Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, micro-rheology, molecular motors, and microfluidic / nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, biophysical systems, osmotic phenomena, molecular motor proteins, and microfluidic / nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic effects in physical systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, biophysical systems, osmotic phenomena, molecular motor proteins, and microfluidic / nanofluidic devices. \\

Mathematical and computational approaches for the study of the role of stochastic effects in physical systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, biophysical systems, osmotic phenomena, molecular motor proteins, and microfluidic / nanofluidic devices. \\

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    - Jon Karl Sigurdsson

1. Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (submitted). [PDF]

Mathematical and computational approaches for the study of stochastic phenomena. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, biophysical systems, osmotic phenomena, molecular motor proteins, and microfluidic / nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena arising in a variety of physical systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic / nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of the role of stochastic effects in biology, physics, and engineering. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena in biology, physics, and engineering. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Grant DMS-0635535*.

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We gratefully acknowledge the following sources of support:\\

1. Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, Vol. 106 No. 9 pp. 2989-2994, (2009), [PDF] [DOI]

1. Hybrid Elastic / Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins, Naji, A., Atzberger, P.J. and Brown, F.L.H., Phys. Rev. Lett. 102, 138102, (2009). [PDF] [DOI]

1. Hybrid Elastic / Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins, Naji, A., Atzberger, P.J. and Brown, F.L.H., Phys. Rev. Lett. 102, 138102, (2009). [PDF] [PRL] [DOI]

• Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (preprint). [PDF]

1. Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (preprint). [PDF]

- Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations (October 2009) [IMA video / slides].
- Multiscale Modeling and Simulation of Soft Materials (November 2008) [IMA video / slides].\\

- Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations [IMA video / slides] (October 2009).\\

1. Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (preprint). [PDF] Δ

Talks (Video)

- Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations [IMA video / slides] [Slides PDF] (October 2009).\\

- Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations[IMA video / slides] [Slides PDF] (October 2009).
- Multiscale Modeling and Simulation of Soft Materials [IMA video / slides] (November 2008).\\

- Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations [IMA video / slides] [Slides PDF] (October 2009).\\

- Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations [IMA video / slides] PDF Δ (October 2009).\\

- Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations [IMA video / slides] (October 2009).\\

Talks on Research (Video)

Talks on Video

Talks (Video)

Talks Videoed

Video of Talks

- Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations [IMA video / slides] (October 2009).\\

- Multiscale Modeling and Simulation of Soft Materials [IMA video / slides] (November 2008).\\

- Multiscale Modeling and Simulation of Soft Materials [IMA video / slides] (November 2009).
- Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations [IMA video / slides] (October 2009).

Multiscale Modeling and Simulation of Soft Materials (March 2009)[IMA video / slides].

Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations (October 2009) [IMA video / slides].

Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations [IMA video / slides].

Mathematical and computational approaches for the study of stochastic phenomena. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

1. Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, Vol. 106 No. 9 pp. 2989-2994, (2009), [DOI] [PDF]

    - Justin Shrake

• Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2009), (submitted). [PDF]

Mathematical and computational approaches for the study of stochastic phenomena in physical systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of the stochastic phenomena in physical systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Application areas in which we work include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena in physics, biology, and engineering. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Application areas in which we work include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena in systems from physics, biology, and engineering. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Application areas in which we work include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Application areas in which we work include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

For a more detailed discussion of our work see the research summaries. \\

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. We work in application areas which include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Our work is motivated by specific problems from application areas which include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Our work is motivated by specific problems from application areas which include: soft condensed materials and complex fluids, osmotic phenomena, molecular motor proteins, and microfluidic/nanofluidic devices.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Our work is motivated by specific problems from application areas which include: soft condensed materials and complex fluids, osmotic phenomena, and microfluidic/nanofluidic devices.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical physics, stochastic analysis, and scientific computing. Our work is motivated by specific application areas, including: soft condensed materials and complex fluids, osmotic phenomena, and microfluidic/nanofluidic devices.

Mathematical and computational approaches for the study of the statistical physics of biological and engineered systems. We work in the areas of stochastic analysis, numerical analysis, and scientific computing. Specific application areas include: Computational Fluid Dynamics, Soft Condensed Materials and Complex Fluids, Osmotic Phenomena, and Microfluidic Devices.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are essentially spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we use notions related to the fluctuation-dissipation principle of statistical mechanics. We are using these approaches to develop a variety of stochastic numerical methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are essentially spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we use notions related to the fluctuation-dissipation principle of statistical mechanics. We are utilizing these approaches to develop numerical methods for the study of spatially extended stochastic systems.

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions or domains of rather simple geometry. We are developing alternative finite difference numerical methods for the approximation of SPDEs utilizing ideas from statistical mechanics. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge these methods address for adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. As a demonstration we present results for a stochastic reaction-diffusion system with two chemical species undergoing Gray-Scott reactions subject to intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize a quad-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: Computational Fluid Dynamics, Soft Condensed Materials and Complex Fluids, Osmotic Phenomena, and Microfluidic Devices.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: Computational Fluid Dynamics, Soft Condensed Materials and Complex Fluids, Osmotic Phenomena, Microfluidic Devices.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include:

• Soft Condensed Materials and Complex Fluids
• Computational Fluid Dynamics
• Osmotic Phenomena
• Microfluidic Devices.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft condensed materials, osmotic phenomena, and microfluidic devices. \\

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft materials (polymeric fluids, lipid bilayer membranes), osmotic phenomena, and microfluidic devices. \\

1. Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2009), (submitted). [PDF]

Multiscale Modeling and Simulation of Soft Materials [IMA video / slides].

\\

See the research summaries for more details.\\

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, molecular motor proteins, and microfluidic devices.

See the research summaries for more details.

1. Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (preprint). [PDF] Δ

1. Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (preprint). [PDF] Δ

Office of Research (Click on File Below)

Office of Research Δ \\

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....

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we use notions related to the fluctuation-dissipation principle of statistical mechanics. We are utilizing these approaches to develop numerical methods for the study of spatially extended stochastic systems.

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions and domains of rather simple geometry. We are developing alternative finite difference numerical methods for the approximation of SPDEs utilizing ideas from statistical mechanics. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge these methods address for adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. As a demonstration we present results for a stochastic reaction-diffusion system with two chemical species undergoing Gray-Scott reactions subject to intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize a quad-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

The Stochastic Immersed Boundary Method (SIB) is a numerical approach for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes.

To study systems in regimes applicable to biological systems and microscopic devices we are developing theory to describe microscopic osmotic phenomena both in equilibrium and non-equlibrium settings. Some recent work includes studying how osmotic pressures generated by polymer solutes depend on polymer topology, stiffness, and excluded volume. In related work, a pumping mechanism for a microfluidic device has been proposed which uses reversible chemical reactions to drive fluid flows.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we use notions related to the fluctuation-dissipation principle of statistical mechanics. We are utilizing these approaches broadly to develop numerical methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we use notions related to the fluctuation-dissipation principle of statistical mechanics. These approaches are also being used more broadly to develop numerical methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we use notions related to the fluctuation-dissipation principle of statistical mechanics. We are also utilizing these approaches more broadly to develop numerical methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we utilize ideas related to the fluctuation-dissipation principle of statistical mechanics. We are also utilizing these approaches more broadly to develop numerical methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +.

Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we utilize ideas related to the fluctuation-dissipation principle of statistical mechanics. We are also utilizing these approaches more broadly to develop numerical methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we utilize ideas related to the fluctuation-dissipation principle of statistical mechanics. We are also utilizing these approaches more broadly to develop numerical methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we utilize an approach similar to the fluctuation-dissipation principle of statistical mechanics. We are also utilizing these approaches more broadly to develop numerical methods for the study of spatially extended stochastic systems.

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions and domains of rather simple geometry. We are developing alternative finite difference numerical methods for the approximation of SPDEs utilizing ideas from statistical mechanics. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge these methods address for adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. As a demonstration we present results for a stochastic reaction-diffusion system with two chemical species undergoing Gray-Scott reactions subject to intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize an oct-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we utilize an approach similar to the fluctuation-dissipation principle of statistical mechanics. This approach is being applied more broadly to develop numerical methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. We utilize an approach similar to the fluctuation-dissipation principle of statistical mechanics to derive consistent discretizations of the stochastic driving field at coarse-refined interfaces. This approach is being applied more broadly to develop methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipation based discretized stochastic driving field. By using the fluctuation-dissipation principle we obtain discretizations for stochastic numerical methods which exhibit the required uncorrelated equilibrium fluctuations at coarse-refined interfaces. This approach is being applied more broadly to develop methods for the study of spatially extended stochastic systems.

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions and domains of rather simple geometry. We are developing alternative finite difference numerical methods for the approximation of SPDEs utilizing ideas from statistical mechanics. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge these methods address for adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. As a demonstration we present results for a stochastic reaction-diffusion system with intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize an oct-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions and domains of rather simple geometry. We are developing alternative finite difference numerical methods for the approximation of SPDEs utilizing ideas from statistical mechanics. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge these methods address for adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. Below we present results for a stochastic reaction-diffusion system with intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize an oct-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions and domains of rather simple geometry. We are developing alternative finite difference numerical methods for the approximation of SPDEs utilizing ideas from statistical mechanics. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge we address for adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. Below we present results for a stochastic reaction-diffusion system with intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize an oct-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions and domains of rather simple geometry. We are developing alternative finite difference numerical methods for the approximation of SPDEs utilizing ideas from statistical mechanics. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge we address when utilizing adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. Below we present results for a stochastic reaction-diffusion system with intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize an oct-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We work in the areas of statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, molecular motor proteins, and microfluidic devices. See our research summaries for more details.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We utilize ideas from statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, molecular motor proteins, and microfluidic devices. See our research summaries for more details.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We develop approaches using ideas from statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, molecular motor proteins, and microfluidic devices. See our research summaries for more details.

Mathematical and computational approaches for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, molecular motor proteins, and microfluidic devices. See our research summaries for more details.

• On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF] [DOI]

Publications

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Publications\\\\

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Publications\\

Publications\\

[AVI Movie Unknot] [AVI Movie Trefoil Knot] [AVI Movie Figure Eight Knot], (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling [AVI Movie Membrane Sheet], [AVI Movie Membrane Vesicle], [AVI Movie Membrane Vesicle Formation], (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load [AVI Movie Motor with No Load], [AVI Movie Motor with Intermediate Load], [AVI Movie Motor with Large Load]. In collaboration with the Brown Group, Department of Chemistry, dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations.

• Hybrid Elastic / Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins, Naji, A., Atzberger, P.J. and Brown, F.L.H., Phys. Rev. Lett. 102, 138102, (2009). [PDF] [DOI]

• On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF] [DOI]
• A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Atzberger, P.J., Kramer, P.R., and Peskin, C.S., J. Comp. Phys., Vol. 224, Iss. 2, (2007). [PDF] [DOI]
• Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations (brief introduction), Atzberger, P.J., Kramer, P.R., and Peskin, C.S., (proceedings of ICIAM 2007). [PDF] [DOI]
• A Note on the Correspondence of the Immersed Boundary Method with Thermal Fluctuations To Stokesian-Brownian Dynamics, Atzberger, P.J., Physica D, Vol. 226, Iss. 2, 15, pg. 144-150, (2007). [PDF] [DOI]
• Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (preprint). [PDF] Δ

• A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., Physica D: Nonlinear Phenomena, Vol. 238, Iss. 14, July, pp. 1168-1179, (2009),[PDF] [DOI]
• Theoretical Framework for Microscopic Osmotic Phenomena, Atzberger, P.J. and Kramer, P.R. , Phys. Rev. E, 75, 1, (2007). PDF] [DOI]

• Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, Vol. 106 No. 9 pp. 2989-2994, (2009), [DOI] [PDF]
• A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether, Atzberger, P.J. and Peskin, C.S., Bull. Math. Biol., vol. 68, no. 1, pp. 131-160, (2006). [PDF] [DOI]

• Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2009), (submitted). [PDF]

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces if the discretization of the stochastic driving field is not handled carefully this introduces artificial spatial correlations. Below is show the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipation based discretized stochastic driving field. By using the fluctuation-dissipation principle we obtain discretizations for stochastic numerical methods which exhibit the required uncorrelated equilibrium fluctuations at coarse-refined interfaces. This approach is being applied more broadly to develop methods for the study of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces if the discretization of the stochastic driving field is not handled carefully this introduces artificial spatial correlations. Below is show the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipation based discretized stochastic driving field. By using the fluctuation-dissipation principle we obtain discretizations for stochastic numerical methods which exhibit the required uncorrelated equilibrium fluctuations at coarse-refined interfaces. This approach is being applied more broadly to develop methods for the study of a wide variety of spatially extended stochastic systems.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces if the discretization of the stochastic driving field is not handled carefully this introduces artificial spatial correlations. Below is show the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipation based discretized stochastic driving field. By using the fluctuation-dissipation principle we obtain discretizations for stochastic numerical methods which exhibit the required uncorrelated equilibrium fluctuations at coarse-refined interfaces.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces if the discretization of the stochastic driving field is not handled carefully this introduces artificial spatial correlations. Below is show the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipation based discretized stochastic driving field. By using the fluctuation-dissipation principle we obtain discretizations which exhibit the required uncorrelated equilibrium fluctuations at coarse-refined interfaces.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces if the discretization of the stochastic driving field is not handled carefully this introduces artificial spatial correlations. Below is show the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipation based discretized stochastic driving field. By using the fluctuation-dissipation principle discretizations we obtain the required uncorrelated equilibrium fluctuations at coarse-refined interfaces.

An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces if the discretization of the stochastic driving field is not handled carefully this introduces artificial spatial correlations. Below is show the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipatoin based discretized stochastic driving field. By using the fluctuation-dissipation principle discretizations we obtain the required uncorrelated equilibrium fluctuations at coarse-refined interfaces.

An important feature of the method is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces if the discretization of the stochastic driving field is not handled carefully this introduces artificial spatial correlations. Below is show the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipatoin based discretized stochastic driving field. By using the fluctuation-dissipation principle discretizations we obtain the required uncorrelated equilibrium fluctuations at coarse-refined interfaces.

An important feature of the method is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces if the discretization of the stochastic driving field is not handled carefully this introduces artificial spatial correlations. Below is show the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipatoin based discretized stochastic driving field. By using the fluctuation-dissipation principle discretizations we obtain the required uncorrelated equilibrium fluctuations.

An important feature of the method is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are spatially uncorrelated at any instant. At coarse-refined interfaces if the discretization of the stochastic driving field is not handled carefully this introduces artificial spatial correlations. Below is show the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a "white-noise" discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our fluctuation-dissipatoin based discretized stochastic driving field.

An important feature of the method is how the the stochastic driving field is treated at the coarse-refined interface. Below we show results for using uncorrelated random fluxes, averaged fluxes, and Fluctuation-Dissipation based fluxes:

- Statistical Mechanics.
- Stochastic Analysis.
- Scientific Computing.
- Applications in Physics, Biology, Engineering.\\

• Statistical Mechanics.
• Stochastic Analysis.
• Scientific Computing.
• Applications in Physics, Biology, Engineering.

Methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, molecular motor proteins, and microfluidic devices. See our research summaries for more details.

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Our research concerns the development of methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, molecular motor proteins, and microfluidic devices. See our research summaries for more details.

Our research concerns the development of methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics, stochastic analysis, and scientific computing. Specific application areas include: soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, molecular motor proteins, and microfluidic devices. See our research summaries for more details concerning specific projects.

Our research concerns the development of methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics, stochastic analysis, and scientific computing. Specific application areas in which we work include: molecular motor proteins, soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices. See our research summaries for more details concerning specific projects.

Our research concerns the development of methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics and stochastic analysis. We also are interested in the development of stochastic numerical methods and high-performance scientific computing. Specific application areas in which we work include: molecular motor proteins, soft materials, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices. See our research summaries for more details concerning specific projects.

Our research concerns the development of methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics and stochastic analysis. We also develop stochastic numerical methods for scientific computing. Specific application areas include: molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.

Our research aims to develop methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics and stochastic analysis. We also develop stochastic numerical methods for scientific computing. Specific application areas include: molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.

In our research we develop methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics and stochastic analysis. We also develop stochastic numerical methods for scientific computing. Specific application areas include: molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.

Our research is concerned with the development of methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics and stochastic analysis. We also develop stochastic numerical methods for scientific computing. Specific application areas include: molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.

Our research concerns the development of methodologies for the study of stochastic phenomena in biological and engineered systems. We utilize approaches from statistical mechanics and stochastic analysis. We also develop stochastic numerical methods for scientific computing. Specific application areas include: molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.

Our research aims to develop methodologies for the study of stochastic phenomena in biological and engineered systems through the utilization of approaches from stochastic analysis and through the development of stochastic numerical methods for scientific computing. Specific application areas include: molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.

Our research aims to develop methodologies for the study of stochastic phenomena in a wide range of physical systems through the utilization of approaches from stochastic analysis and through the development of stochastic numerical methods for scientific computing. Specific application areas include: molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.

Mathematical and computational approaches for the study of the statistical mechanics of biological and engineered systems. Our research aims to develop methodologies for the study of stochastic phenomena in a wide range of physical systems through the utilization of approaches from stochastic analysis and through the development of stochastic numerical methods for scientific computing. Specific application areas motivating this work include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.

Mathematical and computational approaches for the study of the statistical mechanics of biological and engineered systems. Application areas include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices. Our mathematical work aims to develop general methodologies for the study of stochastic phenomena in physical systems through the utilization of approaches from stochastic analysis and through the development of stochastic numerical methods for scientific computing.

Mathematical and computational approaches for the study of the statistical mechanics of biological and engineered systems. Application areas include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices. Our mathematical work aims to develop general methodologies for the study of stochastic phenomena in physical systems through the introduction of new approaches from stochastic analysis and through the new stochastic numerical methods for scientific computing.

Mathematical and computational approaches for the study of the statistical mechanics of biological and engineered systems. Application areas include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices. Our work also aims to develop general scientific computing methodology through the introduction of new stochastic numerical methods. \\

Mathematical and computational approaches for the study of the statistical mechanics of biological and engineered systems. Application areas include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices. \\

Mathematical and computational approaches to the study of the statistical mechanics of biological and engineered systems. Application areas include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices. \\

Mathematical analysis and computational approaches for the study of the statistical mechanics of biological and engineered systems. Application areas include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices. \\

Mathematical analysis and computational approaches for the study of the statistical mechanics of biological and engineered systems. Application areas include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.\\

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions and domains of rather simple geometry. Utilizing ideas from statistical mechanics we are developing alternative finite difference numerical methods for the approximation of SPDEs. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge we address when utilizing adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. Below we present results for a stochastic reaction-diffusion system with intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize an oct-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions and domains of rather simple geometry. Utilizing ideas from statistical mechanics we are developing alternative finite difference numerical methods for the approximation of SPDEs. Our approach allows for general boundary conditions, complex domain geometries, the use of adaptive spatial meshes. A key challenge we address when utilizing adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. Below we present results for a stochastic reaction-diffusion system with intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize an oct-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions and domains of rather simple geometry. Utilizing ideas from statistical mechanics we are developing alternative finite difference numerical methods for the approximation of SPDEs. Our approach allows for general boundary conditions, complex domain geometries, the use of adaptive spatial meshes. A key challenge we address when utilizing adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. Below we present results for a 2D stochastic reaction-diffusion system with intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize an oct-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

Stochastic partial differential equations pose significant mathematical and computational challenges not present in the corresponding deterministic setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs and often spectral methods are employed. Utilizing ideas from statistical mechanics we are developing alternative finite difference numerical methods for the approximation of SPDEs allowing for adaptive spatial meshes. A key challenge when utilizing adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces. Below we present results for a 2D stochastic reaction-diffusion system with intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize an oct-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.

Stochastic partial differential equations pose significant mathematical and computational challenges not present in the corresponding deterministic setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs and often spectral methods are employed. Utilizing ideas from statistical mechanics we are developing alternative finite difference numerical methods for the approximation of SPDEs allowing for adaptive spatial meshes. A key c

[Research Summaries]

[Research Project Summaries]

Mathematical analysis and computational methods for the study of the statistical mechanics of biological and engineered systems. Application areas include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices.

Mathematical analysis and computational methods for the study of the statistical mechanics of biological and engineered systems. Application areas include molecular motor proteins, soft matter physics, polymeric fluids, lipid bilayer membranes, fluctuating hydrodynamics, osmotic phenomena, and microfluidic devices. \\

1. Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2009), (submitted). [PDF]

1. Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Conditions, Atzberger, P.J., (2009), (preprint). [PDF] Δ

1. A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., Physica D: Nonlinear Phenomena, Vol. 238, Iss. 14, July, pp. 1168-1179, (2009),[PDF] [DOI]
2. Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, Vol. 106 No. 9 pp. 2989-2994, (2009), [DOI] [PDF]

1. A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., Physica D: Nonlinear Phenomena, Vol. 238, Iss. 14, July, pp. 1168-1179, (2009), [PDF] [DOI]
2. Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, Vol. 106 No. 9 pp. 2989-2994, (2009),

[DOI] [PDF]

1. Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Condition, Atzberger, P.J., (2009), (preprint). [PDF] Δ

1. Stochastic Eulerian-Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations and Shear Boundary Condition, Atzberger, P.J., (2009), (preprint / rough draft). [PDF] Δ

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[AVI Movie Unknot] [AVI Movie Trefoil Knot] [AVI Movie Figure Eight Knot], (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling [AVI Movie Membrane Sheet], [AVI Movie Membrane Vesicle], [AVI Movie Membrane Vesicle Formation], (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load [AVI Movie Motor with No Load], [AVI Movie Motor with Intermediate Load], [AVI Movie Motor with Large Load]. In collaboration with the Brown Group, Department of Chemistry, dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see the publications section.

[AVI Movie Unknot] [AVI Movie Trefoil Knot] [AVI Movie Figure Eight Knot], (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling [AVI Movie Membrane Sheet], [AVI Movie Membrane Vesicle], [AVI Movie Membrane Vesicle Formation], (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load [AVI Movie Motor with No Load], http://www.math.ucsb.edu/~atzberg/movies/motor_moderate_load.avi[AVI Movie Motor with Intermediate Load] , [AVI Movie Motor with Large Load]. In collaboration with the Brown Group, Department of Chemistry, dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see the publications section.

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In the figure some simulations demonstrating the methodology for a few basic physical systems are shown. Click on the images to play the associated movies. From top to bottom are: (i) polymer knot simulations showing SIB preservation of knot topology without the need for excluded volume interactions [AVI Movie Unknot] [AVI Movie Trefoil Knot] [AVI Movie Figure Eight Knot].

, (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling, (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load. In collaboration with the Brown Group, Department of Chemistry, dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see the publications section.

    - Per Danzl (co-advisor)
    - David Boy (co-advisor)

    - Per Danzl (co-advisor with ME)
    - David Boy (co-advisor with ME)

    - David Boy (co-advisor)

    - Per Danzl (co-advisor)

Aptamers are short segments of DNA or RNA which bind to proteins. In applications, specific aptamers (nucleic acid sequences) are sought which bind to a desired target protein with high affinity and specificity. Applications include purification methods for proteins, analytics in biosensor devices, and target validation in pharmaceutical drug development. Obtaining aptamers with strong affinity and specificity binding a given target molecule poses a significant challenge in practice. In collaboration with the H.T. Soh Laboratory, Department of Mechanical Engineering along with the undergraduate Joe Rudzinski (CCS Program, UCSB) mathematical approaches are being used to study experimental methods for the selection of DNA-Aptamers from a library of random sequences.

References:
-Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, (2009) (in press) [DOI] [PDF]
- PNAS paper was highlighted by Nature Publishing in Sci-Biz Section [Attach:sciBiz.pdf Δ | PDF].

Aptamers are short segments of DNA or RNA which bind to proteins. In applications, specific aptamers (nucleic acid sequences) are sought which bind to a desired target protein with high affinity and specificity. Applications include purification methods for proteins, analytics in biosensor devices, and target validation in pharmaceutical drug development. Obtaining aptamers with strong affinity and specificity binding a given target molecule poses a significant challenge in practice. In collaboration with the H.T. Soh Laboratory, Department of Mechanical Engineering along with the undergraduate Joe Rudzinski (CCS Program, UCSB) mathematical approaches are being used to study experimental methods for the selection of DNA-Aptamers from a library of random sequences.

1. On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF] [DOI]
2. Error Analysis of a Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations , Atzberger, P.J. and Kramer, P.R., Mathematics and Computers in Simulation, Vol. 79, Iss. 3, pg. 379 -- 408, (2008). [PDF] [DOI]

1. Hybrid Elastic / Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins, Naji, A., Atzberger, P.J. and Brown, F.L.H., Phys. Rev. Lett. 102, 138102, (2009). [PDF] [DOI]
2. Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2009), (submitted). [PDF] Δ
3. A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., (2009), (accepted). [PDF] [DOI]
4. Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, (2009) (in press) [DOI] [PDF]

    - Pat Plunkett

  classes taught 2003 - 2007

  course website

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  course website

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Attach:cv2.pdf Δ

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(:imgmapend:)

• Growth of a noise-induced spatial pattern resolved with adaptive meshing [AVI Movie].

We gratefully acknowledge the following source of support:\\

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For a more in-depth discussion of this work and information about additional projects please see the publications section.

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We gratefully acknowledge the following source of support:

A recent talk on the Stochastic Immersed Boundary Method can be found here [IMA video / slides].

  Postdoctoral Researchers:
    - Chetan Pahlajani

  Graduate Students:
    - David Valdman
    - Per Danzl (co-advisor with Jeff Moehlis, IGERT)

  Undergraduate Students:
    - Joe Rudzinski

  Administration:
    - Mathematical Finance Majors (Undergraduate Faculty Adviser)

  Numerical Analysis I (Fall 2008)
  TR 9:30am - 10:45am.
  course website

  Introduction to Mathematical Biology (Winter 2008)
  TR 8:00am - 9:15pm, Old Little Theatre, Rm. 160B.
  course website

  Finite Element Methods (Fall 2007)
  TR 11:00am - 12:15pm, South Hall Rm. 6635.
  course website

  Introduction to Complex Variables (Fall 2007)
  TR 8:00am - 9:15pm, GIRV Rm. 1116.
  course website

  Additional Class Websites:
  classes taught 2003 - 2007

  Mathematical Finance Course Notes and Resources:

  Numerical Analysis I (Summer 2009)  
  MTWR 12:30pm - 1:35pm.
  course website

  Finite Difference Methods for Partial Differential Equations (Spring 2009)
  TR 11:00am - 12:15pm.
  course website

  Numerical Analysis II (Winter 2009)
  TR 9:30am - 10:45am.
  course website

  Introduction to Stochastic Analysis and Applications (Fall 2008)
  TR 11:00am - 12:15pm.
  course website

   Numerical Analysis I (Fall 2008)
   TR 9:30am - 10:45am.
   course website

   Introduction to Mathematical Biology (Winter 2008)
   TR 8:00am - 9:15pm, Old Little Theatre, Rm. 160B.
   course website

   Finite Element Methods (Fall 2007)
   TR 11:00am - 12:15pm, South Hall Rm. 6635.
   course website

   Introduction to Complex Variables (Fall 2007)
   TR 8:00am - 9:15pm, GIRV Rm. 1116.
   course website

Additional Class Websites:

   classes taught 2003 - 2007

Postdoctoral Researchers:

  - Chetan Pahlajani

Graduate Students:

  - David Valdman
  - Per Danzl (co-advisor with Jeff Moehlis, IGERT)

Undergraduate Students:

  - Joe Rudzinski

Administration:

  - Mathematical Finance Majors (Undergraduate Faculty Adviser)

   Mathematical Finance Course Notes and Resources: 
  course notes and supplemental materials

   Numerical Analysis I 
        (Summer 2009)
        course website
        MTWR 12:30pm - 1:35pm.

Mathematical Finance Course Notes and Resources:

course notes and supplemental materials

Paul J. Atzberger

+Paul J. Atzberger+

Stochastic Immersed Boundary Methods / Computational Fluid Dynamics:

Osmotic Phenomena / Microfluidics:

Attach:osmotic_montage.png Δ

Applications in Molecular Biology and Biotechnology:

Attach:molBio_montage.png Δ

Spatially Adaptive Numerical Methods for Stochastic Partial Differential Equations:

Numerical Codes:

Stochastic Immersed Boundary Method (SIB) : Release 1.0 (alpha)

[Research Summary]

1. On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF] [DOI]

1. A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., (2008), (accepted). [PDF] [DOI]
2. Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, (in press) [DOI] [PDF]
3. On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF]? [DOI]
4. Error Analysis of a Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations , Atzberger, P.J. and Kramer, P.R., Mathematics and Computers in Simulation, Vol. 79, Iss. 3, pg. 379 -- 408, (2008). [PDF] [DOI]
5. A Note on the Correspondence of the Immersed Boundary Method with Thermal Fluctuations To Stokesian-Brownian Dynamics, Atzberger, P.J., Physica D, Vol. 226, Iss. 2, 15, pg. 144-150, (2007). [PDF] [DOI]
6. Velocity Correlations of a Thermally Fluctuating Brownian Particle: A Novel Model of the Hydrodynamic Coupling , Atzberger, P.J., Phys. Lett. A, Vol. 351, Iss. 4-5, 6, March, pp. 225-230, (2006). [PDF] [DOI]
7. A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether, Atzberger, P.J. and Peskin, C.S., Bull. Math. Biol., vol. 68, no. 1, pp. 131-160, (2006). [PDF] [DOI]

1. Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations (brief introduction), Atzberger, P.J., Kramer, P.R., and Peskin, C.S., (proceedings of ICIAM 2007). [PDF] [DOI]
2. Theoretical Framework for Microscopic Osmotic Phenomena, Atzberger, P.J. and Kramer, P.R. , Phys. Rev. E, 75, 1, (2007). PDF] [DOI]
3. Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2008), (submitted). [PDF] Δ
4. Hybrid Elastic / Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins, Naji, A., Atzberger, P.J. and Brown, F.L.H., Phys. Rev. Lett. 102, 138102, (2009). [PDF] [DOI]

Attach:NSF_logo.gif Δ

*This material is based upon work supported by the National Science Foundation under Grant No. DMS-0635535. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

Numerical Codes:

Class Websites and Course Notes:

Advising:

Research Summaries:

1. A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Atzberger, P.J., Kramer, P.R., and Peskin, C.S., J. Comp. Phys., Vol. 224, Iss. 2, (2007). [PDF] [DOI]

1. A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Atzberger, P.J., Kramer, P.R., and Peskin, C.S., J. Comp. Phys., Vol. 224, Iss. 2, (2007). [PDF] [DOI]
2. Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations (brief introduction), Atzberger, P.J., Kramer, P.R., and Peskin, C.S., (proceedings of ICIAM 2007). [PDF] [DOI]
3. Theoretical Framework for Microscopic Osmotic Phenomena, Atzberger, P.J. and Kramer, P.R. , Phys. Rev. E, 75, 1, (2007). [PDF] [DOI]
4. Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2008), (submitted). [PDF]
5. Hybrid Elastic / Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins, Naji, A., Atzberger, P.J. and Brown, F.L.H., Phys. Rev. Lett. 102, 138102, (2009). [PDF] [DOI]
6. A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., (2008), (accepted). [PDF] [DOI]
7. Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, (in press) [DOI]
8. On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF] [DOI]
9. Error Analysis of a Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations , Atzberger, P.J. and Kramer, P.R., Mathematics and Computers in Simulation, Vol. 79, Iss. 3, pg. 379 -- 408, (2008). [PDF] [DOI]
10. A Note on the Correspondence of the Immersed Boundary Method with Thermal Fluctuations To Stokesian-Brownian Dynamics, Atzberger, P.J., Physica D, Vol. 226, Iss. 2, 15, pg. 144-150, (2007). [PDF] [DOI]
11. Velocity Correlations of a Thermally Fluctuating Brownian Particle: A Novel Model of the Hydrodynamic Coupling , Atzberger, P.J., Phys. Lett. A, Vol. 351, Iss. 4-5, 6, March, pp. 225-230, (2006). [PDF] [DOI]
12. A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether, Atzberger, P.J. and Peskin, C.S., Bull. Math. Biol., vol. 68, no. 1, pp. 131-160, (2006). [PDF] [DOI]

Attach:SIB_Schematic.png Δ

Multiscale Modeling and Simulation of Soft-Matter Materials [IMA video / slides].

1. item
2. item
3. item

Stochastic Analysis, Numerical Analysis, Scientific Computing. Applications in Physics, Engineering, and Biology.
[Research Summary]

Department of Mathematics
Department of Mechanical Engineering
PhD Mathematics · Courant Institute, New York University · 2003

Department of Mathematics
Department of Mechanical Engineering \\

Assistant Professor\\

Department of Mathematics
Department of Mechanical Engineering \\

Paul J. Atzberger

Assistant Professor

PhD Mathematics · Courant Institute, New York University · 2003

Research Interests

Recent Talks

Publications:

Blank homepage.

Paul J. Atzberger

Assistant Professor

Department of Mathematics Department of Mechanical Engineering

PhD Mathematics · Courant Institute, New York University · 2003

Research Interests

Stochastic Analysis, Numerical Analysis, Scientific Computing. Applications in Physics, Engineering, and Biology.

[Research Summary]

SIB Schematic

Recent Talks

Multiscale Modeling and Simulation of Soft-Matter Materials [IMA video / slides] .

Publications:

1. A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Atzberger, P.J., Kramer, P.R., and Peskin, C.S., J. Comp. Phys., Vol. 224, Iss. 2, (2007). [PDF] [DOI]

2. Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations (brief introduction), Atzberger, P.J., Kramer, P.R., and Peskin, C.S., (proceedings of ICIAM 2007). [PDF] [DOI]

3. Theoretical Framework for Microscopic Osmotic Phenomena, Atzberger, P.J. and Kramer, P.R. , Phys. Rev. E, 75, 1, (2007). [PDF] [DOI]

4. Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2008), (submitted). [PDF]

5. Hybrid Elastic / Discrete-Particle Approach to Biomembrane Dynamics with Application to the Mobility of Curved Integral Membrane Proteins, Naji, A., Atzberger, P.J. and Brown, F.L.H., Phys. Rev. Lett. 102, 138102, (2009). [PDF] [DOI]

6. A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., (2008), (accepted). [PDF] [DOI]

7. Micromagnetic Selection of Aptamers in Microfluidic Channels, Lou, X., Qian, J., Yi, X., Viel, L., Gerdon, A.E, Lagally, E.T, Atzberger, P.J., Heeger, A.J., and Soh, H.T., PNAS, (in press) [DOI]

8. On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF] [DOI]

9. Error Analysis of a Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations , Atzberger, P.J. and Kramer, P.R., Mathematics and Computers in Simulation, Vol. 79, Iss. 3, pg. 379 -- 408, (2008). [PDF] [DOI]

10. A Note on the Correspondence of the Immersed Boundary Method with Thermal Fluctuations To Stokesian-Brownian Dynamics, Atzberger, P.J., Physica D, Vol. 226, Iss. 2, 15, pg. 144-150, (2007). [PDF] [DOI]

11. Velocity Correlations of a Thermally Fluctuating Brownian Particle: A Novel Model of the Hydrodynamic Coupling , Atzberger, P.J., Phys. Lett. A, Vol. 351, Iss. 4-5, 6, March, pp. 225-230, (2006). [PDF] [DOI]

12. A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether, Atzberger, P.J. and Peskin, C.S., Bull. Math. Biol., vol. 68, no. 1, pp. 131-160, (2006). [PDF] [DOI]

We gratefully acknowledge the following source of support: SIB Schematic Grant DMS-0635535*.

• This material is based upon work supported by the National Science Foundation under Grant No. DMS-0635535. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

Numerical Codes:

Stochastic Immersed Boundary Method (SIB) : Release 1.0 (alpha)

Class Websites and Course Notes:

Mathematical Finance Course Notes and Resources:

    *

          course notes and supplemental materials



    *

      Numerical Analysis I

      (Summer 2009)
    *

          course website
    *

          MTWR 12:30pm - 1:35pm.


    *

      Finite Difference Methods for Partial Differential Equations

      (Spring 2009)
    *

          course website
    *

          TR 11:00am - 12:15pm.


    *

      Numerical Analysis II

      (Winter 2009)
    *

          course website
    *

          TR 9:30am - 10:45am.


    *

      Introduction to Stochastic Analysis and Applications

      (Fall 2008)
    *

          course website
    *

          TR 11:00am - 12:15pm.


    *

      Numerical Analysis I

      (Fall 2008)
    *

          course website
    *

          TR 9:30am - 10:45am.


    *

      Introduction to Mathematical Biology

      (Winter 2008)
    *

          course website
    *

          TR 8:00am - 9:15pm, Old Little Theatre, Rm. 160B.


    *

      Finite Element Methods

      (Fall 2007)
    *

          course website
    *

          TR 11:00am - 12:15pm, South Hall Rm. 6635.


    *

      Introduction to Complex Variables

      (Fall 2007)
    *

          course website
    *

          TR 8:00am - 9:15pm, GIRV Rm. 1116.

Additional Class Websites:

    *

          classes taught 2003 - 2007

Advising:

Postdoctoral Researchers:

    *

      Chetan Pahlajani

Graduate Students:

    *

      Lacey Heubel
    *

      David Valdman
    *

      Per Danzl (co-advisor with Jeff Moehlis, IGERT)

Undergraduate Students:

    *

      Joe Rudzinski

Administration:

    *

      Mathematical Finance Majors (Undergraduate Faculty Adviser)

Research Summaries:

Paul J. Atzberger

Stochastic Immersed Boundary Methods / Computational Fluid Dynamics:

A recent talk on the Stochastic Immersed Boundary Method can be found here [IMA video / slides] .

The Stochastic Immersed Boundary Method (SIB) is a numerical approach for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see papers. SIB Schematic

Immersed structures in SIB can be used to represent the mechanics of a variety of microscopic hydrodynamic systems, for example, in a complex fluid the structures could represent solute particles, polymers, or membrane structures. In the figure some simulations demonstrating the methodology for a few basic physical systems are shown. Click on the images to play the associated movies. From top to bottom are: (i) polymer knot simulations showing SIB preservation of knot topology without the need for excluded volume interactions, (ii) simulations demonstrating a tethered membrane model using SIB for the hydrodynamic coupling, (iii) simulations showing how the methodology may be used to simulate more complex mechanical systems subject to thermal fluctuations, in this case a basic model from biology of a motor protein transporting a cargo vesicle under an imposed hydrodynamic load. In collaboration with the Brown Group, Department of Chemistry, dynamic coarse-grained models of lipid bilayer membranes are being developed using the SIB formalism to account for molecular level interactions, lipid-lipid and lipid-solvent hydrodynamic coupling, and thermal fluctuations. For a more in-depth discussion see the publications section.

Osmotic Phenomena / Microfluidics:

Osmosis refers to a general phenomenon in mechanical systems where pressures and forces are generated in a fluid or gas solvent by solute particles which are confined by semi-permeable membrane boundaries or electric fields. Such forces and pressures play a fundamental role in many biological systems and technological devices. Some examples in biology include mechanisms responsible for exchange of nutrients and wastes in microscopic capillaries in tissues and means of propulsion in cell motility. In technological applications, microscopic devices have been designed which utilize osmotic effects to pump fluids or actuate forces via swelling. The classical theories for osmotic pressure, such as van't Hoff's relation, become inaccurate for microscopic systems where solute particles interact on length scales comparable to the size of the confining chamber. Classical theories of osmosis also typically assume systems are in thermodynamic equilibrium. Osmotic Phenomena

To study systems in regimes applicable to biological systems and microscopic devices we are developing theory to describe microscopic osmotic phenomena both in equilibrium and non-equlibrium settings. Some recent work includes studying how osmotic pressures generated by polymer solutes depend on polymer topology, stiffness, and excluded volume. In related work, a pumping mechanism for a microfluidic device has been proposed which uses reversible chemical reactions to drive fluid flows, see the above papers.

Applications in Molecular Biology and Biotechnology:

Technological advances such as optical traps and molecular tagging are yielding a wealth of quantitative information about the mechanics and localization of processes in biological systems at the single molecule level. In cell biology an interesting class of molecules are motor proteins which act essentially as microscopic machines within cells to form cellular structures, transport materials, or generate forces. We are using stochastic modeling, analysis, and simulation to study the basic mechanisms underlying how molecular motor proteins operate with an effort toward making connections with available experimental data. In previous work we have studied the kinesin motor protein and developed a coarse-grained mechanical model consistent with available optical trap experimental data, see papers. Molecular Biology

Aptamers are short segments of DNA or RNA which bind to proteins. In applications, specific aptamers (nucleic acid sequences) are sought which bind to a desired target protein with high affinity and specificity. Applications include purification methods for proteins, analytics in biosensor devices, and target validation in pharmaceutical drug development. Obtaining aptamers with strong affinity and specificity binding a given target molecule poses a significant challenge in practice. In collaboration with the H.T. Soh Laboratory, Department of Mechanical Engineering along with the undergraduate Joe Rudzinski (CCS Program, UCSB) mathematical approaches are being used to study experimental methods for the selection of DNA-Aptamers from a library of random sequences.

Spatially Adaptive Numerical Methods for Stochastic Partial Differential Equations:

Results for a stochastic reaction-diffusion system with intrinsic concentration fluctuations: Adaptive Num. for SPDEs

• Growth of a noise-induced spatial pattern resolved with adaptive meshing [AVI Movie].

For a more in-depth discussion of this work and information about additional projects please see the publications section.

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