TY - JOUR
T1 - Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation
AU - Kalinay, Pavol
AU - Percus, Jerome K.
N1 - Funding Information:
Acknowledgements Support from VEGA grant No. 2/0049/12 and CE SAS QUTE project is gratefully acknowledged. P.K. thanks CIMS, New York University for kind hospitality. We are also grateful to the referee for several suggestions and for alerting us to Ref. [20].
PY - 2012/9
Y1 - 2012/9
N2 - A point-like particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered. We present a rigorous mapping of the 1D Fokker-Planck (Kramers) equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x. The result is the Smoluchowski equation, valid in the overdamped limit, m → 0, with a series of corrections expanded in powers of m/γ, γ denotes the friction coefficient. The corrections are determined unambiguously within the recurrence mapping procedure. The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator.
AB - A point-like particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered. We present a rigorous mapping of the 1D Fokker-Planck (Kramers) equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x. The result is the Smoluchowski equation, valid in the overdamped limit, m → 0, with a series of corrections expanded in powers of m/γ, γ denotes the friction coefficient. The corrections are determined unambiguously within the recurrence mapping procedure. The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator.
KW - Confined diffusion
KW - Inertial effects
KW - Mapping
KW - Smoluchowski equation
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U2 - 10.1007/s10955-012-0570-2
DO - 10.1007/s10955-012-0570-2
M3 - Article
AN - SCOPUS:84866563049
SN - 0022-4715
VL - 148
SP - 1135
EP - 1155
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 6
ER -