The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.

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This diploma thesis deals with the influence of long fiber optic paths on the polarization state of light and their use for powering polarization sensors. The aim of the diploma thesis was to design the arrangement of optical fiber components so that the polarization properties of light at the end of the path are further usable for sensory purposes. Four partial measurements with a long path (in a laboratory setting, laying in the ground, on a curtain, under the influence of external influence) and one measurement without a path under the influence of external influence were designed. The overall measurement results demonstrate that for the functional power supply of long-distance sensor systems, an existing single-mode fiber can be used (laid by laying in the ground), into which a light source for a given sensor system would be multiplexed. In the discussion, all types of measurements are analyzed and compared with each other. Finally, the overall result is summarized and applications are described in which such a sensor system could be used.

birefringance, degreeofpolarization, polarization, Stokesvectors, Poincarésphere, state of polarization, optical power, sensoric fibre, polarization-maintaining fiber, polaroid, poalrimeter

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