Quantum effective potential from discarded degrees of freedom

I obtain the quantum correction $\Delta V_\mathrm{eff}= (\hbar^2/8m) [(1- 4\xi \frac{d+1}{d})(\mathcal{S}')^2 + 2(1-4\xi)\mathcal{S}"]$ that appears in the effective potential whenever a compact $d$-dimensional subspace (of volume $\propto \exp[\mathcal{S}(x)]$) is discarded from the configuration space of a nonrelativistic particle of mass $m$ and curvature coupling parameter $\xi$. This correction gives rise to a force $-\langle\Delta V_\mathrm{eff}'\rangle$ that pushes the expectation value $\langle x\rangle$ off its classical trajectory. Because $\Delta V_\mathrm{eff}$ does not depend on the details of the discarded subspace, these results constitute a generic model of the quantum effect of discarded variables with maximum entropy/information capacity $\mathcal{S}(x)$.Comment: 7 pages, 1 figure; published version with appendi

Categorical colormap optimization with visualization case studies

—Mapping a set of categorical values to different colors is an elementary technique in data visualization. Users of visualization software routinely rely on the default colormaps provided by a system, or colormaps suggested by software such as ColorBrewer. In practice, users often have to select a set of colors in a semantically meaningful way (e.g., based on conventions, color metaphors, and logological associations), and consequently would like to ensure their perceptual differentiation is optimized. In this paper, we present an algorithmic approach for maximizing the perceptual distances among a set of given colors. We address two technical problems in optimization, i.e., (i) the phenomena of local maxima that halt the optimization too soon, and (ii) the arbitrary reassignment of colors that leads to the loss of the original semantic association. We paid particular attention to different types of constraints that users may wish to impose during the optimization process. To demonstrate the effectiveness of this work, we tested this technique in two case studies. To reach out to a wider range of users, we also developed a web application called Colourmap Hospital