Comp. Sci. 1 (2005), pp. 1–27.Google Scholar
  8. [8].
    D. Evans and G. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Academic, New York, 1990.Google Scholar
  9. [9].
    G. Ford, M. Kac, and P. Mazur, Statistical mechanics of assemblies of coupled oscillators, J. Math. Phys. 6 (1965), pp. 504–515.Google Scholar
  10. [10].
    S. Goldstein, On diffusion by discontinuous movements and on the telegraph equation, Q. J. Mech. Appl. Math., 4 (1951), pp. 129–156.Google Scholar
  11. [11].
    D. Givon, R. Kupferman, and A. Stuart, Extracting macroscopic dynamics, model problems and algorithms, Nonlinearity 17 (2004), pp. R55–R127.Google Scholar
  12. [12].
    O. H. Hald and P. Stinis, Optimal prediction and the rate of decay of solutions of the Euler equations in two and three dimensions, Proc. Nat. Acad. Sc. USA 104 (2007), pp. 6527–6532.Google Scholar
  13. [13].
    P. Hohenberg and B. Halperin, Theory of dynamical critical phenomena, Rev. Mod. Phys. 49 (1977) pp. 435–479.Google Scholar
  14. [14].
    M. Kac, A stochastic model related to the telegrapher’s equation, Rocky Mountain J. Math. 4 (1974), pp. 497–509.Google Scholar
  15. [15].
    A. Majda, I. Timofeyev, and E. Vanden Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math. 54, (2001), pp. 891–947.Google Scholar
  16. [16].
    G. Papanicolaou, Introduction to the asymptotic analysis of stochastic equations, in Modern Modeling of Continuum Phenomena, R. DiPrima (ed.), Providence RI, 1974.Google Scholar
  17. [17].
    P. Stinis, Stochastic optimal prediction for the Kuramoto–Sivashinski equation, Multiscale Model. Simul. 2 (2004), pp. 580–612.Google Scholar
  18. [18].
    K. Theodoropoulos, Y.H. Qian and I. Kevrekidis, Coarse stability and bifurcation analysis using timesteppers: a reaction diffusion example, Proc. Natl. Acad. Sci. USA 97 (2000), pp. 9840–9843.Google Scholar
  19. [19].
    R. Zwanzig, Problems in nonlinear transport theory, in Systems Far from Equilibrium, L. Garrido (ed.), Springer-Verlag, New York, 1980.Google Scholar
  20. [20].
    R. Zwanzig, Nonlinear generalized Langevin equations, J. Statist. Phys. 9 (1973), pp. 423–450.Google Scholar
  21. [21].
    R. Zwanzig, Irreversible Statistical Mechanics, Oxford, 2002.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexandre J. Chorin
    • 1
  • Ole H. Hald
    • 2
  1. 1.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA
  2. 2.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

Personalised recommendations