Constructing a statistical mechanics for Beck-Cohen superstatistics

The basic aspects of both Boltzmann-Gibbs (BG) and nonextensive statistical mechanics can be seen through three different stages. First, the proposal of an entropic functional ($S_{BG} =-k\sum_i p_i \ln p_i$ for the BG formalism) with the appropriate constraints ($\sum_i p_i=1$ and $\sum_i p_i E_i = U$ for the BG canonical ensemble). Second, through optimization, the equilibrium or stationary-state distribution ($p_i = e^{-\beta E_i}/Z_{BG}$ with $Z_{BG}=\sum_j e^{-\beta E_j}$ for BG). Third, the connection to thermodynamics (e.g., $F_{BG}= -\frac{1}{\beta}\ln Z_{BG}$ and $U_{BG}=-\frac{\partial}{\partial \beta} \ln Z_{BG}$). Assuming temperature fluctuations, Beck and Cohen recently proposed a generalized Boltzmann factor $B(E) = \int_0^\infty d\beta f(\beta) e^{-\beta E}$. This corresponds to the second stage above described. In this letter we solve the corresponding first stage, i.e., we present an entropic functional and its associated constraints which lead precisely to $B(E)$. We illustrate with all six admissible examples given by Beck and Cohen.Comment: 3 PS figure

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

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

The treatment of mixing in core helium burning models -- II. Constraints from cluster star counts

The treatment of convective boundaries during core helium burning is a fundamental problem in stellar evolution calculations. In Paper~I we showed that new asteroseismic observations of these stars imply they have either very large convective cores or semiconvection/partially mixed zones that trap g-modes. We probe this mixing by inferring the relative lifetimes of asymptotic giant branch (AGB) and horizontal branch (HB) from $R_2$, the observed ratio of these stars in recent HST photometry of 48 Galactic globular clusters. Our new determinations of $R_2$ are more self-consistent than those of previous studies and our overall calculation of $R_2 = 0.117 \pm 0.005$ is the most statistically robust now available. We also establish that the luminosity difference between the HB and the AGB clump is $\Delta \log{L}_\text{HB}^\text{AGB} = 0.455 \pm 0.012$. Our results accord with earlier findings that standard models predict a lower $R_2$ than is observed. We demonstrate that the dominant sources of uncertainty in models are the prescription for mixing and the stochastic effects that can result from its numerical treatment. The luminosity probability density functions that we derive from observations feature a sharp peak near the AGB clump. This constitutes a strong new argument against core breathing pulses, which broaden the predicted width of the peak. We conclude that the two mixing schemes that can match the asteroseismology are capable of matching globular cluster observations, but only if (i) core breathing pulses are avoided in models with a semiconvection/partially mixed zone, or (ii) that models with large convective cores have a particular depth of mixing beneath the Schwarzschild boundary during subsequent early-AGB `gravonuclear' convection.Comment: 22 pages, 14 figures. Accepted for publication in MNRA