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Long-time behavior of a non-Markovian Brownian oscillator

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  • Stewart, Glen R.

Abstract

A study is made of the relaxation process of a Brownian harmonic oscillator based upon the generalized Langevin equation (GLE). A non-Markovian damping term appears in the GLE in order to satisfy a fluctuation-dissipation relation when the stochastic force is not delta-correlated. If the force auto-correlation function is assumed to be approximated by a decaying exponential, then an equivalent Markovian Fokker-Planck equation may be written down in terms of an extended variable set. The extra variable is eliminated by a projection operator technique to obtain a modified Fokker-Planck equation with correction terms in successive powers of the correlation time that are different from those found by van Kampen and by San Miguel and Sancho. In the limit of small damping rate, an energy transport equation is derived which indicates a systematic increase in relaxation time as the force auto-correlation time increases from zero up to about one-third of the oscillation period. The expansion breaks down for longer correlation times.

Suggested Citation

  • Stewart, Glen R., 1982. "Long-time behavior of a non-Markovian Brownian oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 115(3), pages 519-530.
    • Handle: RePEc:eee:phsmap:v:115:y:1982:i:3:p:519-530
    DOI: 10.1016/0378-4371(82)90037-1
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    References listed on IDEAS

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    1. Rodríguez, R.F. & Van Kampen, N.G., 1976. "Systematic treatment of fluctuations in a nonlinear oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 85(2), pages 347-362.
    2. Skinner, James L. & Wolynes, Peter G., 1979. "Derivation of Smoluchowski equations with corrections for Fokker-Planck and BGK collision models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 96(3), pages 561-572.
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