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

Edge of chaos of the classical kicked top map: Sensitivity to initial conditions

We focus on the frontier between the chaotic and regular regions for the classical version of the quantum kicked top. We show that the sensitivity to the initial conditions is numerically well characterised by $\xi=e_q^{\lambda_q t}$, where $e_{q}^{x}\equiv [ 1+(1-q) x]^{\frac{1}{1-q}} (e_1^x=e^x)$, and $\lambda_q$ is the $q$-generalization of the Lyapunov coefficient, a result that is consistent with nonextensive statistical mechanics, based on the entropy $S_q=(1- \sum_ip_i^q)/(q-1) (S_1 =-\sum_i p_i \ln p_i$). Our analysis shows that $q$ monotonically increases from zero to unity when the kicked-top perturbation parameter $\alpha$ increases from zero (unperturbed top) to $\alpha_c$, where $\alpha_c \simeq 3.2$. The entropic index $q$ remains equal to unity for $\alpha \ge \alpha_c$, parameter values for which the phase space is fully chaotic.Comment: To appear in "Complexity, Metastability and Nonextensivity" (World Scientific, Singapore, 2005), Eds. C. Beck, A. Rapisarda and C. Tsalli

Nonextensive statistical mechanics and central limit theorems II - Convolution of q-independent random variables

In this article we review recent generalisations of the central limit theorem for the sum of specially correlated (or q-independent) variables, focusing on q greater or equal than 1. Specifically, this kind of correlation turns the probability density function known as q-Gaussian, which emerges upon maximisation of the entropy Sq, into an attractor in probability space. Moreover, we also discuss a q-generalisation of a-stable Levy distributions which can as well be stable for this special kind of correlation.Within this context, we verify the emergence of a triplet of entropic indices which relate the form of the attractor, the correlation, and the scaling rate, similar to the q-triplet that connects the entropic indices characterising the sensitivity to initial conditions, the stationary state, and relaxation to the stationary state in anomalous systems.Comment: 14 pages, 4 figures, and 1 table. To appear in the Proceedings of the conference CTNEXT07, Complexity, Metastability and Nonextensivity, Catania, Italy, 1-5 July 2007, Eds. S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda and C. Tsallis (American Institute of Physics, 2008) in pres

Nonextensive statistical mechanics and central limit theorems I - Convolution of independent random variables and q-product

In this article we review the standard versions of the Central and of the Levy-Gnedenko Limit Theorems, and illustrate their application to the convolution of independent random variables associated with the distribution known as q-Gaussian. This distribution emerges upon extremisation of the nonadditive entropy, basis of nonextensive statistical mechanics. It has a finite variance for q 5/3. We exhibit that, in the case of (standard) independence, the q-Gaussian has either the Gaussian (if q 5/3) as its attractor in probability space. Moreover, we review a generalisation of the product, the q-product, which plays a central role in the approach of the specially correlated variables emerging within the nonextensive theory.Comment: 13 pages, 4 figures. To appear in the Proceedings of the conference CTNEXT07, Complexity, Metastability and Nonextensivity, Catania, Italy, 1-5 July 2007, Eds. S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda and C. Tsallis (American Institute of Physics, 2008) in pres

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

A new entropy based on a group-theoretical structure

A multi-parametric version of the nonadditive entropy $S_{q}$ is introduced. This new entropic form, denoted by $S_{a,b,r}$, possesses many interesting statistical properties, and it reduces to the entropy $S_q$ for $b=0$, $a=r:=1-q$ (hence Boltzmann-Gibbs entropy $S_{BG}$ for $b=0$, $a=r \to 0$). The construction of the entropy $S_{a,b,r}$ is based on a general group-theoretical approach recently proposed by one of us \cite{Tempesta2}. Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of $S_{a,b,r}$ with respect to the composition of statistically independent subsystems. Depending on the choice of the parameters, the entropy $S_{a,b,r}$ can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles $N$ of the system, or even stabilizes, by increasing $N$, to a limiting value. This paves the way to the use of this entropy in contexts where a system "freezes" some or many of its degrees of freedom by increasing the number of its constituting particles or subsystems.Comment: 12 pages including 1 figur