Stochastic Stokes drift and hypersensitive transport driven by dichotomous noise are theoretically investigated. Explicit mathematical expressions for the asymptotic probability density and drift velocity are derived including the situation in which particles cross unstable fixed points. The results are confirmed by numerical simulations.
We analyse various properties of stochastic Markov processes with multiplicative white noise. We take a single-variable problem as a simple example, and we later extend the analysis to the Landau-Lifshitz-Gilbert equation for the stochastic dynamics
of a magnetic moment. In particular, we focus on the non-equilibrium transfer of angular momentum to the magnetization from a spin-polarised current of electrons, a technique which is widely used in the context of spintronics to manipulate magnetic moments. We unveil two hidden dynamical symmetries of the generating functionals of these Markovian multiplicative white-noise processes. One symmetry only holds in equilibrium and we use it to prove generic relations such as the fluctuation-dissipation theorems. Out of equilibrium, we take profit of the symmetry-breaking terms to prove fluctuation theorems. The other symmetry yields strong dynamical relations between correlation and response functions which can notably simplify the numerical analysis of these problems. Our construction allows us to clarify some misconceptions on multiplicative white-noise stochastic processes that can be found in the literature. In particular, we show that a first-order differential equation with multiplicative white noise can be transformed into an additive-noise equation, but that the latter keeps a non-trivial memory of the discretisation prescription used to define the former.
We perform a time-dependent study of the driven dynamics of overdamped particles which are placed in a one-dimensional, piecewise linear random potential. This set-up of spatially quenched disorder then exerts a dichotomous varying random force on th
e particles. We derive the path integral representation of the resulting probability density function for the position of the particles and transform this quantity of interest into the form of a Fourier integral. In doing so, the evolution of the probability density can be investigated analytically for finite times. It is demonstrated that the probability density contains both a $delta$-singular contribution and a regular part. While the former part plays a dominant role at short times, the latter rules the behavior at large evolution times. The slow approach of the probability density to a limiting Gaussian form as time tends to infinity is elucidated in detail.
In probability theory, the independence is a very fundamental concept, but with a little mystery. People can always easily manipulate it logistically but not geometrically, especially when it comes to the independence relationships among more that tw
o variables, which may also involve conditional independence. Here I am particularly interested in visualizing Markov chains which have the well known memoryless property. I am not talking about drawing the transition graph, instead, I will draw all events of the Markov process in a single plot. Here, to simplify the question, this work will only consider dichotomous variables, but all the methods actually can be generalized to arbitrary set of discrete variables.
We study the arrival time distribution of overdamped particles driven by a constant force in a piecewise linear random potential which generates the dichotomous random force. Our approach is based on the path integral representation of the probabilit
y density of the arrival time. We explicitly calculate the path integral for a special case of dichotomous disorder and use the corresponding characteristic function to derive prominent properties of the arrival time probability density. Specifically, we establish the scaling properties of the central moments, analyze the behavior of the probability density for short, long, and intermediate distances. In order to quantify the deviation of the arrival time distribution from a Gaussian shape, we evaluate the skewness and the kurtosis.
We investigate the all-penetrating drift velocities, due to surface wave motion in an effectively inviscid fluid that overlies a saturated porous bed of finite depth. Previous work in this area either neglects the large-scale flow between layers [Phi
llips (1991)] or only considers the drift above the porous layer [(Monismith (2007)]. We propose a model where flow is described by a velocity potential above the porous layer, and by Darcys law in the porous bed, with derived matching conditions at the interface between the two layers. The damping effect of the porous bed requires a complex wavenumber k and both a vertical and horizontal Stokes drift of the fluid, unlike the solely horizontal drift first derived by Stokes Stokes (1847) in a pure fluid layer. Our work provides a physical model for coral reefs in shallow seas, where fluid drift both above and within the reef is vitally important for maintaining a healthy reef ecosystem [Koehl et al. (1997), Monismith (2007)]. We compare our model with measurements by Koehl & Hadfield (2004) and also explain the vertical drift effects described in Koehl et al. (2007), who measured the exchange between a coral reef layer and the (relatively shallow) sea above.