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Several simple mathematical models for the turbulent di!usion of a passive scalar "eld are developed here with an emphasis on the symbiotic interaction between rigorous mathematical theory (including exact solutions), physical intuition, and numerical simulations. The homogenization theory for periodic velocity "elds and random velocity "elds withâ€¦ (More)

- Paul J. Atzberger, Peter R. Kramer, Charles S. Peskin
- J. Comput. Physics
- 2007

In modeling many biological systems, it is important to take into account flexible structures which interact with a fluid. At the length scale of cells and cell organelles, thermal fluctuations of the aqueous environment become significant. In this work, it is shown how the immersed boundary method of [C.S. Peskin, The immersed boundary method, Acta Num. 11â€¦ (More)

- Jan H. Janse, Tom Aldenberg, Peter R. Kramer
- Hydrobiologia
- 2004

The phosphorus cycle in the ecosystem of the shallow, hypertrophic Loosdrecht lakes (The Netherlands) was simulated by means of the dynamic eutrophication model PCLOOS. The model comprises three algal groups, zooplankton, fish, detritus, zoobenthos, sediment detritus and some inorganic phosphorus fractions. All organic compartments are modelled in twoâ€¦ (More)

- Andrew J. Majda, Peter R. Kramer
- SIAM Journal of Applied Mathematics
- 2004

We illustrate the stochastic mode reduction procedure as formulated recently by Majda, Timofeyev, and Vanden-Eijnden [Comm. Pure Appl. Math., 54 (2001), pp. 891â€“974] (MTV) on the equations of motion underlying various particle-based simulation approaches (such as Stokesian dynamics and Brownian dynamics) and the conceptually distinct dissipative particleâ€¦ (More)

The efficiency with which an incompressible flow mixes a passive scalar field that is continuously replenished by a steady source-sink distribution has been quantified using the suppression of the mean scalar variance below the value it would attain in the absence of the stirring. We examine the relationship this mixing measure has to the effectiveâ€¦ (More)

We examine stochastic coarse-graining strategies for two biomolecular systems. First, we compute the large-scale transport properties of the basic flashing ratchet mathematical model for (Brownian) molecular motors and consider in this light whether the underlying continuous-space, continuous-time Markovian model can be coarse-grained as a discrete-state,â€¦ (More)

We present a new computational method for simulating microscopic processes in physiology which accounts for stochastic thermal uctuations and a dynamically evolving uid environment. This latter feature is a principal advantage of our method relative to Brownian or Stokesian particle dynamics methods. Our approach builds on the \Immersed Boundary Method" ofâ€¦ (More)

- Andrew J. Majda, Peter R. Kramer
- SIAM Journal of Applied Mathematics
- 2004

We apply the formulation of a stochastic mode reduction method developed in a recent paper of Majda, Timofeyev, and Vanden-Eijnden [Comm. Pure Appl. Math., 54 (2001), pp. 891â€“ 974] (MTV) to obtain simplified equations for the dynamics of structures immersed in a thermally fluctuating fluid at low Reynolds (or Kubo) number, as simulated by a recent extensionâ€¦ (More)

Homogenization theory provides a rigorous framework for calculating the effective diffusivity of a decaying passive scalar field in a turbulent or complex flow. We extend this framework to the case where the passive scalar fluctuations are continuously replenished by a source (and/or sink). The basic structure of the homogenized equations carries over, butâ€¦ (More)

- Paul J. Atzberger, Peter R. Kramer
- Physical review. E, Statistical, nonlinear, andâ€¦
- 2007

The basic ingredient of osmotic pressure is a solvent fluid with a soluble molecular species which is restricted to a chamber by a boundary which is permeable to the solvent fluid but impermeable to the solute molecules. For macroscopic systems at equilibrium, the osmotic pressure is given by the classical van 't Hoff law, which states that the pressure isâ€¦ (More)