Author(s): Niannian Fan; Efi Foufoula-Georgiou; Baosheng Wu; Deyu Zhong
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Keywords: Bed load; PDF; Langevin Equation; Fokker-Planck Equation; Hop distance and travel time
Abstract: The Fokker-Planck equation (FPE) has been extensively used to characterize the evolution of the probability density function (PDF) of sediment particles moving above a gravel bed. This equation describes the density of particles at any given point in space and time and its derivation is based on the macroscale statistical properties of particle movement, e. g. the PDF of hop distances. In this study, we use the framework of statistical mechanics to formally connect the motion of individual particles to the behavior of the ensemble. Specially, based on the forces exerted on a single particle, we derive the nonlinear stochastic Langevin equation (LE) to describe the dynamics of a single particle, referring to both the deterministic and stochastic components of forces. Then, the FPE, which describes the evolution of the PDF of the ensemble particle velocities, is derived from the LE. We show that the theoretical PDF obtained by solving the FPE under equilibrium conditions, has exponential form, consistent with the PDF of the simulated particle velocities and that of the experimental data by Roseberry et al. (2012). We also simulate the velocity series of individual particles from the LE and show an emergent nonlinear relationship between hop distances and travel times in agreement with observations.
Year: 2013