Standard 1D solar atmosphere as initial condition for MHD simulations and switch-on effects

Many applications in Solar physics need a 1D atmospheric model as initial condition or as reference for inversions of observational data. The VAL atmospheric models are based on observations and are widely used since decades. Complementary to that, the FAL models implement radiative hydrodynamics and showed the shortcomings of the VAL models since almost equally long time. In this work, we present a new 1D layered atmosphere that spans not only from the photosphere to the transition region, but from the solar interior up to far in the corona. We also discuss typical mistakes that are done when switching on simulations based on such an initial condition and show how the initial condition can be equilibrated so that a simulation can start smoothly. The 1D atmosphere we present here served well as initial condition for HD and MHD simulations and should also be considered as reference data for solving inverse problems.Comment: 10 pages, 3 figures, published versio

Uncertainty Relation for the Discrete Fourier Transform

We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=exp[i phi] VU. Its most important application is to constrain how much a quantum state can be localised simultaneously in two mutually unbiased bases related by a Discrete Fourier Transform. It provides an uncertainty relation which smoothly interpolates between the well known cases of the Pauli operators in 2 dimensions and the continuous variables position and momentum. This work also provides an uncertainty relation for modular variables, and could find applications in signal processing. In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the Harmonic oscillator equation.Comment: Extended Version; 13 pages; In press in Phys. Rev. Let

Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)

We consider the Grothendieck ring of the fusion algebra of the W-extended logarithmic minimal model WLM(1,p). Informally, this is the fusion ring of W-irreducible characters so it is blind to the Jordan block structures associated with reducible yet indecomposable representations. As in the rational models, the Grothendieck ring is described by a simple graph fusion algebra. The 2p-dimensional matrices of the regular representation are mutually commuting but not diagonalizable. They are brought simultaneously to Jordan form by the modular data coming from the full (3p-1)-dimensional S-matrix which includes transformations of the p-1 pseudo-characters. The spectral decomposition yields a Verlinde-like formula that is manifestly independent of the modular parameter $\tau$ but is, in fact, equivalent to the Verlinde-like formula recently proposed by Gaberdiel and Runkel involving a $\tau$-dependent S-matrix.Comment: 13 pages, v2: example, comments and references adde

Temperature and Disorder Chaos in Low Dimensional Directed Paths

The responses of a $1+\epsilon$ dimensional directed path to temperature and to potential variations are calculated exactly, and are governed by the same scaling form. The short scale decorrelation (strong correlation regime) leads to the overlap length predicted by heuristic approaches; its temperature dependence and large absolute value agree with scaling and numerical observations. Beyond the overlap length (weak correlation regime), the correlation decays algebraically. A clear physical mechanism explains the behavior in each case: the initial decorrelation is due to `fragile droplets,' which contribute to the entropy fluctuations as $\sqrt{T}$, while the residual correlation results from accidental intersections of otherwise uncorrelated configurations.Comment: four pages, revtex4; minor modifications in the text and typos correcte