Skip to main content
SpringerLink
Log in
Book cover

Stochastic Tools in Mathematics and Science pp 135–159Cite as

Time-Dependent Statistical Mechanics

  • Chapter
  • First Online:

  • 1863 Accesses

Part of the book series: Surveys and Tutorials in the Applied Mathematical Sciences ((STAMS,volume 1))

Abstract

We now turn to the statistical mechanics of systems not in equilibrium. The first few sections are devoted to special cases, which will be used to build up experience with the questions one can reasonably ask and the kinds of answer one may expect. A general formalism will follow, with applications.

This is a preview of subscription content, 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   34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. B. Alder and T. Wainwright, Decay of the velocity correlation function, Phys. Rev. A 1 (1970), pp. 1–12.

    Article  Google Scholar 

  2. R. Balescu, Statistical Dynamics, Matter out of Equilibrium, Imperial College Press, London, 1997.

    MATH  Google Scholar 

  3. 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.

    Article  Google Scholar 

  4. 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.

    Article  MathSciNet  Google Scholar 

  5. A.J. Chorin, O.H. Hald, and R. Kupferman, Optimal prediction with memory, Physica D, 166 (2002), pp. 239–257.

    Article  MathSciNet  Google Scholar 

  6. A.J. Chorin and P. Stinis, Problem reduction, renormalization, and memory, Comm. Appl. Math. Comp. Sci. 1 (2005), pp. 1–27.

    Google Scholar 

  7. W. E. andB. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci. 1 (2003), pp. 87–133.

    MathSciNet  Google Scholar 

  8. D. Evans and G. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Academic, New York, 1990.

    MATH  Google Scholar 

  9. G. Ford, M. Kac, and P. Mazur, Statistical mechanics of assemblies of coupled oscillators, J. Math. Phys. 6 (1965), pp. 504–515.

    Article  MathSciNet  Google Scholar 

  10. D. Givon, R. Kupferman, and A. Stuart, Extracting macroscopic dynamics, model problems and algorithms, Nonlinearity 17 (2004), pp. R55–R127.

    Article  MathSciNet  Google Scholar 

  11. S. Goldstein, On diffusion by discontinuous movements and on the telegraph equation, Q. J. Mech. Appl. Math. 4 (1951), pp. 129–156.

    Article  Google Scholar 

  12. O.H. Hald and P. Stinis, Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions, Proc. Natl. Acad. Sci. USA 104 (2007), pp. 6527–6532.

    Article  MathSciNet  Google Scholar 

  13. P. Hohenberg and B. Halperin, Theory of dynamical critical phenomena, Rev. Mod. Phys. 49 (1977) pp. 435–479.

    Article  Google Scholar 

  14. I. Horenko and C. Schutte, Likelihood-based estimation of multidimensional Langevin models and its application to biomolecular dynamics, Multiscale Model. Simul. 7 (2008), pp. 731–773.

    Article  MathSciNet  Google Scholar 

  15. M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain J. Math. 4 (1974), pp. 497–509.

    Google Scholar 

  16. S.S. Ma, Modern Theory of Critical Phenomena, Benjamin, Boston, 1976.

    Google Scholar 

  17. A. Majda and X. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, 2006.

    Google Scholar 

  18. A. Majda, I. Timofeyev, and E. Vanden-Eijnden, A mathematical framework for stochastic climate models, Commun. Pure Appl. Math. 54 (2001), pp. 891–947.

    Article  MathSciNet  Google Scholar 

  19. G. Papanicolaou, Introduction to the asymptotic analysis of stochastic equations, in Modern Modeling of Continuum Phenomena, R. DiPrima (ed.), Providence RI, 1974.

    Google Scholar 

  20. C. Schutte, J. Walters, C. Hartmann, and W. Huisinga, An averaging principle for fast degrees of freedom exhibiting long-term correlations, Multiscale Model. Simul. 2 (2004), pp. 501–526.

    Article  Google Scholar 

  21. P. Stinis, Stochastic optimal prediction for the Kuramoto-Sivashinksi equation, Multiscale Model. Simul. 2 (2004), pp. 580–612.

    Article  MathSciNet  Google Scholar 

  22. 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.

    Article  Google Scholar 

  23. R. Zwanzig, Problems in nonlinear transport theory, in Systems Far from Equilibrium, L. Garrido (ed.), Springer-Verlag, New York, 1980.

    Google Scholar 

  24. R. Zwanzig, Nonlinear generalized Langevin equations, J. Statist. Phys. 9 (1973), pp. 423–450.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Californian at Berkeley, 970 Evans Hall, Berkeley, CA, 94720-3840, USA

    Alexandre J. Chorin & Ole H. Hald

Authors
  1. Alexandre J. Chorin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ole H. Hald
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandre J. Chorin .

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag New York

About this chapter

Cite this chapter

Chorin, A.J., Hald, O.H. (2009). Time-Dependent Statistical Mechanics. In: Stochastic Tools in Mathematics and Science. Surveys and Tutorials in the Applied Mathematical Sciences, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1002-8_6

Download citation

Publish with us

Policies and ethics

Search

Navigation