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The development of multi-model ensembles for reliable predictions of inter-annual climate fluctuations and climate change, and their application to health, agronomy and water management, are discussed.

We give a self-contained (though sketchy) derivation of the charged multiplicity distribution of final hadrons in a simplified (only pions) version of the statistical bootstrap model (SBM). The analytical form of this distribution is determined by two prominent features of SBM: a singularity at some temperatureT0 (related to the phase transition hadrons ? quark-gluon plasma) and the production of Poisson-distributed hadron clusters with well-defined decay structure. The multiplicity distribution of final hadrons has two free parameters: an average temperatureT and the mean number of clusters, both depending on the collision energy vs such that for vs?8, the temperature tends to the limitT0 while may grow (very slowly) without bound. With these two free parameters we obtain reasonable to good fits to experimental data at energies ranging from vs=7.87 to 900 GeV (full and in some cases cut pseudorapidity). The average cluster sizes found in our fits are compatible with those inferred from (pseudo) rapidity correlations in the data. We discuss the relation of this model to the negative binomial distribution (NBD).

The equationz=2G(z)-expG(z)+1 (and similar ones obtained from it by substitutions) appears in connection with a variety of problems ranging from pure mathematics (combinatorics; some first order, nonlinear differential equations) over statistical thermodynamics to renormalization theory. It is therefore of interest to solve this equation forG(z) explicitly. It turns out, after study of the complex structure of thez andG planes, that an explicit integral representation ofG(z) can be given, which may be directly used for numerical calculations of high precision.

Do we see the phase transition hadron → quark-gluon plasma (predicted by so many models) at pp collider energies?

We consider relativistic statistical thermodynamics of an ideal Boltzmann gas consisting of the particlesK, N, ?, S and their antiparticles. Baryon number (B) and strangeness (S) are conserved. While any relativistic gas is necessarily grand canonical with respect to particle numbers, conservation laws can be treated canonically or grand canonically. We construct the partition function for canonicalB×S conservation and compare it with the grand canonical one. It is found that the grand canonical partition function is equivalent to a largeB approximation of the canonical one. The relative difference between canonical and grand canonical quantities seems to decrease like const/B (two numerical examples) and from this a simple thumb rule for computing canonical quantities from grand canonical ones is guessed. For precise calculations, an integral representation is given.

We apply the technique of Gaussian regularization to the grand canonical pressure partition function to show that under physically meaningful boundary conditions (non-existence of external confining vessels, i.e., no fixed volumes) the energy density and similar intensive quantities have, in a statistical bootstrap model of extended hadrons (Van der Waals type volume corrections), indeed the singularity claimed in previous papers. Earlier results obtained with an entirely different technique (which had been criticized) are recovered and shown to be correct. The technique used here is useful in all cases where the volume is not imposed from the outside, but results from the internal dynamics of the system as is generally the case in high energy physics and astrophysics.

The statistical model of particle production makes definite predictions about «compound-elastic» scattering cross-sections. Such predictions are available in a numerical form for π-N and p-p collisions with primary energy (laboratory-system) between zero and about 20 to 30 GeV. The result is that the «compound-elastic» cross-section for π-N should show the following dependence on the total centre-of-momentum energyE(measured in GeV):wheref(E)is a slowly varying function representing the «inelasticity». Experimental results are compared to this prediction and qualitative agreement is found.

Zusammenfassung In dieser Arbeit wird versucht, an einem BaTiO3-Eindomänenkristall, von einem einfachen Modell ausgehend, die dielektrischen Eigenschaften bei Zimmertemperatur zu verstehen. Es wird benutzt, daß Ti und das durch die Polarisation ausgezeichnete Sauerstoffion Oz aus ihren Symmetrielagen in Zellen- bzw. Flächenmitte ein Stück aufeinander zu parallel zur Polarisationsrichtung verschoben sind. Für diese Verschiebungen gibt es gute Messungen. Nimmt man an, daß in einem idealen Eindomänenkristall diese Verschiebungen in jeder Zelle dieselben sind, so folgt daraus, daß diese beiden Ionen an Orten sind, wo die inneren elektrostatischen Felder, hervorgerufen von den übrigen Ionen, nicht verschwinden. Andererseits entstehen damit auch an den Orten der übrigen Ionen solche Felder. Da alle Ionen polarisierbar sind, erhalten sie durch diese Felder Dipolmomente, die zu dem Dipolmoment hinzutreten, welches allein schon durch die unsymmetrische Lage von Ti und Oz auftritt. Durch diese sämtlichen Dipole entsteht ein weiteres inneres Feld, welches sich dem von den Ladungen hervorgerufenen überlagert. Dieses Gesamtfeld erst ruft die eigentlich vorhandenen Dipolmomente hervor. Die ganze Wechselwirkung zwischen Polen und Dipolen sowie die Entstehung der inneren Gesamtfelder werden durch ein System von vier Gleichungen mit vier Unbekannten beschrieben, die es gestatten, die Dipolmomente, inneren Felder und die Dielektrizitätskonstante bei gegebenen Ionenladungen, Polarisierbarkeiten und äußerem Feld zu berechnen. Dabei werden Funktionen benutzt und numerisch berechnet, die in einem von Punktladungen bzw. Dipolen besetzten primitiv-tetragonalen Gitter diez-Komponente der Feldstärke als analytische Funktion (Fourier- bzw. Potenzreihen) längs gewisser Gittergeraden angeben.