Quantitative derivation of effective evolution equations for the dynamics of Bose-Einstein condensates.

摘要

This thesis proves certain results concerning an important question in nonequilibrium quantum statistical mechanics which is the derivation of effective evolution equations approximating the dynamics of a system of large number of bosons initially at equilibrium (ground state at very low temperatures). The dynamics of such systems are governed by the time-dependent linear many-body Schrodinger equation from which it is typically difficult to extract useful information due to the number of particles being large. We will study quantitatively (i.e. with explicit bounds on the error) how a suitable one particle non-linear Schrodinger equation arises in the mean field limit as number of particles N → infinity and how the appropriate corrections to the mean field will provide better approximations of the exact dynamics.;In the first part of this thesis we consider the evolution of N bosons, where N is large, with two-body interactions of the form N3betaupsilon(Nbeta·), 0 ≤ beta ≤ 1. The parameter beta measures the strength and the range of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [18,19] by Grillakis-Machedon-Margetis. We extend the results for 0 ≤ beta < 1/3 in [19,20] to the case of beta 0 and p( t) is a polynomial, which implies a specific rate of convergence as N → infinity.;In the second part, utilizing estimates of the type discussed in the first part, we compare the exact evolution with the mean field approximation in the sense of marginals. We prove that the exact evolution is close to the approximate in trace norm for times of the order o(1)√ N compared to log(o(1)N) as obtained in Chen-Lee-Schlein [6] for the Hartree evolution. Estimates of similar type are obtained for stronger interactions as well.;[6] L. Chen, J. O. Lee and B. Schlein, Rate of convergence towards Hartree dynamics, J. Stat. Phys. 144, 872--903 (2011).;[18] M. Grillakis, M. Machedon and D. Margetis, Second-order corrections to mean field evolution of weakly interacting Bosons. I, Commun. Math. Phys. 294, 273--301 (2010).;[19] M. Grillakis, M. Machedon and D. Margetis, Second-order corrections to mean field evolution of weakly interacting Bosons. II, Adv. in Math. 228, 1788--1815 (2011).;[20] M. Grillakis and M. Machedon, Pair excitations and the mean field approximation of interacting Bosons, I, Commun. Math. Phys. 324, 601--636 (2013).