The world is constantly abuzz with particles in motion. Peter Kotelenez, professor of mathematics in the College of Arts and Sciences at Case Western Reserve University, has applied his mathematical knowledge to explain this motion.
Recently Kotelenez finished constructing proofs of models to describe time-dependent physical phenomena as large and small particles interact in the transition from the microscopic, mesoscopic to the macroscopic levels. He provides detailed analyses of this work in his newly published book, Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations (Springer).
The application of stochastic equations that Kotelenez has derived has potential uses for researchers in the field of mathematical biology who might want to describe how proteins move through fluids or chemists and chemical engineers interested in the diffusion of particles in chemical reactions.
Kotelenez has shown how stochastic ordinary and partial differential equations also offer an understanding of how large and small particles transition from the discrete microscopic level to large bulk matter at the macroscopic level. He defines stochastic ordinary and partial differential equations as a mesoscopic level in the transition from microscopic to macroscopic levels.
His book, like the levels he describes in his proofs, has made a transition, over the past 12 years from three chapters that began with an exploration of the mesoscopic world to the finished copy of 14 chapters and an appendix.
As his National Science Foundation-funded research continued, he advanced his proofs to include both microscopic and macroscopic equations and showed the consistency through mesoscopic, continuum and macroscopic limit theorems.
"In the classical picture we know exactly where atoms and molecules move" at the microscopic level, Kotelenez said. It is based primarily on Newton's laws of motion of large and small particles which are coupled through pair-wise repulsion and attraction.
But the question he raised was what pattern of motion of the large solute particles emerges when the infinitely many small solvent particle become a random medium in the transition to coarser scales. "Letting the initial velocities of the small particles go to infinity, and using other scaling techniques, one can show that the larger particles change in time randomly in a stochastic way so that each large particle performs a Brownian motion," said Kotelenez.
Brownian motion is a type of rapid and vibrant movement of particles when suspended in water or other fluids. Brownian motion was described by Einstein in 1905 when he studied large solute particles immersed in a fluid and developed a convincing argument for the existence of molecules and atoms, explained Kotelenez.
Kotelenez has dedicated his book to Donald Dawson, a mathematician from Carleton University in Ottawa, Canada, who has performed highly recognized work in the area of stochastic particle systems and their mass distributions. Kotelenez said he generalized some of Dawson's ideas for the transition from stochastic ordinary to stochastic partial differential equations.
The book's underlining research evolved from his thesis work for his doctoral degree from the University of Bremen in Germany and it was influenced by his thesis advisor, Ludwig Arnold, and some leading physicists of that time, including Nobel Laureate Il'ya Prigogine, as well as N. van Kampen, H. Haken and G. Nicolis. His encounter with those ideas during graduate studies set a course that he has continued to follow in his pursuit to understand stochastic ordinary and stochastic partial differential equations.
Geared for researchers and graduate students in science and math, the book includes a 100-page appendix that provides a host of information from definitions of what stochastic means (randomness in time) to proofs and other background material to make the topic accessible to those unfamiliar with stochastic equations.
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