Monte Carlo Physarum Machine: Characteristics of Pattern Formation in Continuous Stochastic Transport Networks

We present Monte Carlo Physarum Machine (MCPM): a computational model suitable for reconstructing continuous transport networks from sparse 2D and 3D data. MCPM is a probabilistic generalization of Jones's (2010) agent-based model for simulating the growth of Physarum polycephalum (slime mold). We compare MCPM to Jones's work on theoretical grounds, and describe a task-specific variant designed for reconstructing the large-scale distribution of gas and dark matter in the Universe known as the cosmic web. To analyze the new model, we first explore MCPM's self-patterning behavior, showing a wide range of continuous network-like morphologies—called polyphorms—that the model produces from geometrically intuitive parameters. Applying MCPM to both simulated and observational cosmological data sets, we then evaluate its ability to produce consistent 3D density maps of the cosmic web. Finally, we examine other possible tasks where MCPM could be useful, along with several examples of fitting to domain-specific data as proofs of concept.

Anomalous Stochastic Transport of Particles with Self-Reinforcement and Mittag–Leffler Distributed Rest Times

We introduce a persistent random walk model for the stochastic transport of particles involving self-reinforcement and a rest state with Mittag–Leffler distributed residence times. The model involves a system of hyperbolic partial differential equations with a non-local switching term described by the Riemann–Liouville derivative. From Monte Carlo simulations, we found that this model generates superdiffusion at intermediate times but reverts to subdiffusion in the long time asymptotic limit. To confirm this result, we derived the equation for the second moment and find that it is subdiffusive in the long time limit. Analyses of two simpler models are also included, which demonstrate the dominance of the Mittag–Leffler rest state leading to subdiffusion. The observation that transient superdiffusion occurs in an eventually subdiffusive system is a useful feature for applications in stochastic biological transport.

Empirical Lagrangian parametrization for wind-driven mixing of buoyant particles at the ocean surface

Turbulent mixing is a vital component of vertical particulate transport, but ocean global circulation models (OGCMs) generally have low resolution representations of near-surface mixing. Furthermore, turbulence data is often not provided in reanalysis products. We present 1D parametrizations of wind-driven turbulent mixing in the ocean surface mixed layer, which are designed to be easily included in 3D Lagrangian model experiments. Stochastic transport is computed by Markov-0 or Markov-1 models, and we discuss the advantages/disadvantages of two vertical profiles for the vertical diffusion coefficient Kz. All vertical diffusion profiles and stochastic transport models lead to stable concentration profiles for buoyant particles, which for particles with rise velocities of 0.03 and 0.003 m s−1 agree relatively well with concentration profiles from field measurements of microplastics. Markov-0 models provide good model performance for integration timesteps of Δt ≈ 30 seconds, and can be readily applied in studying the behaviour of buoyant particulates in the ocean. Markov-1 models do not consistently improve model performance relative to Markov-0 models, and require an additional parameter that is poorly constrained.

An improved characterisation of regular generalised functions of white noise and an application to singular SPDEs

A characterisation of the spaces $${\mathcal {G}}_K$$ G K and $${\mathcal {G}}_K'$$ G K ′ introduced in Grothaus et al. (Methods Funct Anal Topol 3(2):46–64, 1997) and Potthoff and Timpel (Potential Anal 4(6):637–654, 1995) is given. A first characterisation of these spaces provided in Grothaus et al. (Methods Funct Anal Topol 3(2):46–64, 1997) uses the concepts of holomorphy on infinite dimensional spaces. We, instead, give a characterisation in terms of U-functionals, i.e., classic holomorphic function on the one dimensional field of complex numbers. We apply our new characterisation to derive new results concerning a stochastic transport equation and the stochastic heat equation with multiplicative noise.

Membrane-Suspended Nanopores in Microchip Arrays for Stochastic Transport Recording and Sensing

The transport of nutrients, xenobiotics, and signaling molecules across biological membranes is essential for life. As gatekeepers of cells, membrane proteins and nanopores are key targets in pharmaceutical research and industry. Multiple techniques help in elucidating, utilizing, or mimicking the function of biological membrane-embedded nanodevices. In particular, the use of DNA origami to construct simple nanopores based on the predictable folding of nucleotides provides a promising direction for innovative sensing and sequencing approaches. Knowledge of translocation characteristics is crucial to link structural design with function. Here, we summarize recent developments and compare features of membrane-embedded nanopores with solid-state analogues. We also describe how their translocation properties are characterized by microchip systems. The recently developed silicon chips, comprising solid-state nanopores of 80 nm connecting femtoliter cavities in combination with vesicle spreading and formation of nanopore-suspended membranes, will pave the way to characterize translocation properties of nanopores and membrane proteins in high-throughput and at single-transporter resolution.

Perspectives on the formation of peakons in the stochastic Camassa–Holm equation

A famous feature of the Camassa–Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite-element discretization for it, which we use to explore the formation of peakons. Our simulations using this discretization reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

Higher order schemes in time for the surface quasi-geostrophic system under location uncertainty

<p>In this work we consider the surface quasi-geostrophic (SQG) system under location uncertainty (LU) and propose a Milstein-type scheme for these equations. The LU framework, first introduced in [1], is based on the decomposition of the Lagrangian velocity into two components: a large-scale smooth component and a small-scale stochastic one. This decomposition leads to a stochastic transport operator, and one can, in turn, derive the stochastic LU version of every classical fluid-dynamics system.<span> </span></p><p>    SQG is a simple 2D oceanic model with one partial differential equation, which models the stochastic transport of the buoyancy, and an operator which relies the velocity and the buoyancy.</p><p><span>    </span>For this kinds of equations, the Euler-Maruyama scheme converges with weak order 1 and strong order 0.5. Our aim is to develop higher order schemes in time: the first step is to consider Milstein scheme, which improves the strong convergence to the order 1. To do this, it is necessary to simulate or estimate the Lévy area [2].</p><p><span>    </span>We show with some numerical results how the Milstein scheme is able to capture some of the smaller structures of the dynamic even at a poor resolution.<span> </span></p><p><strong>References</strong></p><p>[1] E. Mémin. Fluid flow dynamics under location uncertainty. <em>Geophysical & Astrophysical Fluid Dynamics</em>, 108.2 (2014): 119-146.<span> </span></p><p>[2] J. Foster, T. Lyons and H. Oberhauser. An optimal polynomial approximation of Brownian motion. <em>SIAM Journal on Numerical Analysis</em> 58.3 (2020): 1393-1421.</p>