Generalizing the Planck distribution

Along the lines of nonextensive statistical mechanics, based on the entropy $S_q = k(1- \sum_i p_i^q)/(q-1) (S_1=-k \sum_i p_i \ln p_i)$, and Beck-Cohen superstatistics, we heuristically generalize Planck's statistical law for the black-body radiation. The procedure is based on the discussion of the differential equation $dy/dx=-a_{1}y-(a_{q}-a_{1}) y^{q}$ (with $y(0)=1$), whose $q=2$ particular case leads to the celebrated law, as originally shown by Planck himself in his October 1900 paper. Although the present generalization is mathematically simple and elegant, we have unfortunately no physical application of it at the present moment. It opens nevertheless the door to a type of approach that might be of some interest in more complex, possibly out-of-equilibrium, phenomena.Comment: 6 pages, including 2 figures. To appear in {\it Complexity, Metastability and Nonextensivity}, Proc. 31st Workshop of the International School of Solid State Physics (20-26 July 2004, Erice-Italy), eds. C. Beck, A. Rapisarda and C. Tsallis (World Scientific, Singapore, 2005

The treatment of mixing in core helium burning models -- III. Suppressing core breathing pulses with a new constraint on overshoot

Theoretical predictions for the core helium burning phase of stellar evolution are highly sensitive to the uncertain treatment of mixing at convective boundaries. In the last few years, interest in constraining the uncertain structure of their deep interiors has been renewed by insights from asteroseismology. Recently, Spruit (2015) proposed a limit for the rate of growth of helium-burning convective cores based on the higher buoyancy of material ingested from outside the convective core. In this paper we test the implications of such a limit for stellar models with a range of initial mass and metallicity. We find that the constraint on mixing beyond the Schwarzschild boundary has a significant effect on the evolution late in core helium burning, when core breathing pulses occur and the ingestion rate of helium is fastest. Ordinarily, core breathing pulses prolong the core helium burning lifetime to such an extent that models are at odds with observations of globular cluster populations. Across a wide range of initial stellar masses ($0.83 \leq M/\text{M}_\odot \leq 5$), applying the Spruit constraint reduces the core helium burning lifetime because core breathing pulses are either avoided or their number and severity reduced. The constraint suggested by Spruit therefore helps to resolve significant discrepancies between observations and theoretical predictions. Specifically, we find improved agreement for $R_2$, the observed ratio of asymptotic giant branch to horizontal branch stars in globular clusters; the luminosity difference between these two groups; and in asteroseismology, the mixed-mode period spacing detected in red clump stars in the \textit{Kepler} field.Comment: Accepted for publication in MNRAS; 11 pages, 6 figure

An evaluation of biotic ligand models predicting acute copper toxicity to Daphnia magna in wastewater effluent

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ 2010 SETAC.The toxicity of Cu to Daphnia magna was investigated in a series of 48-h immobilization assays in effluents from four wastewater treatment works. The assay results were compared with median effective concentration (EC50) forecasts produced by the HydroQual biotic ligand model (BLM), the refined D. magna BLM, and a modified BLM that was constructed by integrating the refined D. magna biotic ligand characterization with the Windermere humic aqueous model (WHAM) VI geochemical speciation model, which also accommodated additional effluent characteristics as model inputs. The results demonstrated that all the BLMs were capable of predicting toxicity by within a factor of two, and that the modified BLM produced the most accurate toxicity forecasts. The refined D. magna BLM offered the most robust assessment of toxicity in that it was not reliant on the inclusion of effluent characteristics or optimization of the dissolved organic carbon active fraction to produce forecasts that were accurate by within a factor of two. The results also suggested that the biotic ligand stability constant for Na may be a poor approximation of the mechanisms governing the influence of Na where concentrations exceed the range within which the biotic ligand stability constant value had been determined. These findings support the use of BLMs for the establishment of site-specific water quality standards in waters that contain a substantial amount of wastewater effluent, but reinforces the need for regulators to scrutinize the composition of models, their thermodynamic and biotic ligand parameters, and the limitations of those parameters.EPSRC and Severn Trent Water

Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy

The $q$-sum $x \oplus_q y \equiv x+y+(1-q) xy$ ($x \oplus_1 y=x+y$) and the $q$-product $x\otimes_q y \equiv [x^{1-q} +y^{1-q}-1]^{\frac{1}{1-q}}$ ($x\otimes_1 y=x y$) emerge naturally within nonextensive statistical mechanics. We show here how they lead to two-parameter (namely, $q$ and $q^\prime$) generalizations of the logarithmic and exponential functions (noted respectively $\ln_{q,q^\prime}x$ and $e_{q,q^\prime}^{x}$), as well as of the Boltzmann-Gibbs-Shannon entropy $S_{BGS}\equiv -k \sum_{i=1}^Wp_i \ln p_i$ (noted $S_{q,q^\prime}$). The remarkable properties of the $(q,q^\prime)$-generalized logarithmic function make the entropic form $S_{q,q^\prime} \equiv k \sum_{i=1}^W p_i \ln_{q,q^\prime}(1/p_i)$ to satisfy, for large regions of $(q,q^\prime)$, important properties such as {\it expansibility}, {\it concavity} and {\it Lesche-stability}, but not necessarily {\it composability}.Comment: 9 pages, 4 figure

El fundador del Opus Dei. III. Los caminos divinos de la tierra. [Reseña]

Reseña de Andrés VÁZQUEZ DE PRADA, El fundador del Opus Dei. III. Los caminos divinos de la tierra, Rialp, Madrid 2003, 800 pp., 17 x 25, ISBN 84-321-3462-7