Ziel der vorliegenden Arbeit ist es, das Preisach-Modell bezüglich stochastischer äußerer Felder und temperaturbezogener Aspekte zu untersuchen. Das phänomenologische Preisach-Modell wird oft erfolgreich angewendet, um Systeme mit Hysterese zu beschreiben. Im ersten Teil der Arbeit wird die Antwort des Preisach-Modells auf stochastische äußere Felder untersucht. Hier liegt das Augenmerk hauptsächlich auf der Autokorrelation; sie dient dazu den Einfluss des hysteretischen Gedächtnisses zu quantifizieren. Mit analytischen Methoden wird gezeigt, dass sich ein Langzeitgedächtnis, sichtbar in der Autokorrelation der Systemantwort, entwickeln kann, selbst wenn das treibende Feld unkorreliert ist. Im Anschluss werden diese Resultate, m.H. von Simulationen, auf äußere Felder ausgeweitet, die selbst Korrelationen aufweisen können. Der zweite Teil der Arbeit befasst sich mit dem Einfluss einer endlichen Temperatur auf das Preisach-Modell. Es werden unterschiedliche Methoden besprochen, wie das Nichtgleichgewichtsmodell in seiner mikromagnetischen Interpretation mit Temperatur als Gleichgewichtseigenschaft verknüpft werden kann. Eine Formulierung wird genutzt, um die Magnetisierung von Nickelnanopartikeln in einer Fullerenmatrix zu simulieren und mit Experimenten zu vergleichen. Des Weiteren wird die Relaxationsdynamik des Gedächtnisses des Preisach-Modells bei endlichen Temperaturen untersucht. / The aim of this thesis is to investigate the Preisach model in regard to stochastically driving and temperature-related aspects. The Preisach model is a phenomenological model for systems with hysteresis which is often successfully applied. Hysteresis is a widespread phenomenon which is observed in nature and the key feature of certain technological applications. Further, it contributes to phenomena of interest in social science and economics as well. Prominent examples are the magnetization of ferromagnetic materials in an external magnetic field or the adsorption-desorption hysteresis observed in porous media. Hysteresis involves the development of a hysteresis memory, and multistability in the interrelations between external driving fields and system response. In the first part, we mainly investigate the response of Preisach hysteresis models driven by stochastic input processes with regard to autocorrelation functions to quantify the influence of the system’s memory. Using rigorous methods, it is shown that the development of a hysteresis memory is reflected in the possibility of long-time tails in the autocorrelation functions, even for uncorrelated driving fields. In the case of uncorrelated driving, these long-time tails in the autocorrelations of the system’s response are determined only by the tails of the involved densities. They will be observed if there are broad Preisach densities assigning a high weight to elementary loops of large width and narrow input densities such that rare extreme events of the input time series contribute significantly to the output for a long period of time. Afterwards, these results are extended by simulations to driving fields which themselves show correlations. It is shown that the autocorrelation of the output does not decay faster than the autocorrelation of the input process. Further, there is a possibility that long-term memory in the hysteretic response is more pronounced in the case of uncorrelated driving than in the case of correlated driving. The behavior of the output probability distribution at the saturation values is quite universal. It is not affected by the presence of correlations and allows conclusions whether the input density is much more narrow than the Preisach density or not. Moreover, the existence of effective Preisach densities is shown which define equivalence classes of systems of input and Preisach densities which lead to realizations of the same output variable. The asymptotic behavior of an effective Preisach density determines the asymptotic correlation decay of the system’s response in the case of uncorrelated driving. In the second part, temperature-related effects are considered. It is reviewed how the non-equilibrium Preisach model in its micromagnetic picture can be related to temperature within the framework of extended irreversible thermodynamics. The irreversible response of a ferromagnetic material, namely, Nickel nanoparticles in a fullerene matrix, is simulated. The model includes superparamagnetism where ferromagnetism breaks down at temperatures lower than the Curie temperature and the results are compared to experimental data. Furthermore, we adapt known results for the thermal relaxation of the system’s memory in the form of a front propagation in the Preisach plane derived basically from solving a master equation and by the use of a contradictory assumption. A closer look is taken at short time scales which dissolves the contradiction and shows that the known results apply, taking into account the fact that the dividing line propagation starts with an additional delay time depending on the front coordinates in the Preisach plane. Additionally, it is outlined how thermal relaxation behavior in the Preisach model of hysteresis can be studied using a Fokker-Planck equation. The latter is solved analytically in the non-hysteretic limit using eigenfunction methods. The results indicate a change in the relaxation behavior, especially on short time scales.

Ziel der vorliegenden Arbeit ist es, das Preisach-Modell bezüglich stochastischer äußerer Felder und temperaturbezogener Aspekte zu untersuchen. Das phänomenologische Preisach-Modell wird oft erfolgreich angewendet, um Systeme mit Hysterese zu beschreiben. Im ersten Teil der Arbeit wird die Antwort des Preisach-Modells auf stochastische äußere Felder untersucht. Hier liegt das Augenmerk hauptsächlich auf der Autokorrelation; sie dient dazu den Einfluss des hysteretischen Gedächtnisses zu quantifizieren. Mit analytischen Methoden wird gezeigt, dass sich ein Langzeitgedächtnis, sichtbar in der Autokorrelation der Systemantwort, entwickeln kann, selbst wenn das treibende Feld unkorreliert ist. Im Anschluss werden diese Resultate, m.H. von Simulationen, auf äußere Felder ausgeweitet, die selbst Korrelationen aufweisen können. Der zweite Teil der Arbeit befasst sich mit dem Einfluss einer endlichen Temperatur auf das Preisach-Modell. Es werden unterschiedliche Methoden besprochen, wie das Nichtgleichgewichtsmodell in seiner mikromagnetischen Interpretation mit Temperatur als Gleichgewichtseigenschaft verknüpft werden kann. Eine Formulierung wird genutzt, um die Magnetisierung von Nickelnanopartikeln in einer Fullerenmatrix zu simulieren und mit Experimenten zu vergleichen. Des Weiteren wird die Relaxationsdynamik des Gedächtnisses des Preisach-Modells bei endlichen Temperaturen untersucht. / The aim of this thesis is to investigate the Preisach model in regard to stochastically driving and temperature-related aspects. The Preisach model is a phenomenological model for systems with hysteresis which is often successfully applied. Hysteresis is a widespread phenomenon which is observed in nature and the key feature of certain technological applications. Further, it contributes to phenomena of interest in social science and economics as well. Prominent examples are the magnetization of ferromagnetic materials in an external magnetic field or the adsorption-desorption hysteresis observed in porous media. Hysteresis involves the development of a hysteresis memory, and multistability in the interrelations between external driving fields and system response. In the first part, we mainly investigate the response of Preisach hysteresis models driven by stochastic input processes with regard to autocorrelation functions to quantify the influence of the system’s memory. Using rigorous methods, it is shown that the development of a hysteresis memory is reflected in the possibility of long-time tails in the autocorrelation functions, even for uncorrelated driving fields. In the case of uncorrelated driving, these long-time tails in the autocorrelations of the system’s response are determined only by the tails of the involved densities. They will be observed if there are broad Preisach densities assigning a high weight to elementary loops of large width and narrow input densities such that rare extreme events of the input time series contribute significantly to the output for a long period of time. Afterwards, these results are extended by simulations to driving fields which themselves show correlations. It is shown that the autocorrelation of the output does not decay faster than the autocorrelation of the input process. Further, there is a possibility that long-term memory in the hysteretic response is more pronounced in the case of uncorrelated driving than in the case of correlated driving. The behavior of the output probability distribution at the saturation values is quite universal. It is not affected by the presence of correlations and allows conclusions whether the input density is much more narrow than the Preisach density or not. Moreover, the existence of effective Preisach densities is shown which define equivalence classes of systems of input and Preisach densities which lead to realizations of the same output variable. The asymptotic behavior of an effective Preisach density determines the asymptotic correlation decay of the system’s response in the case of uncorrelated driving. In the second part, temperature-related effects are considered. It is reviewed how the non-equilibrium Preisach model in its micromagnetic picture can be related to temperature within the framework of extended irreversible thermodynamics. The irreversible response of a ferromagnetic material, namely, Nickel nanoparticles in a fullerene matrix, is simulated. The model includes superparamagnetism where ferromagnetism breaks down at temperatures lower than the Curie temperature and the results are compared to experimental data. Furthermore, we adapt known results for the thermal relaxation of the system’s memory in the form of a front propagation in the Preisach plane derived basically from solving a master equation and by the use of a contradictory assumption. A closer look is taken at short time scales which dissolves the contradiction and shows that the known results apply, taking into account the fact that the dividing line propagation starts with an additional delay time depending on the front coordinates in the Preisach plane. Additionally, it is outlined how thermal relaxation behavior in the Preisach model of hysteresis can be studied using a Fokker-Planck equation. The latter is solved analytically in the non-hysteretic limit using eigenfunction methods. The results indicate a change in the relaxation behavior, especially on short time scales.