# Physical Modeling

Mathematical research in physical modeling focuses on the formulation and analysis of mathematical representations of problems motivated by other branches of science and engineering. In addition to generating novel problems with new computational and analytical challenges, constructing accurate models for complex systems may uncover the need for fundamental extensions to the governing equations.

Multiscale modeling is one very active area of current research, which focuses on bridging results from different types of mathematical models applicable to microscopic- vs. large-scale properties of problems. Examples include: estimates of macroscopic properties of materials from molecular structure and quantum mechanics, descriptions of population dynamics from behaviors of individuals, and homogenized effective properties of media derived from details of spatially varying in homogeneities.

Some of the other challenging aspects in current studies in physical modeling include: the influence of randomness and stochastic processes, simplifying high-dimensional models, and understanding the influences of complex and dynamic problem geometries.

Some of the primary areas of application of physical modeling at Duke include: quantum mechanics, fluid dynamics, astrophysics and biological/physiological systems. Studies in physical modeling draw extensively on techniques from analysis, probability, partial differential equations, dynamical systems and computational methods.

## Faculty

#### J. Thomas Beale

###### Professor Emeritus of Mathematics

**Keywords in this area**

boundary integral methods, motion of fluid interfaces, equations of incompressible flow, convergence of numerical methods for fluid flow

#### Robert Calderbank

###### Charles S. Sydnor Professor of Computer Science

**Keywords in this area**

wireless communications, data storage, detection and estimation

#### John Everett Dolbow

###### Professor of Civil and Environmental Engineering

**Keywords in this area**

fracture, lipid membranes

#### Gregory Joseph Herschlag

###### Visiting Assistant Professor of Mathematics

**Keywords in this area**

physical modeling, fluids flow across dynamic channels, surface catalysis, molecular dynamics, stochastic boundary conditions

#### Jianfeng Lu

###### Associate Professor of Mathematics

**Keywords in this area**

electronic structure models, multiscale modeling and simulations, rare events simulation, computational physics, kinetic equations, nonlinear Schroedinger equations, quantum chemistry, computational statistical mechanics

#### Jonathan Christopher Mattingly

###### Professor of Mathematics

**Keywords in this area**

Stochastic and Random PDEs, Stochastic Dynamical Systems, Fluid Mechanics

#### James H. Nolen

###### Associate Professor of Mathematics

**Keywords in this area**

asymptotic analysis

#### Arlie O. Petters

###### Benjamin Powell Professor of Mathematics in Trinity College of Arts and Sciences

**Keywords in this area**

gravity, light, geometric lensing, stochastic lensing, black holes, extra dimensions

#### Thomas Peter Witelski

###### Professor in the Department of Mathematics

**Keywords in this area**

fluid dynamics

#### Haizhao Yang

###### Visiting Assistant Professor in the Department of Mathematics

**Keywords in this area**

Modeling in materials science at atomic scale

#### Zhennan Zhou

###### William W. Elliott Assistant Research Professor

**Keywords in this area**

semi-classical methods in quantum dynamics, multi-scale methods for physics models, band crossing for Bloch electrons, kinetic theory, Born-Oppenheimer approximation, numerical methods for conservation law