Stochastic interpretation of the advection diffusion equation and its relevance to bed load transport
The advection diffusion equation is one of the most widespread equations in physics. It arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Phenomenological laws are usually sufficient to derive this equation and interpret its terms. Stochastic models can also be used to derive it with the significant advantage that they provide information on the statistical properties of particle activity. These models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. Among these stochastic models, the most common approach consists of random walk models. For instance, they have been used to model the random displacement of tracers in river streams. Here we explore an alternative approach, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. Birth-death Markov processes are well-suited to this objective. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received no attention. We therefore look into the possibility of deriving the advection diffusion equation (with a source term) within the framework of birth-death Markov processes. We show that in the continuum limit (when the cell size becomes vanishingly small), we can derive an advection diffusion equation for particle activity. Yet, while this derivation is formally valid in the continuum limit, it runs into difficulty in the practical applications involving cells or meshes of finite length. Indeed, within our stochastic framework, particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particles velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due to velocity fluctuations), with the important consequence that local measurements depend on both the intrinsic properties of particles displacement and the dimensions of the measurement system.