Download PDF by Russel E. Caflisch (auth.), George Papanicolaou (eds.): Hydrodynamic Behavior and Interacting Particle Systems

By Russel E. Caflisch (auth.), George Papanicolaou (eds.)

This IMA quantity in arithmetic and its purposes HYDRODYNAMIC habit AND INTERACTING PARTICLE platforms is partly the lawsuits of a workshop which used to be an essential component of the 1985-86 IMA software on STOCHASTIC DIFFERENTIAL EQUATIONS AND THEIR purposes. we're thankful to the clinical Committee: Daniel Stroock (Chairman) Wendell Fleming Theodore Harris Pierre-Louis Lions Steven Orey George Papanicolaou for making plans and enforcing a thrilling and stimulating year-long software. We specially thank this system Organizer, George Papanicolaou for orga­ nizing a workshop which introduced jointly scientists and mathematicians in numerous components for a fruitful trade of rules. George R. promote Hans Weinberger PREFACE A workshop at the hydrodynamic habit of interacting particle structures used to be held on the Institute for arithmetic and its purposes on the collage of Minnesota throughout the week of March 17, 1986. Fifteen papers offered on the workshop are amassed during this quantity. They include examine in numerous varied instructions which are presently being pursued. The paper of Chaikin, Dozier and Lindsay is anxious with experimental effects on suspensions in regimes the place glossy mathematical tools will be necessary. The paper of Fritz offers an advent to those tools as does the paper of Spohn. Analytical equipment at the moment utilized by within the physics and chemistry literature are awarded within the paper of Freed, Wang and Douglas. The paper of Caflisch bargains with time established results in sedimentation.

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M S As the parameters are obtained by solving a system of five self-consistency equations. a, c, p multiple equilibria. ted near a are varied this system can exhibit bifurcation of In particular there can exist equilibrium densities concentra- (non-global) local maximum. Since the empirical distribution of searchers follows the McKean-Vlasov equation for long periods of time when N is large, this means that in such a case the finite searcher system can get trapped at a local maximum. 3• TRAPPING AT A LOCAL MAXIMUM In this section we analyse in detail the phenomenon of trapping in a local maximum for a simplified model in which we replace the weighted mean-field inter- 30 action with a simple mean-field interaction.

Da a~O and there is a pair of equilibrium measures concentrated near a > 1, there is a unique equilibrium On the other hand if the two global maxima. measure. (b) i c 0 but Nalcl is sufficiently small. It can be shown that the conclusion of (a) remains valid in this case. (c) Nalcl large. ) that there exists a unique equilibrium near the global maximum of Case (iii) . KN- 1 and P ~ Pc' c In this case O. K8N 1- a dm(a) I da ~ a~O where REMARK. a > In the case t and P ~ Pc ~ - mu )4 K(x is a constant.

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