Abstract:
Random-matrix theory is applied to transition-rate matrices in the Pauli master equation. We study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems. Both the cases of identical and of independent rates of forward and backward transitions are considered. The first case leads to symmetric transition-rate matrices, whereas the second corresponds to general, asymmetric matrices. The resulting matrix ensembles are different from the standard ensembles and show different eigenvalue distributions. For example, the fraction of real eigenvalues scales anomalously with matrix dimension in the asymmetric case.

Abstract:
Spontaneously created vortex-antivortex pairs are the predominant source of flux noise in high-temperature superconductors. In principle, flux noise measurements allow to check theoretical predictions for both the distribution of vortex-pair sizes and for the vortex diffusivity. In this paper the flux-noise power spectrum is calculated for the highly anisotropic high-temperature superconductor Bi-2212, both for bulk crystals and for ultra-thin films. The spectrum is basically given by the Fourier transform of the temporal magnetic-field correlation function. We start from a Berezinskii-Kosterlitz-Thouless type theory and incorporate vortex diffusion, intra-pair vortex interaction, and annihilation of pairs by means of a Fokker-Planck equation to determine the noise spectrum below and above the superconducting transition temperature. We find white noise at low frequencies omega and a spectrum proportional to 1/omega^(3/2) at high frequencies. The cross-over frequency between these regimes strongly depends on temperature. The results are compared with earlier results of computer simulations.

Abstract:
Based on the dissipative Landau-Lifshitz equation, the spatiotemporal structure formation problem is investigated in the region far above the transverse ferromagnetic resonance instability. Apart from the external fields, the model contains an isotropic exchange field, a shape demagnetisation field and an anisotropy field. Numerical simulations exhibit in the rotating frame a stationary domain structure with a precessing motion in the walls regimes. Employing analytical methods, characteristic elements of this structure are explained. This driven dissipative system shows similarities to equilibrium systems of coexisting phases and organises itself in such a way that the local dynamics tends to become hamiltonian.

Abstract:
A multiple-time scaling analysis of the dissipative, transversely driven Landau-Lifshitz equation in presence of exchange, shape demagnetisation and week anisotropy fields is performed for a dynamic domain state. Stationary solutions of the resulting equations explain the spatiotemporal structure of the walls and are in agreement with previous simulations.

Abstract:
Systems of coupled rate equations are ubiquitous in many areas of science, for example in the description of electronic transport through quantum dots and molecules. They can be understood as a continuity equation expressing the conservation of probability. It is shown that this conservation law can be implemented by constructing a gauge theory akin to classical electrodynamics on the network of possible states described by the rate equations. The properties of this gauge theory are analyzed. It turns out that the network is maximally connected with respect to the electromagnetic fields even if the allowed transitions form a sparse network. It is found that the numbers of degrees of freedom of the electric and magnetic fields are equal. The results shed light on the structure of classical abelian gauge theory beyond the particular motivation in terms of rate equations.

Abstract:
Inelastic tunneling through magnetically anisotropic molecules is studied theoretically in the presence of a strong magnetic field. Since the molecular orientation is not well controlled in tunneling experiments, we consider arbitrary angles between easy axis and field. This destroys all conservation laws except that of charge, leading to a rich fine structure in the differential conductance. Besides single molecules we also study monolayers of molecules with either aligned or random easy axes. We show that detailed information on the molecular transitions and orientations can be obtained from the differential conductance for varying magnetic field. For random easy axes, averaging over orientations leads to van Hove singularities in the differential conductance. Rate equations in the sequential-tunneling approximation are employed. An efficient approximation for their solution for complex molecules is presented. The results are applied to Mn12-based magnetic molecules.

Abstract:
An important class of approaches to the description of electronic transport through molecules and quantum dots is based on the master equation. We discuss various formalisms for deriving a master equation and their interrelations. It is shown that the master equations derived by Wangsness, Bloch, and Redfield and by Koenig et al. are equivalent. The roles of the large-reservoir and Markov approximations are clarified. The Markov approximation is traced back to nonzero bias voltage and temperature, whereas interactions and the corresponding rapid relaxation in the leads are shown to be irrelevant for the transport under certain conditions. It is explained why the T-matrix formalism gives incomplete results except for diagonal density operators and to second order in the tunneling amplitudes. The time-convolutionless master equation is adapted to tunneling problems and a diagrammatic scheme for generating arbitrary orders in the tunneling amplitudes is developed.

Abstract:
A renormalization group theory for a system consisting of coupled superconducting layers as a model for typical high-temperature superconducters is developed. In a first step the electromagnetic interaction over infinitely many layers is taken into account, but the Josephson coupling is neglected. In this case the corrections to two-dimensional behavior due to the presence of the other layers are very small. Next, renormalization group equations for a layered system with very strong Josephson coupling are derived, taking into account only the smallest possible Josephson vortex loops. The applicability of these two limiting cases to typical high-temperature superconductors is discussed. Finally, it is argued that the original renormalization group approach by Kosterlitz is not applicable to a layered system with intermediate Josephson coupling.