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Brownian motion of a nonlinear oscillator

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Abstract

Starting with the Langevin equation for a nonlinear oscillator (the “Duffing oscillator”) undergoing ordinary Brownian motion, we derive linear transport laws for the motion of the average position and velocity of the oscillator. The resulting linear equations are valid for only small deviations of average values from thermal equilibrium. They contain a renormalized oscillator frequency and a renormalized and non-Markovian friction coefficient, both depending on the nonlinear part of the original equation of motion. Numerical computations of the position correlation function and its spectral density are presented. The spectral density compares favorably with experimental results obtained by Morton using an analog computer method.

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References

  1. R. Zwanzig,Lectures Theoret. Phys. (Boulder) 3:106 (1961).

    Google Scholar 

  2. H. Mori,Progr. Theoret. Phys. 33:423 (1965).

    Google Scholar 

  3. N. Van Kampen, inStochastic Processes in Chemical Physics, K. E. Shuler, ed. (Interscience Publishers, New York, 1969), p. 76.

    Google Scholar 

  4. L. D. Landau and E. M. Lifshitz,Fluid Mechanics (Pergamon Press, London, 1959), Chapter XVII.

    Google Scholar 

  5. L. P. Kadanoff and J. Swift,Phys. Rev. 166:89 (1968); K. Kawasaki,Phys. Rev. 150:291 (1966).

    Google Scholar 

  6. G. W. Ford, M. Kac, and P. Mazur,J. Math. Phys. 6:504 (1965).

    Google Scholar 

  7. J. B. Morton, Ph.D. thesis, The Johns Hopkins University, 1967.

  8. J. B. Morton and S. Corrsin,J. Math. Phys. 10:361 (1969).

    Google Scholar 

  9. J. B. Morton and S. Corrsin,J. Stat. Phys. 2(2):153 (1970).

    Google Scholar 

  10. H. Mori,Progr. Theoret. Phys. 34:399 (1965).

    Google Scholar 

  11. M. Abramovitz and I. A. Stegun (eds.),Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series-25 (U.S. Govt. Printing Office, Washington, D.C., 1964), pp. 686–720.

    Google Scholar 

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Author notes

  1. Mordechai Bixon

    Present address: Department of Chemistry, University of Tel Aviv, Israel

Authors and Affiliations

  1. Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Maryland

    Mordechai Bixon & Robert Zwanzig

Authors
  1. Mordechai Bixon
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  2. Robert Zwanzig
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Additional information

Technical Note BN-674. Research supported in part by NSF grant GP-12591, and in part by PHS Research Grant No. MG16426-02 from the National Institute of General Medical Sciences.

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Bixon, M., Zwanzig, R. Brownian motion of a nonlinear oscillator. J Stat Phys 3, 245–260 (1971). https://doi.org/10.1007/BF01011383

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  • DOI: https://doi.org/10.1007/BF01011383

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