Finsler Geometry Modeling Research Articles

Numerical study of anisotropic diffusion in Turing patterns based on Finsler geometry modeling.

We numerically study the anisotropic Turing patterns (TPs) of an activator-inhibitor system described by the reaction-diffusion (RD) equationof Turing, focusing on anisotropic diffusion using the Finsler geometry (FG) modeling technique. In FG modeling, the diffusion coefficients are dynamically generated to be direction dependent owing to an internal degree of freedom (IDOF) and its interaction with the activator and inhibitor. Because of this dynamical diffusion coefficient, FG modeling of the RD equationsharply contrasts with the standard numerical technique in which direction-dependent coefficients are manually assumed. To find the solution of the RD equationsin FG modeling, we use a hybrid numerical technique combining the Metropolis Monte Carlo method for IDOF updates and discrete RD equationsfor steady-state configurations of the activator-inhibitor variables. We find that the newly introduced IDOF and its interaction are a possible origin of spontaneously emergent anisotropic patterns of living organisms, such as zebra and fishes. Moreover, the IDOF makes TPs controllable by external conditions if the IDOF is identified with the direction of cell diffusion accompanied by thermal fluctuations.

The stability of 3D skyrmions under mechanical stress studied via Monte Carlo calculations

Using Monte Carlo (MC) simulations, we study the skyrmion stability/instability as a response to uniaxial mechanical stresses. Skyrmions emerge in chiral magnetic materials as a stable spin configuration under external magnetic field B→ with the competition of ferromagnetic interaction and Dzyaloshinskii–Moriya interaction (DMI) at low temperature T. Skyrmion configurations are also known to be stable (unstable) under a compressive stress applied parallel (perpendicular) to B→. To understand the origin of such experimentally confirmed stability/instability, we use the Finsler geometry modeling technique with a new degree of freedom for strains, which plays an essential role in DMI being anisotropic. We find from MC data that the area of the skyrmion state on the B−T phase diagram increases (decreases) depending on the direction of applied stresses, in agreement with reported experimental results. This change in the area of the skyrmion state indicates that skyrmions become more (less) stable if the tensile strain direction is parallel (perpendicular) to B→. From the numerical data in this paper, we find that the so-called magneto-elastic effect is suitably implemented in the effective DMI theory with the strain degree of freedom without complex magneto-elastic coupling terms for chiral magnetic materials. This result confirms that experimentally-observed skyrmion stability and instability are caused by DMI anisotropy.

Monte Carlo studies on shape deformation and stability of 3D skyrmions under mechanical stresses

We study the stability/instability of skyrmions under mechanical stresses by Monte Carlo simulations in a 3D disk composed of tetrahedrons. Skyrmions emerge in chiral magnetic materials, such as FeGe and MnSi, under the competition of ferromagnetic interaction (FMI) and Dzyaloshinskii-Moriya interaction (DMI) and are stabilized by the external magnetic field. Recent experimental studies show that skyrmions are also stabilized/destabilized by uniaxial compressive stress perpendicular to or along the magnetic field direction. These phenomena are studied by using a 3D Finsler geometry (FG) model. In this 3D FG model, the DMI coefficient is automatically anisotropic by a geometrically implemented coupling of strains and electronic spins. We find that skyrmions are stabilized (destabilized) by extension (compression) stress along the direction of the applied magnetic field consistent with reported experimental data. This consistency implies that the 3D FG model successfully implements the magnetostrictive or magneto-elastic effect of external mechanical stresses on chiral magnetic orders, including the skyrmion configuration.

Finsler geometry modeling and Monte Carlo study of skyrmion shape deformation by uniaxial stress

Skyrmions in chiral magnetic materials are topologically stable and energetically balanced spin configurations appearing under the presence of ferromagnetic interaction (FMI) and Dzyaloshinskii-Moriya interaction (DMI). Much of the current interest has focused on the effects of magneto-elastic coupling on these interactions under mechanical stimuli, such as uniaxial stresses for future applications in spintronics devices. Recent studies suggest that skyrmion shape deformations in thin films are attributed to an anisotropy in the coefficient of DMI, such that $D_{x}\!\not=\!D_{y}$, which makes the ratio $\lambda/D$ anistropic, where the coefficient of FMI $\lambda$ is isotropic. It is also possible that $\lambda_{x}\!\not=\!\lambda_{y}$ while $D$ is isotropic for $\lambda/D$ to be anisotropic. In this paper, we study this problem using a new modeling technique constructed based on Finsler geometry (FG). Two possible FG models are examined: In the first (second) model, the FG modeling prescription is applied to the FMI (DMI) Hamiltonian. We find that these two different FG models' results are consistent with the reported experimental data for skyrmion deformation. We also study responses of helical spin orders under lattice deformations corresponding to uniaxial extension/compression and find a clear difference between these two models in the stripe phase, elucidating which interaction of FMI and DMI is deformed to be anisotropic by uniaxial stresses.

Electromechanical properties of ferroelectric polymers: Finsler geometry modeling and a Monte Carlo study

Polyvinylidene difluoride (PVDF) is a ferroelectric polymer characterized by negative strain along the direction of the applied electric field. However, the electromechanical response mechanism of PVDF remains unclear due to the complexity of the hierarchical structure across the length scales. In this letter, we employ the Finsler geometry model as a new solution to the aforementioned problem and demonstrate that the deformations observed through Monte Carlo simulations on 3D tetrahedral lattices are nearly identical to those of real PVDF. Specifically, the simulated mechanical deformation and polarization are similar to those observed experimentally.

Finsler geometry modeling for anisotropic diffusion in Turing patterns

Turing patterns are known to be described by diffusion reaction (DR) equations, and the patterns become zebra-like anisotropic if the diffusion constant is directionally dependent. However, the origin of this dependence is unclear. In this study, we report that Turing patterns can be studied with the Finsler geometry (FG) modeling technique by applying a hybrid numerical technique, which combines DR equations and the Monte Carlo (MC) technique. In the DR equations and the Hamiltonian for MC, the Finsler metric is introduced using internal degrees of freedom. We numerically show that anisotropic patterns appear according to a constraint given by some external forces applied to the internal degrees of freedom. In this FG modeling technique, direction-dependent diffusion constants are unnecessary, and these constants automatically or effectively become anisotropic. We consider that the internal degrees of freedom introduced for the Finsler metric play an essential role in the anisotropic patterns.

Finsler geometry modeling of complex fluids: reduction in viscous resistance

Complex fluids refer to a broad range of materials that contain two different phases, such as solid-liquid and fluid-gas mixtures. Generally, in fluids with dispersed matter, the viscous resistance is not always proportional to the velocity of the fluids, and a fluid exhibiting such behavior is called a non-Newtonian fluid. The viscous resistance is considerably decreased in a certain range of velocities compared to the case without macromolecules. This change in the macroscopic viscous resistance is expected to originate from the interactions between fluids and dispersed matter. In this study, we apply the Finsler geometry (FG) modelling technique to implement this complex interaction, and we numerically show a mechanism for the reduction in viscous resistance. In the FG modeling, a new dynamical variable corresponding to the directional degrees of freedom of dispersed matter is introduced and updated by the Monte Carlo simulation technique during iterations of the Navier-Stokes equation. The tentative results indicate that the distribution of the viscous force and its position dependence are considerably different from those in standard Newtonian fluids. This finding indicates the possibility that the FG modeling technique can properly describe the effects of dispersed matter on the flow behavior.

Modeling shape and volume transitions in liquid crystal elastomers

Shape and volume phase transitions of liquid crystal elastomers (LCEs) immersed in a solvent are studied within the framework of the Finsler geometry (FG). In the FG model of an LCE, an internal variable is introduced to describe the directional degrees of freedom of liquid crystals. We also introduce an Ising-like variable to effectively describe the swelling behavior of the LCE. From the results of Monte Carlo simulations, we find that the model successfully reproduces changes in the swelling degree as well as the smooth change in the shape anisotropy with increasing temperature. Moreover, the maximal value of the shape anisotropy is close to the experimentally observed value.

Finsler geometry modeling of reverse piezoelectric effect in PVDF

We apply the Finsler geometry (FG) modeling technique to study the electric field-induced strain in ferroelectric polymers. Polyvinylidene difluoride (PVDF) has a negative longitudinal piezoelectric coefficient, which is unusual in ferroelectrics, and therefore the shape changes in this material are hard to predict. We find that the results of Monte Carlo simulations for the FG model are in good agreement with experimental strain-electric field curves of PVDF-based polymers in both longitudinal and transverse directions. This implies that FG modeling is suitable for reproducing the reverse piezoelectric effect in PVDF.

Mathematical modeling of skyrmion shape deformation under uni-axial stresses

Skyrmion is a topologically stable spin texture and expected to be applied to the future computer memory. On the semiconductors such as FeGe and MnSi, the skyrmion configuration is stable in the sense that it is not strongly affected by a small variation of external stimuli such as temperature, magnetic field etc. In recent experiments, it was reported that the skymion shape is deformed from isotropic (or circular shape) to anisotropic (or elliptic shape) by an external mechanical stress. This shape deformation is caused by the so-called “strain-induced anisotropy (SIA)” of Dzyaloshinskii-Moriya interaction (DMI). In this presentation, we study the reason why this SIA appears in the DMI coefficient on the basis of Finsler geometry modeling technique by introducing a microscopic strain field τ, which is caused by or interacts with the applied external mechanical force.

Monte Carlo Study of Rubber Elasticity on the Basis of Finsler Geometry Modeling

Configurations of the polymer state in rubbers, such as so-called isotropic (random) and anisotropic (almost aligned) states, are symmetric/asymmetric under space rotations. In this paper, we present numerical data obtained by Monte Carlo simulations of a model for rubber formulations to compare these predictions with the reported experimental stress–strain curves. The model is defined by extending the two-dimensional surface model of Helfrich–Polyakov based on the Finsler geometry description. In the Finsler geometry model, the directional degree of freedom σ → of the polymers and the polymer position r are assumed to be the dynamical variables, and these two variables play an important role in the modeling of rubber elasticity. We find that the simulated stresses τ sim are in good agreement with the reported experimental stresses τ exp for large strains of up to 1200 % . It should be emphasized that the stress–strain curves are directly calculated from the Finsler geometry model Hamiltonian and its partition function, and this technique is in sharp contrast to the standard technique in which affine deformation is assumed. It is also shown that the obtained results are qualitatively consistent with the experimental data as influenced by strain-induced crystallization and the presence of fillers, though the real strain-induced crystallization is a time-dependent phenomenon in general.

Mathematical Modeling of Rubber Elasticity

A mathematical modeling, the Finsler geometry (FG) technique, is applied to study the rubber elasticity. Existing experimental data of stress-strain (SS) diagrams, which are highly non-linear, are numerically reproduced. Moreover, the strain induced crystallization (SIC), typical of some rubbers like Natural Rubber (NR), which is known to play an important role in the mechanical property of rubbers, is partly implemented in the model. Indeed, experimentally observed hysteresis of SS curve can be reproduced if the parameter a of non-polar (or polar) interaction energy is increased for the unloading or shrinkage process in the Monte Carlo (MC) simulations, and at the same time we find that the order parameter M of the directional degrees of freedom σ of polymer show a hysteresis behavior which is compatible with that of the crystallization ratio. In addition, rupture phenomena, which are accompanied by a necking phenomenon observed in the plastic deformation region, can also be reproduced. Thus we find that the interaction implemented in the FG model via the Finsler metric is suitable in describing the mechanical property of rubbers.

Finsler geometry modeling and Monte Carlo study of liquid crystal elastomers under electric fields

The shape transformation of liquid crystal elastomers (LCEs) under external electric fields is studied through Monte Carlo simulations of models constructed on the basis of Finsler geometry (FG). For polydomain side-chain-type LCEs, it is well known that the external-field-driven alignment of the director is accompanied by an anisotropic shape deformation. However, the mechanism of this deformation is quantitatively still unclear in some part and should be studied further from the microscopic perspective. In this paper, we evaluate whether this shape deformation is successfully simulated, or in other words, quantitatively understood, by the FG model. The FG assumed inside the material is closely connected to the directional degrees of freedom of LC molecules and plays an essential role in the anisotropic transformation. We find that the existing experimental data on the deformations of polydomain LCEs are in good agreement with the Monte Carlo results. It is also found that experimental diagrams of strain versus external voltage of a monodomain LCE in the nematic phase are well described by the FG model.

Bending of Thin Liquid Crystal Elastomer under Irradiation of Visible Light: Finsler Geometry Modeling.

In this paper, we show that the 3D Finsler geometry (FG) modeling technique successfully explains a reported experimental result: a thin liquid crystal elastomer (LCE) disk floating on the water surface deforms under light irradiation. In the reported experiment, the upper surface is illuminated by a light spot, and the nematic ordering of directors is influenced, but the nematic ordering remains unchanged on the lower surface contacting the water. This inhomogeneity of the director orientation on/inside the LCE is considered as the origin of the shape change that drives the disk on the water in the direction opposite the movement of the light spot. However, the mechanism of the shape change is still insufficiently understood because to date, the positional variable for the polymer has not been directly included in the interaction energy of the models for this system. We find that this shape change of the disk can be reproduced using the FG model. In this FG model, the interaction between , which represents the director field corresponding to the directional degrees of LC, and the polymer position is introduced via the Finsler metric. This interaction, which is a direct consequence of the geometry deformation, provides a good description of the shape deformation of the LCE disk under light irradiation.

Mathematical Modeling and Simulations for Large-Strain J-Shaped Diagrams of Soft Biological Materials.

Herein, we study stress–strain diagrams of soft biological materials such as animal skin, muscles, and arteries by Finsler geometry (FG) modeling. The stress–strain diagram of these biological materials is always J-shaped and is composed of toe, heel, linear, and failure regions. In the toe region, the stress is almost zero, and the length of this zero-stress region becomes very large (≃150%) in, for example, certain arteries. In this paper, we study long-toe diagrams using two-dimensional (2D) and 3D FG modeling techniques and Monte Carlo (MC) simulations. We find that, except for the failure region, large-strain J-shaped diagrams are successfully reproduced by the FG models. This implies that the complex J-shaped curves originate from the interaction between the directional and positional degrees of freedom of polymeric molecules, as implemented in the FG model.

Finsler Geometry Modeling of an Orientation-Asymmetric Surface Model for Membranes

In this paper, a triangulated surface model is studied in the context of Finsler geometry (FG) modeling. This FG model is an extended version of a recently reported model for two-component membranes, and it is asymmetric under surface inversion. We show that the definition of the model is independent of how the Finsler length of a bond is defined. This leads us to understand that the canonical (or Euclidean) surface model is obtained from the FG model such that it is uniquely determined as a trivial model from the viewpoint of well definedness.

J-shaped stress-strain diagram of collagen fibers: Frame tension of triangulated surfaces with fixed boundaries.

We present Monte Carlo data of the stress-strain diagrams obtained using two different triangulated surface models. The first is the canonical surface model of Helfrich and Polyakov (HP), and the second is a Finsler geometry (FG) model. The shape of the experimentally observed stress-strain diagram is called J-shaped. Indeed, the diagram has a plateau for the small strain region and becomes linear in the relatively large strain region. Because of this highly nonlinear behavior, the J-shaped diagram is far beyond the scope of the ordinary theory of elasticity. Therefore, the mechanism behind the J-shaped diagram still remains to be clarified, although it is commonly believed that the collagen degrees of freedom play an essential role. We find that the FG modeling technique provides a coarse-grained picture for the interaction between the collagen and the bulk material. The role of the directional degrees of freedom of collagen molecules or fibers can be understood in the context of FG modeling. We also discuss the reason for why the J-shaped diagram cannot (can) be explained by the HP (FG) model.

Orientation Asymmetric Surface Model for Membranes: Finsler Geometry Modeling

We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping r from a two-dimensional parameter space M to the three-dimensional Euclidean space R 3 . The metric variable g a b , which is always fixed to the Euclidean metric δ a b , can be extended to a more general non-Euclidean metric on M in the continuous model. The problem we focus on in this paper is whether such an extension is well defined or not in the discrete model. We find that a discrete surface model with a nontrivial metric becomes well defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in M depends on the direction. It is also shown that the discrete FG model is orientation asymmetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some asymmetries for orientation-changing transformations.

Finsler geometry modeling of elongation of flexible materials under external electromagnetic field

ABSTRACTWe study numerically elongation phenomena of flexible materials, such as liquid crystal (LC) elastomers (or ferroelectric polymers FEP), under external electromagnetic fields using Finsler geometry (FG) model. In this model, the interaction between LC molecules (or monomers) and the bulk material is implemented in the Finsler metric, the elements of which are considered to be dependent on the LC molecule direction (or the polarization of monomers in FEP). We have found out from Monte Carlo data that dependence of strain versus external field is consistent with existing experimental data.

Finsler Geometry Modeling of Phase Separation in Multi-Component Membranes.

A Finsler geometric surface model is studied as a coarse-grained model for membranes of three components, such as zwitterionic phospholipid (DOPC), lipid (DPPC) and an organic molecule (cholesterol). To understand the phase separation of liquid-ordered (DPPC rich) and liquid-disordered (DOPC rich) , we introduce a binary variable into the triangulated surface model. We numerically determine that two circular and stripe domains appear on the surface. The dependence of the morphological change on the area fraction of is consistent with existing experimental results. This provides us with a clear understanding of the origin of the line tension energy, which has been used to understand these morphological changes in three-component membranes. In addition to these two circular and stripe domains, a raft-like domain and budding domain are also observed, and the several corresponding phase diagrams are obtained.