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.

Connecting the dots: Examining the role of parental beliefs and preschoolers’ affect and engagement in predicting parent-child number exploration during a meaningful math experience

The current study examined the frequency and quality of how 3- to 4-year-old children and their parents explore the relations between symbolic and non-symbolic quantities in the context of a playful math experience, as well as the role of both parent and child factors in this exploration. Preschool children’s numerical knowledge was assessed while parents completed a survey about the number-related experiences they share with their children at home, and their math-related beliefs. Parent-child dyads were then videotaped playing a modified version of the card game War. Results suggest that parents and children explored quantity explicitly on only half of the cards and card pairs played, and dyads of young children and those with lower number knowledge tended to be most explicit in their quantity exploration. Dyads with older children, on the other hand, often completed their turns without discussing the numbers at all, likely because they were knowledgeable enough about numbers that they could move through the game with ease. However, when dyads did explore the quantities explicitly, they focused on identifying numbers symbolically, used non-symbolic card information interchangeably with symbolic information to make the quantity comparison judgments, and in some instances, emphasized the connection between the symbolic and non-symbolic number representations on the cards. Parents reported that math experiences such as card game play and quantity comparison occurred relatively infrequently at home compared to activities geared towards more foundational practice of number, such as counting out loud and naming numbers. However, parental beliefs were important in predicting both the frequency of at-home math engagement as well as the quality of these experiences. In particular, parents’ specific beliefs about their children’s abilities and interests were associated with the frequency of home math activities, while parents’ math-related ability beliefs and values along with children’s engagement in the card game were associated with the quality of dyads’ number exploration during the card game. Taken together, these findings suggest that card games can be an engaging context for parent-preschooler exploration of numbers in multiple representations, and suggests that parents’ beliefs and children’s level of engagement are important predictors of this exploration.