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G Lozano - One of the best experts on this subject based on the ideXlab platform.
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magnetization dynamics path integral formalism for the stochastic landau lifshitz Gilbert Equation
Journal of Statistical Mechanics: Theory and Experiment, 2014Co-Authors: Camille Aron, Daniel G Barci, Leticia F Cugliandolo, Zochil Gonzalez Arenas, G LozanoAbstract:We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the Landau?Lifshitz?Gilbert Equation?proposed by Brown (1963 Phys. Rev. 130 1677), with the possible addition of spin-torque terms. In the process of constructing this functional in the Cartesian coordinate system, we critically revisit this stochastic Equation. We present it in a form that accommodates for any discretization scheme thanks to the inclusion of a drift term. The generalized Equation?ensures the conservation of the magnetization modulus and the approach to the Gibbs?Boltzmann equilibrium in the absence of non-potential and time-dependent forces. The drift term vanishes only if the mid-point Stratonovich prescription is used. We?next reset the problem in the more natural spherical coordinate system. We show that the noise transforms non-trivially to spherical coordinates acquiring a non-vanishing mean value in this coordinate system, a fact that has been often overlooked in the literature. We next construct the generating functional formalism in this system of coordinates for any discretization prescription. The functional formalism in Cartesian or spherical coordinates should serve as a starting point to study different aspects of the out-of-equilibrium dynamics of magnets. Extensions to colored noise, micro-magnetism and disordered problems are straightforward.
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numerical integration of the stochastic landau lifshitz Gilbert Equation in generic time discretization schemes
Physical Review E, 2014Co-Authors: F Roma, Leticia F Cugliandolo, G LozanoAbstract:We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert Equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.
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magnetization dynamics path integral formalism for the stochastic landau lifshitz Gilbert Equation
arXiv: Statistical Mechanics, 2014Co-Authors: Camille Aron, Daniel G Barci, Leticia F Cugliandolo, Zochil Gonzalez Arenas, G LozanoAbstract:We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the Landau-Lifshitz-Gilbert Equation proposed by Brown, with the possible addition of spin-torque terms. In the process of constructing this functional in the Cartesian coordinate system, we critically revisit this stochastic Equation. We present it in a form that accommodates for any discretization scheme thanks to the inclusion of a drift term. The generalized Equation ensures the conservation of the magnetization modulus and the approach to the Gibbs-Boltzmann equilibrium in the absence of non-potential and time-dependent forces. The drift term vanishes only if the mid-point Stratonovich prescription is used. We next reset the problem in the more natural spherical coordinate system. We show that the noise transforms non-trivially to spherical coordinates acquiring a non-vanishing mean value in this coordinate system, a fact that has been often overlooked in the literature. We next construct the generating functional formalism in this system of coordinates for any discretization prescription. The functional formalism in Cartesian or spherical coordinates should serve as a starting point to study different aspects of the out-of-equilibrium dynamics of magnets. Extensions to colored noise, micro-magnetism and disordered problems are straightforward.
Changjian Xie - One of the best experts on this subject based on the ideXlab platform.
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a second order numerical method for landau lifshitz Gilbert Equation with large damping parameters
Journal of Computational Physics, 2021Co-Authors: Yongyong Cai, Jingrun Chen, Cheng Wang, Changjian XieAbstract:Abstract A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert Equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method are associated with the following features: (1) It only solves linear systems of Equations with coefficient matrices independent of the magnetization, and fast solvers are available, so that the numerical efficiency has been greatly improved, in comparison with the existing Gauss-Seidel project method. (2) The second-order accuracy in time is achieved, and it is unconditionally stable for large damping parameters. Moreover, both the second-order accuracy and the great efficiency improvement will be verified by several numerical examples in the 1D and 3D simulations. In the presence of large damping parameters, it is observed that this method is unconditionally stable and finds physically reasonable structures while many existing methods have failed. For the domain wall dynamics, the linear dependence of wall velocity with respect to the damping parameter and the external magnetic field will be obtained through the reported simulations.
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a second order numerical method for landau lifshitz Gilbert Equation with large damping parameters
arXiv: Computational Physics, 2021Co-Authors: Yongyong Cai, Jingrun Chen, Cheng Wang, Changjian XieAbstract:A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert Equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method are associated with the following features: (1) It only solves linear systems of Equations with constant coefficients where fast solvers are available, so that the numerical efficiency has been greatly improved, in comparison with the existing Gauss-Seidel project method. (2) The second-order accuracy in time is achieved, and it is unconditionally stable for large damping parameters. Moreover, both the second-order accuracy and the great efficiency improvement will be verified by several numerical examples in the 1D and 3D simulations. In the presence of large damping parameters, it is observed that this method is unconditionally stable and finds physically reasonable structures while many existing methods have failed. For the domain wall dynamics, the linear dependence of wall velocity with respect to the damping parameter and the external magnetic field will be obtained through the reported simulations.
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two improved gauss seidel projection methods for landau lifshitz Gilbert Equation
Journal of Computational Physics, 2020Co-Authors: Changjian Xie, Jingrun Chen, Xiaoping WangAbstract:Abstract Micromagnetic simulation is an important tool to study various dynamic behaviors of magnetic order in ferromagnetic materials. The underlying model is the Landau-Lifshitz-Gilbert Equation, where the magnetization dynamics is driven by the gyromagnetic torque term and the Gilbert damping term. Numerically, considerable progress has been made in the past decades. One of the most popular methods is the Gauss-Seidel projection method developed by Xiao-Ping Wang, Carlos Garcia-Cervera, and Weinan E in 2001. It first solves a set of heat Equations with constant coefficients and updates the gyromagnetic term in the Gauss-Seidel manner, and then solves another set of heat Equations with constant coefficients for the damping term. Afterwards, a projection step is applied to preserve the length constraint in the pointwise sense. This method has been verified to be unconditionally stable numerically and successfully applied to study magnetization dynamics under various controls. In this paper, we present two improved Gauss-Seidel projection methods with unconditional stability. The first method updates the gyromagnetic term and the damping term simultaneously and follows by a projection step. The second method introduces two sets of approximate solutions, where we update the gyromagnetic term and the damping term simultaneously for one set of approximate solutions and apply the projection step to the other set of approximate solutions in an alternating manner. Compared to the original Gauss-Seidel projection method which has to solve heat Equations 7 times at each time step, the improved methods solve heat Equations 5 times and 3 times, respectively. First-order accuracy in time and second-order accuracy in space are verified by examples in both 1D and 3D. In addition, unconditional stability with respect to both the grid size and the damping parameter is confirmed numerically. Application of both methods to a realistic material is also presented with hysteresis loops and magnetization profiles. Compared with the original method, the recorded running times suggest that savings of both methods are about 2/7 and 4/7 for the same accuracy requirement, respectively.
Dieter Suess - One of the best experts on this subject based on the ideXlab platform.
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learning magnetization dynamics
Journal of Magnetism and Magnetic Materials, 2019Co-Authors: Alexander Kovacs, Lukas Exl, Johann Fischbacher, Harald Oezelt, Markus Gusenbauer, Florian Bruckner, Dieter Suess, T SchreflAbstract:Abstract Deep neural networks are used to model the magnetization dynamics in magnetic thin film elements. The magnetic states of a thin film element can be represented in a low dimensional space. With convolutional autoencoders a compression ratio of 1024:1 was achieved. Time integration can be performed in the latent space with a second network which was trained by solutions of the Landau-Lifshitz-Gilbert Equation. Thus the magnetic response to an external field can be computed quickly.
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coupling of dynamical micromagnetism and a stationary spin drift diffusion Equation a step towards a fully self consistent spintronics framework
Physica B-condensed Matter, 2016Co-Authors: Michele Ruggeri, G Hrkac, Dieter Suess, Claas Abert, Dirk PraetoriusAbstract:Abstract We consider the coupling of the Landau–Lifshitz–Gilbert Equation with a quasilinear diffusion Equation to describe the interplay of magnetization and spin accumulation in magnetic-nonmagnetic multilayer structures. For this problem, we propose and analyze a convergent finite element integrator, where, in contrast to prior work, we consider the stationary limit for the spin diffusion. Numerical experiments underline that the new approach is more effective, since it leads to the same experimental results as for the model with time-dependent spin diffusion, but allows for larger time-steps of the numerical integrator.
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efficient energy minimization in finite difference micromagnetics speeding up hysteresis computations
Journal of Applied Physics, 2014Co-Authors: Claas Abert, Florian Bruckner, Gregor Wautischer, Armin Satz, Dieter SuessAbstract:We implement an efficient energy-minimization algorithm for finite-difference micromagnetics that proofs especially useful for the computation of hysteresis loops. Compared to results obtained by time integration of the Landau-Lifshitz-Gilbert Equation, a speedup of up to two orders of magnitude is gained. The method is implemented in a finite-difference code running on central processing units (CPUs) as well as graphics processing units (GPUs). This setup enables us to compute accurate hysteresis loops of large systems with a reasonable computational effort. As a benchmark, we solve the μMag standard problem #1 with a high spatial resolution and compare the results to the solution of the Landau-Lifshitz-Gilbert Equation in terms of accuracy and computing time.
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spin polarized transport in ferromagnetic multilayers an unconditionally convergent fem integrator
Computers & Mathematics With Applications, 2014Co-Authors: Claas Abert, G Hrkac, Michele Ruggeri, Dirk Praetorius, Marcus Page, Dieter SuessAbstract:We propose and analyze a decoupled time-marching scheme for the coupling of the Landau–Lifshitz–Gilbert Equation with a quasilinear diffusion Equation for the spin accumulation. This model describes the interplay of magnetization and electron spin accumulation in magnetic and nonmagnetic multilayer structures. Despite the strong nonlinearity of the overall PDE system, the proposed integrator requires only the solution of two linear systems per time-step. Unconditional convergence of the integrator towards weak solutions is proved.
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efficient energy minimization in finite difference micromagnetics speeding up hysteresis computations
arXiv: Computational Physics, 2014Co-Authors: Claas Abert, Florian Bruckner, Gregor Wautischer, Armin Satz, Dieter SuessAbstract:We implement an efficient energy-minimization algorithm for finite-difference micromagnetics that proofs especially useful for the computation of hysteresis loops. Compared to results obtained by time integration of the Landau-Lifshitz-Gilbert Equation, a speedup of up to two orders of magnitude is gained. The method is implemented in a finite-difference code running on CPUs as well as GPUs. This setup enables us to compute accurate hysteresis loops of large systems with a reasonable computational effort. As a benchmark we solve the {\mu}Mag Standard Problem #1 with a high spatial resolution and compare the results to the solution of the Landau-Lifshitz-Gilbert Equation in terms of accuracy and computing time.
Camille Aron - One of the best experts on this subject based on the ideXlab platform.
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magnetization dynamics path integral formalism for the stochastic landau lifshitz Gilbert Equation
Journal of Statistical Mechanics: Theory and Experiment, 2014Co-Authors: Camille Aron, Daniel G Barci, Leticia F Cugliandolo, Zochil Gonzalez Arenas, G LozanoAbstract:We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the Landau?Lifshitz?Gilbert Equation?proposed by Brown (1963 Phys. Rev. 130 1677), with the possible addition of spin-torque terms. In the process of constructing this functional in the Cartesian coordinate system, we critically revisit this stochastic Equation. We present it in a form that accommodates for any discretization scheme thanks to the inclusion of a drift term. The generalized Equation?ensures the conservation of the magnetization modulus and the approach to the Gibbs?Boltzmann equilibrium in the absence of non-potential and time-dependent forces. The drift term vanishes only if the mid-point Stratonovich prescription is used. We?next reset the problem in the more natural spherical coordinate system. We show that the noise transforms non-trivially to spherical coordinates acquiring a non-vanishing mean value in this coordinate system, a fact that has been often overlooked in the literature. We next construct the generating functional formalism in this system of coordinates for any discretization prescription. The functional formalism in Cartesian or spherical coordinates should serve as a starting point to study different aspects of the out-of-equilibrium dynamics of magnets. Extensions to colored noise, micro-magnetism and disordered problems are straightforward.
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magnetization dynamics path integral formalism for the stochastic landau lifshitz Gilbert Equation
arXiv: Statistical Mechanics, 2014Co-Authors: Camille Aron, Daniel G Barci, Leticia F Cugliandolo, Zochil Gonzalez Arenas, G LozanoAbstract:We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the Landau-Lifshitz-Gilbert Equation proposed by Brown, with the possible addition of spin-torque terms. In the process of constructing this functional in the Cartesian coordinate system, we critically revisit this stochastic Equation. We present it in a form that accommodates for any discretization scheme thanks to the inclusion of a drift term. The generalized Equation ensures the conservation of the magnetization modulus and the approach to the Gibbs-Boltzmann equilibrium in the absence of non-potential and time-dependent forces. The drift term vanishes only if the mid-point Stratonovich prescription is used. We next reset the problem in the more natural spherical coordinate system. We show that the noise transforms non-trivially to spherical coordinates acquiring a non-vanishing mean value in this coordinate system, a fact that has been often overlooked in the literature. We next construct the generating functional formalism in this system of coordinates for any discretization prescription. The functional formalism in Cartesian or spherical coordinates should serve as a starting point to study different aspects of the out-of-equilibrium dynamics of magnets. Extensions to colored noise, micro-magnetism and disordered problems are straightforward.
Zdzislaw Brzeźniak - One of the best experts on this subject based on the ideXlab platform.
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weak solutions of a stochastic landau lifshitz Gilbert Equation driven by pure jump noise
Communications in Mathematical Physics, 2019Co-Authors: Zdzislaw Brzeźniak, Utpal MannaAbstract:In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert Equation perturbed by pure jump noise in the Marcus canonical form. We show the existence of a weak martingale solution taking values in a two-dimensional sphere $${\mathbb{S}^2}$$ and discuss certain regularity results. The construction of a solution is based on the classical Faedo–Galerkin approximation, the compactness methods and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.
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stochastic landau lifshitz Gilbert Equation with anisotropy energy driven by pure jump noise
Computers & Mathematics With Applications, 2019Co-Authors: Zdzislaw Brzeźniak, Utpal MannaAbstract:Abstract In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert Equation with non-zero anisotropy energy, which is drive by pure jump noise. We show existence of weak martingale solutions taking values in a two-dimensional sphere S 2 . The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.
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large deviations and transitions between equilibria for stochastic landau lifshitz Gilbert Equation
Archive for Rational Mechanics and Analysis, 2017Co-Authors: Zdzislaw Brzeźniak, Beniamin Goldys, Terence JegarajAbstract:We study a stochastic Landau–Lifshitz Equation on a bounded interval and with finite dimensional noise. We first show that there exists a pathwise unique solution to this Equation and that this solution enjoys the maximal regularity property. Next, we prove the large deviations principle for the small noise asymptotic of solutions using the weak convergence method. An essential ingredient of the proof is the compactness, or weak to strong continuity, of the solution map for a deterministic Landau–Lifschitz Equation when considered as a transformation of external fields. We then apply this large deviations principle to show that small noise can cause magnetisation reversal. We also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise. The problem is motivated by applications from ferromagnetic nanowires to the fabrication of magnetic memories.
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computational studies for the stochastic landau lifshitz Gilbert Equation
SIAM Journal on Scientific Computing, 2013Co-Authors: ľubomir Baňas, Zdzislaw Brzeźniak, Andreas ProhlAbstract:The stochastic Landau--Lifshitz--Gilbert Equation describes the thermally induced dynamics of magnetic moments in ferromagnetic materials. Solutions of this highly nonlinear stochastic PDE are unit vector fields and satisfy an energy estimate. These are crucial properties to construct a convergent discretization in space and time. We propose a convergent finite element approximation of the problem based on the midpoint rule. The numerical scheme preserves the underlying properties of the continuous problem. Further, we construct a robust and efficient Newton-multigrid solver for the solution of the nonlinear systems associated with the discretized problems at each time level. Computational studies show the optimal convergence behavior of the scheme in the case of smooth solutions. Long-time dynamics for finite ensembles of spins evidence the ergodicity of an invariant measure of the continuum model. Numerical experiments in two dimensions demonstrate pathwise finite time blow-up behavior of the solution w...
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large deviations and transitions between equilibria for stochastic landau lifshitz Gilbert Equation
arXiv: Probability, 2012Co-Authors: Zdzislaw Brzeźniak, Beniamin Goldys, Terence JegarajAbstract:We study a stochastic Landau-Lifshitz Equation on a bounded interval and with finite dimensional noise. We first show that there exists a pathwise unique solution to this Equation and that this solution enjoys the maximal regularity property. Next, we prove the large deviations principle for small noise asymptotic of solutions using the weak convergence method. An essential ingredient of the proof is compactness, or weak to strong continuity, of the solution map for a deterministic Landau-Lifschitz Equation, when considered as a transformation of external fields. We then apply this large deviations principle to show that small noise can cause magnetisation reversal. We also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise. The problem is motivated by applications of ferromagnetic nanowires to the fabrication of magnetic memories. This is an updated version of the previous version of this paper.