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Ergodic properties of classical dissipative systems I

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References

  • [AM]Abraham, R. &Marsden, J. E.,Foundations of Mechanics, 2nd edition. Benjamin/Cummings Publishing Co., Reading, MA, 1978.

    Google Scholar 

  • [C3P]Casetti, L., Cerruti-Sola, M., Cohen, E. G. D. &Pettini, M., The Fermi-Pasta-Ulam problem revisited.Phys. Rev. E, 55 (1997), 2539.

    Article  MathSciNet  Google Scholar 

  • [CFS]Cornfeld, I. P. Fomin, S. V. &Sinai, Ya. G.,Ergodic Theory. Springer-Verlag, New York-Berlin, 1982.

    Google Scholar 

  • [De]Devinatz, A., The factorization of operator valued functions.Ann. of Math. (2), 73 (1961), 458–495.

    Article  MATH  MathSciNet  Google Scholar 

  • [DM]Dym, H. &McKean, H. P.,Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Academic Press, New York-London, 1976.

    Google Scholar 

  • [FKM]Ford, G. W., Kac, M. &Mazur, P., Statistical mechanics of assemblies of coupled oscillators.J. Math. Phys., 6 (1965), 504–515.

    Article  MathSciNet  Google Scholar 

  • [FPU]Fermi, E., Pasta, J. &Ulam, S., Los Alamos Report LA-1940 (1955), inCollected Papers of Enrico Fermi, Vol. 2 (E. Segré, ed.), Univ. of Chicago, Chicago, 1965.

    Google Scholar 

  • [GJ]Glimm, J. &Jaffe, A.,Quantum Physics, 2nd edition. Springer-Verlag, New York-Berlin, 1987.

    Google Scholar 

  • [He]Helson, H.,Lectures on Invariant Subspaces. Academic Press, New York-London, 1964.

    Google Scholar 

  • [Hö]Hörmander, L.,The Analysis of Linear Partial Differential Operators, I Springer-Verlag, Berlin-New York, 1983.

    Google Scholar 

  • [JP1]Jakšić, V. & Pillet, C.-A., Ergodic properties of classical dissipative systems, II. In preparation.

  • [JP2]— Ergodic properties of the non-Markovian Langevin equation.Lett. Math. Phys., 41 (1997), 49–57.

    Article  MathSciNet  Google Scholar 

  • [JP3]— On a model for quantum friction, II. Fermi's golden rule and dynamics at positive temperature.Comm. Math. Phys., 176 (1996), 619–643.

    Article  MathSciNet  Google Scholar 

  • [JP4]— On a model for quantum friction, III. Ergodic properties of the spin-boson system.Comm. Math. Phys., 178 (1996), 627–651.

    Article  MathSciNet  Google Scholar 

  • [JP5]Jakšić, V. On a model for quantum friction IV. In preparation.

  • [KKS]Komech, A., Kunze, M. &Spohn, H., Long time asymptotics for a classical particle interacting with a scalar field.Comm. Partial Differential, Equations, 22 (1997), 307–335.

    MathSciNet  Google Scholar 

  • [Le]Lebowitz, J. L., Microscopic reversibility and macroscopic behavior: Physical explanations and mathematical derivations, in 25Years of Non-Equilibrium Statistical Mechanics (Sitges, 1994), pp. 1–20. Lecture Notes in Phys., 445. Springer-Verlag, Berlin, 1995.

    Google Scholar 

  • [LL]Lanford, O. E., III &Lebowitz, J. L., Time evolution and ergodic properties of harmonic systems, inDynamical Systems, Theory and Applications (Seattle, WA, 1974), pp. 144–177. Lecture Notes in Phys., 38. Springer-Verlag, Berlin, 1975.

    Google Scholar 

  • [LM]Levinson, N. &McKean, H. P., Jr., Weighted trigonometrical approximation on R1 with application to the germ field of a stationary Gaussian noise.Acta Math., 112 (1964), 99–143.

    MathSciNet  Google Scholar 

  • [LP]Lax, P. D. &Phillips, R. S.,Scattering Theory. Academic Press, New York-London, 1967.

    Google Scholar 

  • [LT]Lewis, J. T., &Thomas, L. C., How to make a heat bath, inFunctional Integration and Its Applications (London, 1974), pp. 97–123. Clarendon Press, Oxford, 1975.

    Google Scholar 

  • [M]Mañé, R.,Ergodic Theory and Differentiable Dynamics Springer-Verlag, Berlin-New York, 1987.

    Google Scholar 

  • [Ne]Nelson, E.,Dynamical Theories of Brownian Motion. Princeton Univ. Press, Princeton, NJ, 1976.

    Google Scholar 

  • [Ni]Nikol'skiî N. K.,Treatise on the Shift Operator. Springer-Verlag, Berlin-New York, 1986.

    Google Scholar 

  • [P]Parisi, G., On the approach to equilibrium of an Hamiltonian chain of anharmonic oscillators. Preprint. 1997.

  • [Ro]Robinson, D. W.,C *-algebras and quantum statistical mechanics, inC *-Algebras and Their Applications to Statistical Mechanics and Quantum Field Theory (Varenna, 1973), pp. 235–252. North-Holland, Amsterdam, 1976.

    Google Scholar 

  • [RS1]Reed, M. &Simon, B.,Methods of Modern Mathematical Physics, I.Functional Analysis, 2nd edition. Academic Press, London, 1980.

    Google Scholar 

  • [RS2]—Methods of Modern Mathematical Physics, II.Fourier Analysis, Self-Adjointness, Academic Press, London, 1975.

    Google Scholar 

  • [RS3]—Methods of Modern Mathematical Physics, III.Scattering Theory. Academic Press, London, 1978.

    Google Scholar 

  • [Ru]Ruelle, D.,Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, Boston, MA, 1989.

    Google Scholar 

  • [S]Simon, B., TheP(φ) 2 Euclidean (Quantum) Field Theory. Princeton Univ. Press, Princeton, NJ, 1974.

    Google Scholar 

  • [T]Tropper, M. M., Ergodic and quasi-deterministic properties of finite-dimensional stochastic systems.J. Statist. Phys., 17 (1977), 491–509.

    Article  MathSciNet  Google Scholar 

  • [UF]Uhlenbeck, G. E. &Ford, G. W.,Lectures in Statistical Mechanics, Amer. Math. Soc., Providence, RI, 1963.

    Google Scholar 

  • [UO]Uhlenbeck, G. E. &Ornstein, L. S., On the theory of Brownian motion.Phys. Rev., 36 (1930), 823.

    Article  Google Scholar 

  • [Wa]Walters, P.,An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.

    Google Scholar 

  • [Wx]Wax, N. Ed.,Selected Papers on Noise and Stochastic Processes Dover Publications, New York, 1954.

    Google Scholar 

  • [Wi]Wiener, N., On the factorization of matrices.Comment. Math. Helv., 29 (1955), 97–111.

    MATH  MathSciNet  Google Scholar 

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Jakšić, V., Pillet, CA. Ergodic properties of classical dissipative systems I. Acta Math. 181, 245–282 (1998). https://doi.org/10.1007/BF02392587

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