19 July 2012
A researcher at New York University (NYU)’s Courant Institute of Mathematical Sciences and two collaborators from University of Toronto have solved a decades-old equation that models several real-world systems, such as the development of cracks in materials, the formation of bacteria, and the growth of liquid crystals.
The Kardar-Parisi-Zhang (KPZ) equation, first published in 1986, describes the temporal evolution of an interface growing in a disordered environment. Solving the equation means computing exact statistics which describe the random fluctuations in the interface growth.
The success came from collaboration between NYU’s Ivan Corwin and the University of Toronto’s Jeremy Quastel and Gideon Amir. At the time of their work on the KPZ equation, Corwin was a doctoral student at Courant and partially supported by a grant from National Science Foundation’s Partnerships for International Research and Education (PIRE) program. Quastel, a professor at the University of Toronto, received his Ph.D. from Courant in 1990.
Solving the KPZ equation gives researchers another tool to analyze physical disordered systems, with implications in the fields of physics, engineering, materials science, biology, and ecology. The statistics developed in the work will allow for better modeling of such complex systems.
The work also marks the first time mathematicians have obtained an exact solution to a non-linear stochastic partial differential equation—an equation that describes random development in systems dependent on both location and time. In non-linear systems, doubling an input does not necessarily double the output—for example, doubling a patient’s medication may more than double the effect on the body. Non-linear systems are recognized as of great importance, yet relatively little theory has been developed to describe them. The work represents a step in that direction.