Brownian motion of a nonlinear oscillator

Mordechai Bixon*, Robert Zwanzig

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)245-260
Number of pages16
JournalJournal of Statistical Physics
Issue number3
StatePublished - Sep 1971
Externally publishedYes


  • Brownian motion
  • noise
  • nonlinear
  • oscillator
  • transport


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