Sperm cells are guided to the egg by chemoattractants in many species. Sperm cells are propelled in a liquid by the regular beat of their flagellum. In the presence of a concentration gradient of a chemoattractant, they can steer upwards the concentration gradient, a process called chemotaxis. Eggs release chemoattractants to guide the sperm cells to the egg. Sperm chemotaxis is best studied experimentally in the sea urchin. There, specific receptors in the flagellar membrane of the sperm cells are activated upon binding of chemoattractant molecules and trigger a signaling cascade which ultimately changes the activity of the molecular motors which drive the flagellar beat and result in a swimming response. Sea urchin sperm cells swim along circular and helical paths. Sperm cells of the sea urchin and several other species swim along helical paths far from boundary surfaces in the absence of chemoattractant. In a two-dimensional experimental geometry, sperm swimming paths are planar circles. The non-zero curvature of their swimming paths is a direct consequence of an asymmetry of their flagellar beat. In a concentration gradient of chemoattractant, sperm swimming path are drifting circles in two dimensions and bend helices in three dimensions. What is the working mechanism of sperm chemotaxis? In this thesis, we develop a theoretical description of sperm chemotaxis which can be subsumed as follows: While swimming along an approximately circular path in a concentration gradient a sperm cell traces a periodic concentration stimulus from the concentration field that has the frequency of circular swimming. The chemotactic signaling system processes this stimulus and causes a periodic modulation of the curvature of the swimming path which then gives rise to a swimming path which is a drifting circle. The relative direction of the drift with respect to the gradient direction is determined by the phase shift between the stimulus and the curvature oscillations. This picture is in perfect agreement with recent experimental findings. The mechanism is more general and also works in three dimensions for swimming along helical paths. Our results. Our theoretical description of sperm chemotaxis clarifies the concepts underlying sperm chemotaxis. In particular, we derive the role of internal timing of the chemotactic signaling system for sperm chemotaxis. We conclude that sampling a concentration field along circular and helical paths is a robust strategy for chemotaxis that does not require fine-tuning of parameters and which works reliable also in the presence of fluctuations. In a last chapter of this thesis, we discuss sperm chemotaxis in the more general context of an abstract search problem.

The asymptotic state of a quantum system, which is in contact with a heat bath, is strongly disturbed by a time-periodic driving in comparison to a time-independent system. In this thesis an extensive picture of the asymptotic state of time-periodic quantum systems is drawn by relating it to the structure of the corresponding classical phase space. To this end the occupation probabilities of the Floquet states are analyzed with respect to their semiclassical property of being either regular or chaotic. The regular Floquet states are occupied with exponential weights e^{-betaeff Ereg} similar to the canonical weights e^{-beta E} of time-independent systems. The regular energies Ereg are defined by the quantization of the time-periodic system, whose classical properties also determine the effective temperature 1/betaeff. In contrast, the chaotic Floquet states acquire almost equal probabilities, irrespective of their time-averaged energy. Beyond these semiclassical properties the existence of avoided crossings in the spectrum is an intrinsic quantum property of time-periodic systems. Avoided crossings can strongly influence the entire occupation distribution. As an impressive application a novel switching mechanism is proposed in a periodically driven double well potential coupled to a heat bath. By a weak variation of the driving amplitude its asymptotic state is switched from the ground state in one well to a state with higher average energy in the other well. / Der asymptotische Zustand eines Quantensystems, das in Kontakt mit einem Wärmebad steht, wird durch einen zeitlich periodischen Antrieb gegenüber einem zeitunabhängigen System nachhaltig verändert. In dieser Arbeit wird ein umfassendes Bild über den asymptotischen Zustand zeitlich periodischer Quantensysteme entworfen, indem es diesen zur Struktur des zugehörigen klassischen Phasenraums in Beziehung setzt. Dazu werden die Besetzungswahrscheinlichkeiten der Floquet-Zustände hinsichtlich ihrer semiklassischen Eigenschaft analysiert, nach welcher sie entweder regulär oder chaotisch sind. Die regulären Floquet-Zustände sind mit exponentiellen Gewichten e^{-betaeff Ereg} ähnlich der kanonischen Verteilung e^{-beta E} zeitunabhängiger Systeme besetzt. Dabei sind die reguläre Energien Ereg durch die Quantisierung des Systems vorgegeben, dessen klassische Eigenschaften auch die effektive Temperatur 1/betaeff bestimmen. Die chaotischen Zustände dagegen haben fast einheitliche Besetzungswahrscheinlichkeiten, welche unabhängig von ihrer mittleren Energie sind. Über diese semiklassischen Eigenschaften hinaus ist das Auftreten von vermiedenen Kreuzungen im Spektrum eine intrinsisch quantenmechanische Eigenschaft zeitlich periodischer Systeme. Diese können die gesamte Besetzungsverteilung nachhaltig beeinflussen und finden eine eindrucksvolle Anwendung in Form eines neuartigen Schaltmechanismus in einem harmonisch modulierten Doppelmuldenpotential in Kontakt mit einem Wärmebad. Der asymptotische Zustand kann unter geringer Variation der Antriebsamplitude vom Grundzustand der einen Mulde in einen Zustand höherer mittlerer Energie in der anderen Mulde geschaltet werden.