arXiv subjects

Isshin Arai

Publications and source records attributed to Isshin Arai.

4 recordsLinked to original sources

arXiv2025

Group-Theoretic Upper Bounds on Reconstructability in Inverse Problems

Reconstructing the causal structure of physical systems from observational data constitutes a fundamental inverse problem. Here we show that the reconstruction dimension---defined as an upper bound on the number of recoverable components---is determined by the group-representation structure of the observation spaces and reconstruction maps. This formulation provides an explicit and operational characterization of reconstructability and reconstruction dimension, extending ideas that are often understood only intuitively in equivariant representation theory. As a concrete example, we demonstrate the reconstruction of the local velocity-gradient tensor from orientational measurements of particles suspended in flows, where the observation and velocity-gradient tensor spaces form SO(3) representations with constrained equivariant maps between them. Using an SO(3)-equivariant neural network (implemented with e3nn), we show that the reconstructable subspaces predicted by the representation decomposition are qualitatively consistent with those found in practice. Our formulation shows that the representation structure constrains reconstructability by determining an upper bound sector by sector, while our numerical results suggest that the actual saturation of the bound depends on the physics and data geometry. Beyond providing a useful theoretical framework, this work also connects the abstract representation-theoretic structure to concrete inverse reconstruction problems in fluid physics.

math-ph↗

arXiv2025

Nonlinear Aggregation of Phase Elements on the Unit Circle under Parametric External Fields

We investigate nonlinear aggregation dynamics of phase elements distributed on the unit circle under parametrically modulated external fields. Our model, inspired by flaky particle rotation in fluids, employs the equation ${d\alpha/dt} = \lambda(t)\sin 2(\alpha - \phi(t))$ with $\lambda(t) = \cos(\omega_1 t)$ and $\phi(t) = \omega_2 t$, representing a switching rotating attractive device where the attractive strength oscillates while the attractive point rotates at independent frequencies. Through numerical simulations and analytical approaches, we discover Arnold tongue-like structures in parameter space $(\omega_1, \omega_2)$, where initially isotropic phase distributions aggregate into highly anisotropic states. Complete aggregation occurs within wedge-shaped stability regions radiating from bifurcation points, forming band structures with characteristic slope relationships. The dynamics exhibit rich nonlinear behavior including attractors, limit cycles, and quasi-periodic trajectories in reduced indicator space spanned by aggregation degree ($I$), field-alignment measure ($O$), and temporal variation ($P$). Our findings reveal fundamental principles governing collective phase dynamics under competing temporal modulations, with potential applications spanning from biological synchronization to socio-economic dynamics and controllable collective systems.

nlin.CD↗

arXiv2025

Probabilistic description of flake orientation suspended in rotating wave flows

In fluid dynamics experiments, flake-based flow visualization is a common technique to capture flow structures through the rays reflected from flat tracers suspended in the fluid. However, the correspondence between light intensity patterns in visualization images and the underlying physical properties of the flow can only be elucidated when the flow is known {\it a priori}. To reframe this limitation, just as the introduction of spin variable transformed quantum mechanics, we introduced the orientation variable into fluid dynamics and derived the time-dependent equation of the tracer orientation probability density field from an Eulerian perspective. As a first example in which a dimensionless parameter distinguishes the dependency on the initial condition, we illustrated an analytical solution of the orientation probability in a rotating wave flow. With the inclusion of the diffusion term in the governing equation, the probability converged to the flow-determined state with spatially varying anisotropy, eliminating dependency on initial conditions. As a second example, we solved the orientation probability field in the axisymmetric state in spherical Couette flow, to demonstrate independence from initial conditions consistent with experimental observations. An asymmetric pattern in experimental images, unexplained by the dynamics of the tracer orientation, was reproduced from the unique solution of the proposed equations.

physics.flu-dyn↗

arXiv2024

Revisiting visualization of spiral states in a wide-gap spherical Couette flow

A pioneering study conducted by Egbers and Rath [Acta Mech. 111 pp. 125--140 (1995)] experimentally captured spiral waves to elucidate the transition in the wide-gap spherical Couette flow. However, the physical field quantities of the spiral waves corresponding to light patterns of various intensities, as obtained in the experiment, remain unclear, and we have yet to move beyond the understanding that the reflected light from shear-sensitive flake tracers responds to a flow that appears at the transition. In this study, the experiment to visualize spiral waves using aluminum flakes, as performed by Egbers and Rath, was numerically reproduced by solving the translational and rotational motions of the particles in a spiral wave. First, the spiral wave in a spherical Couette flow with an aspect ratio $\eta=1/2$ was numerically calculated using the Newton--Raphson method. Subsequently, the image that was numerically reproduced from the spiral wave was compared with an experimentally visualized image. The torque acting on the inner sphere and the phase angular velocity of the spiral waves with various wavenumbers were provided. Attempts have been made to determine the instantaneous physical quantity to which the light and dark patterns obtained in the visualization corresponded, and the orientation motion of the flakes developed in the advective history of the flow is essential to yield favorable results. Exploring the correlation between flow visualization results and shear structures may provide a new avenue for quantitatively estimating spatial structures and time scales in complex and quickly time-varying flow fields, such as turbulence.

physics.flu-dyn↗