Equivalent power law potentials

It is shown that the radial Schroedinger equation for a power law potential and a particular angular momentum may be transformed using a change of variable into another Schroedinger equation for a different power law potential and a different angular momentum. It is shown that this leads to a mapping of the spectra of the two related power law potentials. It is shown that a similar correspondence between the classical orbits in the two related power law potentials exists. The well known correspondence of the Coulomb and oscillator spectra is a special case of a more general correspondence between power law potentials.Comment: 10 pages. Typographical mistakes in the earlier version are correcte

Quasi-power law ensembles

Quasi-power law ensembles are discussed from the perspective of nonextensive Tsallis distributions characterized by a nonextensive parameter $q$. A number of possible sources of such distributions are presented in more detail. It is further demonstrated that data suggest that nonextensive parameters deduced from Tsallis distributions functions $f\left(p_T\right)$, $q_1$, and from multiplicity distributions (connected with Tsallis entropy), $q_2$, are not identical and that they are connected via $q_1 + q_2 = 2$. It is also shown that Tsallis distributions can be obtained directly from Shannon information entropy, provided some special constraints are imposed. They are connected with the type of dynamical processes under consideration (additive or multiplicative). Finally, it is shown how a Tsallis distribution can accommodate the log-oscillating behavior apparently seen in some multiparticle data.Comment: 20 pages, 4 figures. Based on the material presented as invited talk at 27th Marian Smoluchowski Symposium on Statistical Physics, September 22-26, Zakopane, Polan

Anisotropic Power-law Inflation

We study an inflationary scenario in supergravity model with a gauge kinetic function. We find exact anisotropic power-law inflationary solutions when both the potential function for an inflaton and the gauge kinetic function are exponential type. The dynamical system analysis tells us that the anisotropic power-law inflation is an attractor for a large parameter region.Comment: 14 pages, 1 figure. References added, minor corrections include

Power-law random walks

We present some new results about the distribution of a random walk whose independent steps follow a $q-$Gaussian distribution with exponent $\frac{1}{1-q}; q \in \mathbb{R}$. In the case $q>1$ we show that a stochastic representation of the point reached after $n$ steps of the walk can be expressed explicitly for all $n$. In the case $q<1,$ we show that the random walk can be interpreted as a projection of an isotropic random walk, i.e. a random walk with fixed length steps and uniformly distributed directions.Comment: 5 pages, 4 figure

Power-law Parameterized Quintessence Model

We introduce a power-law parameterized quintessence model for the dark energy which accelerate universe at the low redshifts while behaves as an ordinary matter for the early universe. We construct a unique scalar potential for this parameterized quintessence model. As the observational test, the Supernova Type Ia (SNIa) Gold sample data, size of baryonic acoustic peak from Sloan Digital Sky Survey (SDSS), the position of the acoustic peak from the CMB observations and structure formation from the 2dFGRS survey are used to constrain the parameters of the quintessence model. The best fit parameters indicates that the equation of state of this model at the present time is less than one $(w_0<-1)$ which violates the energy condition in General Relativity. Finally we compare the age of old objects with age of universe in this model.Comment: 11 pages, 17 figures, submitted to Phys. Rev.