Mean field limit of a dynamical model of polymer systems

E Weinan^{1,2*}, SHEN Hao^{3}

1 School of Mathematical Sciences and BICMR, Peking University, Beijing 100871, China;
2 Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544, USA;
3 Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA

This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. Every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations (SPDEs). We show that the mean field limit as N →∞ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDE system of the polymers and the mean field limit equation are well-posed.

This work was supported by National Natural Science Foundation of China (Grant No. 91130005) and the US Army Research Office (Grant No. W911NF-11-1-0101). The authors are grateful to Xiuyuan Cheng for helpful discussions.

Corresponding Authors:
E Weinan
E-mail: weinan@princeton.edu

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