Quantitative Derivation of Effective Evolution Equations for the Dynamics of Bose-Einstein Condensates

This thesis proves certain results concerning an important question in non-equilibrium 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 Schroedinger 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 Schroedinger equation arises in the mean field limit as number of particles N → ∞ 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 N³ᵝv(Nᵝ⋅), 0≤β≤1. The parameter β 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 ≤ β < 1/3 in [19, 20] to the case of β < 1/2 and obtain an error bound of the form p(t)/Nᵅ, where α>0 and p(t) is a polynomial, which implies a specific rate of convergence as N → ∞. 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.

MECHANISM OF UPREGULATION OF PHOSPHATIDYLCHOLINE SYNTHESIS DURING PICORNAVIRUS INFECTION AND ITS ROLE IN THE DEVELOPMENT OF VIRAL REPLICATION ORGANELLES

Picornaviruses are a group of human and animal pathogens capable of inflicting serious public health diseases and economic burdens. Treatments options through vaccines for prevention or antivirals to cure infection are not available for the vast majority of these viruses. These shortcomings, in the development of vaccines or antivirals therapeutic, are linked to the genetic diversity and to an incomplete understanding of the biology of these viruses. Despite the diverse host range, this group of positive-strand RNA viruses shares the same replication mechanisms, including the development of membranous structures (replication organelles) in the cytoplasm of infected cells. The development of these membranous structures, which serve as sites for the replication of the viral RNA genome, has been linked to the hijacking of elements of the cellular membrane metabolism pathways. Here we show that upon picornavirus infection, there is a specific activation of acyl-CoA synthetase enzymes resulting in strong import and accumulation of long chain fatty acids in the cytoplasm of infected cells. We show that the newly imported fatty acids serve as a substrate for the upregulation of phosphatidylcholine synthesis required for the structural development of replication organelles. In this work, we identified that acyl-CoA synthetase long chain 3 (ACSL3) is required for the upregulation of lipids syntheses and the replication of poliovirus. We have shown that the poliovirus protein 2A was required but not sufficient for the activation of import of long chain fatty acids in infected cells. We demonstrated that the fatty acid import is upregulated upon infection by diverse picornaviruses and that such upregulation is not dependent on activation of ER stress response or the autophagy pathways. In this work, we have demonstrated that phosphatidylcholine was required for the structural development of replication organelles. Phosphatidylcholine synthesis was dispensable for the production of infectious particles at high MOI but required at a low MOI for the protection of the replication complexes from the cellular innate immunity mechanisms.