Abstract
We present a purely probabilistic proof of propagation of molecular chaos for N-particle systems in dimension 3 with interaction forces scaling like \(1/\vert q\vert ^{3\lambda - 1}\) with \(\lambda \) smaller but close to one and cut-off at \(q = N^{-1/3}\). The proof yields a Gronwall estimate for the maximal distance between exact microscopic and approximate mean-field dynamics. This can be used to show weak convergence of the one-particle marginals to solutions of the respective mean-field equation without cut-off in a quantitative way. Our results thus lead to a derivation of the Vlasov equation from the microscopic N-particle dynamics with force term arbitrarily close to the physically relevant Coulomb- and gravitational forces.
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Acknowledgments
We originally intended to publish this paper in a special issue dedicated to the 60th anniversary of Herbert Spohn. Unfortunately, substantial corrections delayed the completion of the manuscript. We wish to thank Herbert for his friendship and his support. Furthermore, very helpful discussions with Detlef Dürr and Dustin Lazarovici are gratefully acknowledged. Finally, we would like to thank the referees for carefully proofreading our paper. Their detailed comments helped us to substantially improve our manuscript.
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Boers, N., Pickl, P. On Mean Field Limits for Dynamical Systems. J Stat Phys 164, 1–16 (2016). https://doi.org/10.1007/s10955-015-1351-5
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DOI: https://doi.org/10.1007/s10955-015-1351-5