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Abstract To date, there is no consensus on the probability distribution of particle velocities during bedload transport, with some studies suggesting an exponential‐like distribution while others a Gaussian‐like distribution. Yet, the form of this distribution is key for the determination of sediment flux and the dispersion characteristics of tracers in rivers. Combining theoretical analysis of the Fokker‐Planck equation for particle motions, numerical simulations of the corresponding Langevin equation, and measurements of motion in high‐speed imagery from particle‐tracking experiments, we examine the statistics of bedload particle trajectories, revealing a two‐regime distance‐time (L‐Tp) scaling for the particle hops (measured from start to stop). We show that particles of short hop distances scale asL~giving rise to the Weibull‐like front of the hop distance distribution, while particles of long hop distances transition to a different scaling regime ofL~Tpleading to the exponential‐like tail of the hop distance distribution. By demonstrating that the predominance of mostly long hop particles results in a Gaussian‐like velocity distribution, while a mixture of both short and long hop distance particles leads to an exponential‐like velocity distribution, we argue that the form of the probability distribution of particle velocities can depend on the physical environment within which particle transport occurs, explaining and unifying disparate views on particle velocity statistics reported in the literature.
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The Inert Drift Atlas Model
Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles. This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight in Probab Theory Relat Fields 121:577–598, 2001). We study the long-time behavior of the process (V , Z), where V is the velocity of the inert particle and Z is the vector of gaps between successive particles ordered by their relative positions. The system is not hypoelliptic, not reversible, and has singular form interactions. Thus the study of stability behavior of the system requires new ideas. We show that this process has a unique stationary distribution that takes an explicit product form which is Gaussian in the velocity component and exponential in the other components. We also show that convergence in total variation distance to the stationary distribution happens at an exponential rate. We further obtain certain law of large numbers results for the particle locations and intersection local times.
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- PAR ID:
- 10400124
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- ISSN:
- 0010-3616
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00220-022-04589-2
