Abstract
The contraction of the description of Brownian motion from phase space to position space is discussed by means of non-Markovian Langevin equations in position space. A Fokker-Planck equation valid for any time is derived for the harmonic oscillator, and the overdamped, critical, and infradamped cases are discussed. For anharmonic potentials systematic corrections to the Smoluchowski equation are derived. Such corrections can be interpreted in this context as an expansion in powers of the correlation time of the “colored” stochastic noise appearing in the Langevin equation. The breakdown of the Fokker-Planck approximation is also discussed.
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