A functional calculus for the magnetization dynamics

A functional calculus approach is applied to the derivation of evolution equations for the moments of the magnetization dynamics of systems subject to stochastic fields. It allows us to derive a general framework for obtaining the master equation for the stochastic magnetization dynamics, that is applied to both, Markovian and non-Markovian dynamics. The formalism is applied for studying different kinds of interactions, that are of practical relevance and hierarchies of evolution equations for the moments of the distribution of the magnetization are obtained. In each case, assumptions are spelled out, in order to close the hierarchies. These closure assumptions are tested by extensive numerical studies, that probe the validity of Gaussian or non--Gaussian closure Ans\"atze.Comment: 17 pages, 5 figure

Non-Markovian magnetization dynamics for uniaxial nanomagnets

A stochastic approach for the description of the time evolution of the magnetization of nanomagnets is proposed, that interpolates between the Landau-Lifshitz-Gilbert and the Landau-Lifshitz-Bloch approximations, by varying the strength of the noise. Its finite autocorrelation time, i.e. when it may be described as colored, rather than white, is, also, taken into account and the consequences, on the scale of the response of the magnetization are investigated. It is shown that the hierarchy for the moments of the magnetization can be closed, by introducing a suitable truncation scheme, whose validity is tested by direct numerical solution of the moment equations and compared to the averages obtained from a numerical solution of the corresponding colored stochastic Langevin equation. This comparison is performed on magnetic systems subject to both an external uniform magnetic field and an internal one-site uniaxial anisotropy.Comment: 4 pages, 3 figure

Quantum Magnets and Matrix Lorenz Systems

The Landau--Lifshitz--Gilbert equations for the evolution of the magnetization, in presence of an external torque, can be cast in the form of the Lorenz equations and, thus, can describe chaotic fluctuations. To study quantum effects, we describe the magnetization by matrices, that take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can describe chaotic fluctuations. We identify a criterion, for the appearance of such non-linear terms. This depends on whether an invariant, symmetric tensor of the algebra can vanish or not. This proposal is studied in detail for the fundamental representation of $\mathfrak{u}(2)=\mathfrak{u}(1)\times\mathfrak{su}(2)$. We find a knotted structure for the attractor, a bimodal distribution for the largest Lyapunov exponent and that the dynamics takes place within the Cartan subalgebra, that does not contain only the identity matrix, thereby can describe the quantum fluctuations.Comment: 5 pages, 3 figures. Uses jpconf style. Presented at the ICM-SQUARE 4 conference, Madrid, August 2014. The topic is a special case of the content of 1404.7774, currently under revisio

Knotted Strange Attractors and Matrix Lorenz Systems

A generalization of the Lorenz equations is proposed where the variables take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can describe chaotic fluctuations. We identify a criterion, for the appearance of such non-linear terms. This depends on whether an invariant, symmetric tensor of the algebra can vanish or not. This proposal is studied in detail for the fundamental representation of $\mathfrak{u}(2)$. We find a knotted structure for the attractor, a bimodal distribution for the largest Lyapunov exponent and that the dynamics takes place within the Cartan subalgebra, that does not contain only the identity matrix, thereby can describe the quantum fluctuations.Comment: 10 pages Revtex, 3 figure

Closing the hierarchy for non-Markovian magnetization dynamics

We propose a stochastic approach for the description of the time evolution of the magnetization of nanomagnets, that interpolates between the Landau--Lifshitz--Gilbert and the Landau--Lifshitz--Bloch approximations, by varying the strength of the noise. In addition, we take into account the autocorrelation time of the noise and explore the consequences, when it is finite, on the scale of the response of the magnetization, i.e. when it may be described as colored, rather than white, noise and non-Markovian features become relevant. We close the hierarchy for the moments of the magnetization, by introducing a suitable truncation scheme, whose validity is tested by direct numerical solution of the moment equations and compared to the average deduced from a numerical solution of the corresponding stochastic Langevin equation. In this way we establish a general framework, that allows both coarse-graining simulations and faster calculations beyond the truncation approximation used here.Comment: 5 pages LaTeX2e, 2 EPS figures; uses elsarticle.cls. Presented at HMM 2015, 10th International Symposium on Hysteresis Modeling and Micromagnetic

Colored-noise magnetization dynamics: from weakly to strongly correlated noise

Statistical averaging theorems allow us to derive a set of equations for the averaged magnetization dynamics in the presence of colored (non-Markovian) noise. The non-Markovian character of the noise is described by a finite auto-correlation time, tau, that can be identified with the finite response time of the thermal bath to the system of interest. Hitherto, this model was only tested for the case of weakly correlated noise (when tau is equivalent or smaller than the integration timestep). In order to probe its validity for a broader range of auto-correlation times, a non-Markovian integration model, based on the stochastic Landau-Lifshitz-Gilbert equation is presented. Comparisons between the two models are discussed, and these provide evidence that both formalisms remain equivalent, even for strongly correlated noise (i.e. tau much larger than the integration timestep).Comment: 4 pages LaTeX2e, 3 EPS figures; uses IEEEtran.cl

Frequency-dependent effective permeability tensor of unsaturated polycrystalline ferrites

Frequency-dependent permeability tensor for unsaturated polycrystalline ferrites is derived through an effective medium approximation that combines both domain-wall motion and rotation of domains in a single consistent scattering framework. Thus derived permeability tensor is averaged on a distribution function of the free energy that encodes paramagnetic states for anhysteretic loops. The initial permeability is computed and frequency spectra are given by varying macroscopic remanent field.Comment: 24 pages, 3 figure

El lugar como regalo: el caso de La Martella

[Resumen] El deseo de enfrentarme a esta investigación nace de la necesidad de querer entender el valor que tiene la lectura de los espacios públicos de una ciudad para la generación de nuevos espacios cualitativamente significativos. Se afrontará el argumento trayendo el caso del proyecto del barrio La Martella propuesto por Ludovico Quaroni en el 1951 como respuesta a la necesidad de re-ubicar los habitantes de los Sassi de Matera en el campo limítrofe, y, además, con este análisis se pondrá en luz la importancia de los movimientos de los individuos, voluntarios o inducidos, en la generación de esos mismos espacios. Este estudio se propone analizar aquella parte hermética del territorio italiano, sea por su morfología territorial, sea por su cultura, y a su vez entender, si la forma en que se ha dado la re-colocación ha sido una solución adecuada. La Martella es la extensión física de Matera, pero también es un complejo contenedor de vicisitudes, un puente entre pasado y futuro, entre arquitectura primitiva y moderna. Territorio y sociedad dialogan entre ellas, evolucionan, cambian. Pero, a la vez, siguiendo el proceso de transformación del lugar nos preguntamos si ese dialogo plantea alguna continuidad con la historia y con la cultura de la ciudad originaria. Si la memoria ha sido conservada o si en la sucesión dialógica de los acontecimientos se produce una discontinuidad. Estas preguntas nacen de la hipótesis que un proyecto no se agote solo en la forma, sino que sea un cruce entre mente, territorio y sociedad. Se pondrá el acento sobre la simbología de los lugares religiosos, sobre las fiestas paganas, y sobre el sucederse de espacios intersticiales y sus utilizo social, para compararlos con la estructura urbanística de la nueva aldea de La Martella. En ese sistema de prefiguración, configuración y refiguración del lugar, el fenómeno social hubiera debido estar intrínsecamente relacionado a la arquitectura y al espacio, pero, finalmente, en ese proceso de comunicación falla la comprensión entre autor y usuarios, lo cual producirá un ulterior abandono del pueblo. Por lo tanto se intentará comprender en donde y porque el proceso de re-dibujo no haya tenido éxito en la lectura del nuevo lugar por parte de sus protagonistas