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Entropy Production in Gaussian Thermostats

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Abstract

We show that an arbitrary Anosov Gaussian thermostat on a surface is dissipative unless the external field has a global potential. This result is obtained by studying the cohomological equation of more general thermostats using the methods in [3].

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References

  1. Bonetto F., Gentile G., Mastropietro V. (2000) Electric fields on a surface of constant negative curvature. Ergod. Th. Dynam. Sys. 20, 681–696

    Article  MATH  MathSciNet  Google Scholar 

  2. Croke C.B., Sharafutdinov V.A. (1998) Spectral rigidity of a negatively curved manifold. Topology 37, 1265–1273

    Article  MATH  MathSciNet  Google Scholar 

  3. Dairbekov N.S., Paternain G.P. (2005) Longitudinal KAM-cocycles and action spectra of magnetic flows. Math. Res. Lett. 12, 719–730

    MATH  MathSciNet  Google Scholar 

  4. Dairbekov N.S., Sharafutdinov V.A. (2003) Some problems of integral geometry on Anosov manifolds. Ergod. Th. Dynam. Sys. 23, 59–74

    Article  MATH  MathSciNet  Google Scholar 

  5. Gallavotti G. (1995) Reversible Anosov diffeomorphisms and large deviations. Math. Phys. Electronic J. 1, 1–12

    MATH  MathSciNet  Google Scholar 

  6. Gallavotti G. (1999) New methods in nonequilibrium gases and fluids. Open Sys. Inf. Dynam. 6, 101–136

    Article  MATH  MathSciNet  Google Scholar 

  7. Gallavotti G., Cohen E.G.D. (1995) Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697

    Article  ADS  Google Scholar 

  8. Gallavotti G., Cohen E.G.D. (1995) Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Gallavotti G., Ruelle D. (1997) SRB states and nonequilibrium statistical mechanics close to equilibrium. Commun. Math. Phys. 190, 279–281

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Gentile G. (1998) Large deviation rule for Anosov flows. Forum Math. 10, 89–118

    Article  MATH  MathSciNet  Google Scholar 

  11. Ghys E. (1984) Flots d’Anosov sur les 3-variétés fibrées en cercles. Ergod. Th. Dynam. Sys. 4, 67–80

    Article  MATH  MathSciNet  Google Scholar 

  12. Guillemin V., Kazhdan D. (1980) Some inverse spectral results for negatively curved 2-manifolds. Topology 19, 301–312

    Article  MATH  MathSciNet  Google Scholar 

  13. Hoover W.G. (1986) Molecular Dynamics, Lecture Notes in Phys 258. Berlin Heidelberg New York, Springer

    Google Scholar 

  14. Katok A., Hasselblatt B. (1995) Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications 54. Cambridge, Cambridge University Press

    Google Scholar 

  15. de la Llave R., Marco J.M., Moriyon R. (1986) Canonical perturbation theory of Anosov systems and regularity for the Livsic cohomology equation. Ann. Math. 123, 537–611

    Article  MATH  Google Scholar 

  16. Lopes A.O., Thieullen P. (2005) Sub-actions for Anosov flows. Ergod. Th. Dynam. Sys. 25, 605–628

    Article  MATH  MathSciNet  Google Scholar 

  17. Paternain G.P. (1999) Geodesic flows. Progress in Mathematics 180. Basel-Boston, Birkäuser

    Google Scholar 

  18. Paternain G.P. (2005) The longitudinal KAM-cocycle of a magnetic flow. Math. Proc. Camb. Phil. Soc. 139, 307–316

    Article  MATH  MathSciNet  Google Scholar 

  19. Pollicott M., Sharp R. (2004) Livsic theorems, maximising measures and the stable norm. Dyn. Sys.: An Int. J. 19, 75–88

    Article  MATH  MathSciNet  Google Scholar 

  20. Ruelle D. (1996) Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys. 85, 1–23

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Wojtkowski M.P. (2000) Magnetic flows and Gaussian thermostats on manifolds of negative curvature. Fund. Math. 163, 177–191

    MATH  MathSciNet  Google Scholar 

  23. Wojtkowski M.P. (2000) W-flows on Weyl manifolds and Gaussian thermostats. J. Math. Pures Appl. 79, 953–974

    Article  MATH  MathSciNet  Google Scholar 

  24. Wojtkowski, M.P.: Weyl manifolds and Gaussian thermostats. Proceedings of the International Congress of Mathematicians, Beijing 2002, Vol. III, pp. 511–523 China: Higher Ed. Press/ world Scientific (2002)

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Authors and Affiliations

  1. Kazakh British Technical University, Tole bi 59, 050000, Almaty, Kazakhstan

    Nurlan S. Dairbekov

  2. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, England

    Gabriel P. Paternain

Authors
  1. Nurlan S. Dairbekov
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  2. Gabriel P. Paternain
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Correspondence to Gabriel P. Paternain.

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Communicated by G. Gallavotti

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Dairbekov, N.S., Paternain, G.P. Entropy Production in Gaussian Thermostats. Commun. Math. Phys. 269, 533–543 (2007). https://doi.org/10.1007/s00220-006-0117-y

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  • DOI: https://doi.org/10.1007/s00220-006-0117-y

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