Leading-order term

The leading-order terms (or corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude. The sizes of the different terms in the equation(s) will change as the variables change, and hence, which terms are leading-order may also change. A common and powerful way of simplifying and understanding a wide variety of complicated mathematical models is to investigate which terms are the largest (and therefore most important), for particular sizes of the variables and parameters, and analyse the behaviour produced by just these terms (regarding the other smaller terms as negligible). This gives the main behaviour – the true behaviour is only small deviations away from this. This main behaviour may be captured sufficiently well by just the strictly leading-order terms, or it may be decided that slightly smaller terms should also be included. In which case, the phrase leading-order terms might be used informally to mean this whole group of terms. The behaviour produced by just the group of leading-order terms is called the leading-order behaviour of the model. Consider the equation y = x3 + 5x + 0.1. For five different values of x, the table shows the sizes of the four terms in this equation, and which terms are leading-order. As x increases further, the leading-order terms stay as x3 and y, but as x decreases and then becomes more and more negative, which terms are leading-order again changes. There is no strict cut-off for when two terms should or should not be regarded as approximately the same order, or magnitude. One possible rule of thumb is that two terms that are within a factor of 10 (one order of magnitude) of each other should be regarded as of about the same order, and two terms that are not within a factor of 100 (two orders of magnitude) of each other should not. However, in between is a grey area, so there are no fixed boundaries where terms are to be regarded as approximately leading-order and where not. Instead the terms fade in and out, as the variables change.

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Ground state properties of large atoms and quantum dots

Rico Rueedi

We investigate the ground state properties of large atoms and quantum dots described by a d-dimensional N-body Hamiltonian of confinement ZV. In atoms, d = 3 and V is the Coulomb interaction; in dots, d = 2 and V is phenomenologically determined. We express the grand-canonical partition function in a path integral approach, and evaluate its expansion in Z-1. The problem can be seen as that of field theory possessing a saddle point. This saddle point results in a mean-field contribution to the energy, while the fluctuations result in the correlation energy. The mean-field contribution to the energy is self-consistently determined by the Hartree potential and contains an exchange term. Its smooth contribution is evaluated by a semiclassical method, with ε = Z-1/d in the role of ℏ, while its oscillating contribution can be related to the periodic orbits in the corresponding classical Hamiltonian. In the case of atoms, the leading order in ε of the correlation energy contains a term in Z ln Z1/3, which is essential in reproducing the behaviour shown by reference values, and a term in Z. While we have evaluated the contribution to the Z-term provided by the leading fluctuation order, the numerical evaluation of the contributions provided by higher order fluctuations remains an open problem. The self-consistent contribution to the energy corresponds to the statistical atom, composed of Thomas-Fermi and its corrections, comprehensively analysed, including oscillations, by Schwinger and Englert. In the case of dots, the leading order in ε of the correlation energy is a universal contribution of order Z, which we obtain in closed form. We then determine the expansion in ε of the smooth contributions down to this correlation order. We apply the approach to dots of quadratic and quartic confinement, including the oscillating contribution in the case of a chaotic quartic confinement.

EPFL2009

Couvre la définition des générateurs du groupe conforme et l'application du théorème de Wick.