Skip to main content
SpringerLink
Log in

A Selection of Nonequilibrium Issues

  • Chapter
  • First Online:
Book cover Methods of Contemporary Mathematical Statistical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1970))

Summary

We give a pedagogical introduction to a selection of recently discussed topics in nonequilibrium statistical mechanics, concentrating mostly on formal structures and on general principles. Part I contains an overview of the formalism of lattice gases that we use to explain various symmetries and inequalities generally valid for nonequilibrium systems, including the fluctuation symmetry, Jarzynski equality, and the direction of currents. That mostly concerns the time-antisymmetric part of dynamical fluctuation theory.We also briefly comment on recent attempts to combine that with the time-symmetric sector in a Langrangian or extended Onsager-Machlup approach. In Part II we concentrate on the macroscopic state and how entropy provides a bridge between microscopic dynamics and macroscopic irreversibility; included is a construction of quantum macroscopic states and a result on the equivalence of ensembles.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

  1. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic fluctuation theory for stationary non-equilibrium states, J. Stat. Phys. 107, 635–675 (2002).

    Article  MathSciNet  Google Scholar 

  2. I. Bjelaković, J. D. Deuschel, T. Krüger, S. Seiler, R. Siegmund-Schultze, A. Szkola, A quantum version of Sanov's theorem, Commun. Math. Phys. 260, 659–671 (2005).

    Article  MathSciNet  Google Scholar 

  3. I. Bjelaković, J. D. Deuschel, T. Krüger, S. Seiler, R. Siegmund-Schultze, A. Szkola, The Shannon-McMillan theorem for ergodic quantum lattice systems, Invent. Math. 155 (1), 203–202 (2004).

    Article  MathSciNet  Google Scholar 

  4. A. Bovier, F. den Hollander and F. Nardi, Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary conditions, Prob. Th. Rel. Fields. 135, 265–310 (2006).

    Article  Google Scholar 

  5. O. Bratelli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Volume 2, Springer, Berlin (1979).

    Book  Google Scholar 

  6. J. Bricmont, Science of chaos or chaos in science, In: The Flight from Science and Reason, Ann. N.Y. Academy of Science, 79, 731 (1996).

    Google Scholar 

  7. G. E. Crooks, Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems, J. Stat. Phys. 90, 1481 (1998).

    Article  MathSciNet  Google Scholar 

  8. N. Datta, Quantum entropy and quantum information, In: Les Houches, Session LXXXIII, 2005, Eds. A. Bovier, F. Dunlop, F. den Hollander, A. van Enter, and J. Dalibard, pp. 395–465 (2006).

    Google Scholar 

  9. A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications, Jones and Barlett Publishers, Boston (1993).

    MATH  Google Scholar 

  10. W. De Roeck, C. Maes, and K. Netočný, An extension of the Kac ring model, J. Phys. A: Math. Gen. 36, 1–13 (2003).

    Article  MathSciNet  Google Scholar 

  11. W. De Roeck, C. Maes, and K. Netočný, H-theorems from macroscopic autonomous equations, J. Stat. Phys. 123, 571–584 (2006).

    Article  MathSciNet  Google Scholar 

  12. W. De Roeck, C. Maes, and K. Netočný, Quantum macrostates, equivalence of ensembles and an H-theorem, J. Math. Phys. 47, 073303 (2006).

    Article  MathSciNet  Google Scholar 

  13. B. Derrida, J. L. Lebowitz, and E. R. Speer, Free Energy Functional for Nonequilibrium Systems: An Exactly Soluble Case, Phys. Rev. Lett. 87, 150601 (2001); J. Stat. Phys. 107, 599 (2002).

    Article  MathSciNet  Google Scholar 

  14. M. D. Donsker, S. R. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time I., Comm. Pure Appl. Math. 28:1–47 (1975).

    Article  MathSciNet  Google Scholar 

  15. M. Dresden, F. Feiock, Models in nonequilibrium quantum statistical mechanics, J. Stat. Phys. 4, 111–173 (1972).

    Article  Google Scholar 

  16. D. E. Evans, E. G. D. Cohen, and G. P. Morriss, Probability of second law violations in steady flows, Phys. Rev. Lett. 71, 2401–2404 (1993). cond-mat/9908420

    Article  Google Scholar 

  17. G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett. 74, 2694–2697 (1995); Dynamical ensembles in stationary states, J. Stat. Phys. 80, 931–970 (1995).

    Article  Google Scholar 

  18. P. L. Garrido, S. Goldstein and J. L. Lebowitz, The Boltzmann entropy for dense fluids not in local equilibrium, Phys. Rev. Lett. 92, 050602 (2003).

    Article  Google Scholar 

  19. H.-O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter, New York (1988).

    Google Scholar 

  20. S. Goldstein and J. L. Lebowitz, On the (Boltzmann) entropy of nonequilibrium systems, Physica D 193, 53–56 (2004).

    Article  MathSciNet  Google Scholar 

  21. S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North Holland Publishing Company (1962).

    Google Scholar 

  22. P. Hänggi, P. Talkner, and M. Borkovec, Reaction rate theory: fifty years after Kramers, Rev. of Mod. Phys. 62, 251–341 (1990).

    Article  MathSciNet  Google Scholar 

  23. F. den Hollander, Large Deviations, Field Institute Monographs, Providence, Rhode Island (2000).

    Google Scholar 

  24. R. Israel, Convexity in the Theory of Lattice Gases, Princeton University Press (1978).

    Google Scholar 

  25. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690–2693 (1997).

    Article  Google Scholar 

  26. E. T. Jaynes, The second law as physical fact and as human inference (1990) (unpublished), download from http://www.etjaynescenter.org /bibliography.shtml.

  27. E. T. Jaynes, The evolution of Carnot's principle, In: Maximum-Entropy and Bayesian Methods in Science and Engineering, 1, Eds. G. J. Erickson and C. R. Smith, Kluwer, Dordrecht, pp. 267–281 (1988).

    Chapter  Google Scholar 

  28. E. T. Jaynes, Gibbs vs Boltzmann entropies, In: Papers on Probability, Statistics and Statistical Physics, Reidel, Dordrecht (1983); Am. J. Phys. 33, 391–398 (1965).

    Article  Google Scholar 

  29. M. Kac. Probability and Related Topics in Physical Sciences, Interscience Publishers Inc., New York (1959).

    MATH  Google Scholar 

  30. C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Springer-Verlag, Berlin (1999).

    Book  Google Scholar 

  31. R. Kubo, K. Matsuo, and K. Kitahara, Fluctuation and Relaxation of Macrovariables, J. Stat. Phys. 9, 51–95 (1973).

    Article  Google Scholar 

  32. O. E. Lanford III, Time evolution of large classical systems, Lecture Notes in Phys. 38, 1–111 (1975); The hard sphere gas in the Boltzmann-Grad limit, Physica 106A, 70–76 (1981).

    Article  Google Scholar 

  33. J. L. Lebowitz, Microscoic origins of irreversible macroscopic behavior, Physica A 263, 516–527 (1999).

    Article  MathSciNet  Google Scholar 

  34. J. L. Lebowitz, C. Maes, and E. R. Speer, Statistical mechanics of probabilistic cellular automata, J. Stat. Phys. 59 117–170 (1990).

    Article  MathSciNet  Google Scholar 

  35. J. L. Lebowitz and H. Spohn, A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics, J. Stat. Phys. 95, 333–365 (1999).

    Article  MathSciNet  Google Scholar 

  36. T. M. Liggett, Interacting Particle Systems, Springer (1985).

    Google Scholar 

  37. R. Lima, Equivalence of ensembles in quantum lattice systems: states, Commun. Math. Phys. 24, 180–192 (1972).

    Article  MathSciNet  Google Scholar 

  38. C. Maes, On the origin and the use of fluctuation relations for the entropy, Poincaré Seminar 2003, Eds. J. Dalibard, B. Duplantier and V. Rivasseau, Birkhäuser (Basel), pp. 145–191 (2004).

    Chapter  Google Scholar 

  39. C. Maes. Fluctuation theorem as a Gibbs property, J. Stat. Phys. 95, 367–392 (1999).

    Article  MathSciNet  Google Scholar 

  40. C. Maes and K. Netočný, Time-reversal and entropy, J. Stat. Phys. 110, 269–310 (2003).

    Article  MathSciNet  Google Scholar 

  41. C. Maes and K. Netočný, Static and dynamical nonequilibrium fluctuations, cond-mat/0612525.

    Google Scholar 

  42. C. Maes, K. Netočný, and M. Verschuere, Heat conduction networks, J. Stat. Phys. 111, 1219–1244 (2003).

    Article  MathSciNet  Google Scholar 

  43. C. Maes, F. Redig, and A. Van Moffaert, On the definition of entropy production via examples, J. Math. Phys. 41, 1528–1554 (2000).

    Article  MathSciNet  Google Scholar 

  44. C. Maes, F. Redig, and A. Van Moffaert, The restriction of the Ising model to a layer, J. Stat. Phys. 96, 69–107 (1999).

    Article  MathSciNet  Google Scholar 

  45. C. Maes and K. Netočný, The canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states, Europhysics Letters 82, 30003 (2008).

    Article  MathSciNet  Google Scholar 

  46. C. Maes, K. Netočný and B. Wynants, On and beyond entropy production; the case of Markov jump processes, to appear in Markov Processes and Related Fields (2008).

    Google Scholar 

  47. C. Maes and K. Netočný, Minimum entropy production principle from a dynamical fluctuation law, J.Math.Phys. 48, 053306 (2007).

    Article  MathSciNet  Google Scholar 

  48. C. Maes, K. Netočný and B. Wynants, Steady state statistics of driven diffusion, Physica A 387, 2675–2689 (2008).

    Article  MathSciNet  Google Scholar 

  49. K. Netočný and F. Redig, Large deviations for quantum spin systems, J. Stat. Phys. 117, 521–547 (2004).

    Article  MathSciNet  Google Scholar 

  50. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Translated form the German edition by R. T. Beyer, Princeton University Press, Princeton(1955).

    MATH  Google Scholar 

  51. D. Petz, G. A. Raggio, and A. Verbeure, Asymptotics of Varadhan-type and the Gibbs variational principle, Comm. Math. Phys. 121, 271-?282 (1989) .

    Article  MathSciNet  Google Scholar 

  52. I. Prigogine, Introduction to Non-Equilibrium Thermodynamics, Wiley-Interscience, New York (1962).

    MATH  Google Scholar 

  53. M. Lenci, L. Rey-Bellet, Large deviations in quantum lattice systems: one-phase region, J. Stat. Phys., 119, 715–746 (2005).

    Article  MathSciNet  Google Scholar 

  54. D. Ruelle, Smooth dynamics and new rheoretical ideas in nonequilibrium statistical mechanics, J. Stat. Phys. 95, 393–468 (1999).

    Article  Google Scholar 

  55. B. Simon, The Statistical Mechanics of Lattice Gases, Princeton Univ. Press Princeton, (1993).

    MATH  Google Scholar 

  56. H. Spohn, Large Scale Dynamics of Interacting Particles, Springer Verlag Heidelberg, (1991).

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituut voor Theoretische Fysica, K. U. Leuven, Belgium

    Christian Maes &  Bidzina Shergelashvili

  2. Institute of Physics AS CR, Prague, Czech Republic

    Karel Netoˇcn´y

Authors
  1. Christian Maes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Karel Netoˇcn´y
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bidzina Shergelashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Christian Maes or Karel Netoˇcn´y .

Editor information

Editors and Affiliations

  1. Ctr. for Theoretical Study, Charles University, Jilska 1, Praha 1, 110 00, Czech Republic

    Roman Kotecký

  2. Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK

    Roman Kotecký

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Maes, C., Netoˇcn´y, ., Shergelashvili, . (2009). A Selection of Nonequilibrium Issues. In: Kotecký, R. (eds) Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics(), vol 1970. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92796-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation