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Minicourse: Multiscale Ideas and Dynamical Systems, Prof. Jens Lorenz. In this course I want to give a first introduction to multiscale modeling. It is well recognized that many physical phenomena have important features at multiple temporal and spatial scales. Particular approaches are appropriate for the description of a system on each level, and it is challenging to connect the different descriptions. The whole subject is vast, of course, and I will restrict myself to consider molecular dynamics models and relate them to continuum descriptions. A first result is due to Daniel Bernoulli. In 1738 he derived Boyle's law ($pV=const$) from a particle point--of--view. Bernoulli assumed that all particles have the same speed $|u|$, which is incorrect. However, in 1860 Maxwell introduced his velocity distribution, and if one replaces Bernoulli's $|u|$ with Maxwell's root--mean--square $\langle |u|^2\rangle ^{1/2}$ the result becomes correct for ideal gases. We will take a historical path starting with Bernoulli to relate molecular dynamics and a continuum result. We will then relate Boltzmann's equation to Navier--Stokes, through a simple example leading to a viscous shock in one--space dimension. %My former Ph.D. student, Pavlo Cherepanov, has written his thesis on related %numerical issues. In this course I will describe the basic phenomena and difficulties. The students will run Matlab codes.} Professor Lorenz is on the editorial board of four journals. He has served as Graduate Chair (2008-2011) and as Interim Department Chair (2008-2009). He is writing a book accepted for publication by NOVA, that is based on his lecture notes from the previously NSF funded MCTP workshop. The current course builds upon and expands that material, and may serve as a basis for a new undergraduate mathematics modeling course at UNM. |
