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Revision as of 17:08, 21 September 2011 by Stech (talk | contribs) (Omar Morandi, TU Graz)
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Sigurd Angenent, UW-Madison

Deterministic and random models for polarization in yeast cells

I'll present one of the existing models for "polarization in yeast cells." The heuristic description of the model allows at least two mathematical formulations, one using pdes (a reaction diffusion equation) and one using stochastic particle processes, which give different predictions for what will happen. The model is simple enough to understand and explain why this is so.

John Finn, Los Alamos

Symplectic integrators with adaptive time steps


Jay Bardhan, Rush Univ

Understanding Protein Electrostatics using Boundary-Integral Equations

The electrostatic interactions between biological molecules play important roles determining their structure and function, but are challenging to model because they depend on the collective response of thousands of surrounding water molecules. Continuum electrostatic theory -- e.g., the Poisson equation -- offers a successful and simple theory for biomolecule science and engineering, and boundary-integral equation formulations of the problem offer several theoretical and computational advantages. In this talk, I will highlight some recent modeling advances derived from the boundary-integral perspective, which have important applications in biophysics and whose mathematical foundations may be useful in other domains as well. First, one may derive a fast electrostatic model that resembles Generalized Born theory, but is based on a rigorous operator approximation for rapid, accurate estimation of a Green's function. In addition, we have been exploring a boundary-integral approach to nonlocal continuum theory as a means to model the influence of water structure, an important piece of molecular physics left out of the standard continuum theory.

Omar Morandi, TU Graz

Modeling quantum transport with the phase-space formalism

Quantum modeling is becoming a crucial aspect in nanoelectronics research in perspective of analog and digital applications. Devices like interband tunneling diodes or graphene sheets are examples of solid state structures that are receiving a great importance in the modern nanotechnology for high-speed and miniaturized systems. Differing from the usual transport where the electronic current flows within a single band, the remarkable feature of such solid state structures is the possibility to achieve a sharp coupling among states belonging to different bands. As a consequence, the single band transport or the classical phase-space description of the charge motion based on the Boltzmann equation are not longer accurate. Moreover, in a crystal where the effective Hamiltonian is expressed by a partially diagonalized basis (e. g. in graphene or in semiconductors), the usual definitions of the macroscopic quantities, as for example the mean velocity or the particle density, no longer apply. The theory of Berry phases offers an elegant explanation of this effect in terms of the intrinsic curvature of the perturbed band.

Different approaches have been proposed to achieve a full quantum description of electron transport where the interaction among the different bands can be included. Among them, the phase-space formulation of quantum mechanics based on the concept of “Wigner-Weyl quantization”, offers a framework in which the quantum phenomena can be described with a classical language and the question of the quantum-classical correspondence can be directly investigated. In this contribution, an extension of the original Wigner-Weyl theory based on a suitable projection procedure, is presented. The applications of this formalism span among different subjects: the multi-band transport and applications to nano-devices, the infinite- order 􏰀-approximations of the motion and the characterization of a system in terms of Berry phases or, more generally, the representation of a quantum system as a Riemann manifold with a suitable connection. Furthermore, some asymptotic procedures devised for the approx- imation of the quantum Wigner-Weyl solution have shown a very attractive connection with the Dyson-Feynmann theory of the particle interaction, which allows us to describe quantum transition by means of an effective Boltzmann process.

George Hagedorn, Virginia Tech

Time Dependent Semiclassical Quantum Dynamics: Analysis and Numerical Algorithms

We begin with some elementary comments about time-dependent quantum mechanics and the role of Planck's constant. We then describe several mathematical results about approximate solutions to the Schr\"odinger equation for small values of the Planck constant. Finally, we discuss numerical difficulties of semiclassical quantum dynamics and algorithms that have recently been developed, including some work in progress.

Qiang Deng, UW-Madison



Ray Pierrehumbert, U of Chicago



Jianfeng Lu, Courant Institute



Anne Shiu, U of Chicago



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