arXiv subjects

Saori Morimoto

Publications and source records attributed to Saori Morimoto.

5 recordsLinked to original sources

arXiv2025

Eigenvalue and pseudospectrum processes generated by nonnormal Toeplitz matrices with rank 1 perturbations

We introduce two kinds of matrix-valued dynamical processes generated by nonnormal Toeplitz matrices with the additive rank 1 perturbations $δJ$, where $δ\in {\mathbb{C}}$ and $J$ is the all-ones matrix. For each process, first we report the complicated motion of the numerically obtained eigenvalues. Then we derive the specific equation which determines the motion of non-zero simple eigenvalues and clarifies the time-dependence of degeneracy of the zero-eigenvalue $λ_0=0$. Comparison with the solutions of this equation, it is concluded that the numerically observed non-zero eigenvalues distributing around $λ_0$ are the exact eigenvalues not of the original system, but of the system perturbed by uncontrolled rounding errors of computer. The complex domain in which the eigenvalues of randomly perturbed system are distributed is identified with the pseudospectrum including $λ_0$ of the original system with $δJ$. We characterize the pseudospectrum processes using the symbol curves of the corresponding nonnormal Toeplitz operators without $δJ$. We report new phenomena in our second model such that at each time the outermost closed simple curve cut out from the symbol curve is realized as the exact eigenvalues, but the inner part of symbol curve is reduced in size and embedded in the pseudospectrum including $λ_0$. Such separation of exact simple eigenvalues and a degenerated eigenvalue associated with pseudospectrum will be meaningful for numerical analysis, since the former is stable and robust, but the latter is highly sensitive and unstable with respective to perturbations. The present study will be related to the pseudospectra approaches to non-Hermitian systems developed in quantum physics

math-ph↗

arXiv2025

Generalized Eigenspaces and Pseudospectra of Nonnormal and Defective Matrix-Valued Dynamical Systems

We consider nonnormal matrix-valued dynamical systems with discrete time. For an eigenvalue of matrix, the number of times it appears as a root of the characteristic polynomial is called the algebraic multiplicity. On the other hand, the geometric multiplicity is the dimension of the linear space of eigenvectors associated with that eigenvalue. If the former exceeds the latter, then the eigenvalue is said to be defective and the matrix becomes nondiagonalizable by any similarity transformation. The discrete-time of our dynamics is identified with the geometric multiplicity of the zero eigenvalue $λ_0=0$. Its algebraic multiplicity takes about half of the matrix size at $t=1$ and increases stepwise in time, which keeps excess to the geometric multiplicity until their coincidence at the final time. Our model exhibits relaxation processes from far-from-normal to near-normal matrices, in which the defectivity of $λ_0$ is recovering in time. We show that such processes are realized as size reductions of pseudospectrum including $λ_0$. Here the pseudospectra are the domains on the complex plane which are not necessarily exact spectra but in which the resolvent of matrix takes extremely large values. The defective eigenvalue $λ_0$ is sensitive to perturbation and the eigenvalues of the perturbed systems are distributed densely in the pseudospectrum including $λ_0$. By constructing generalized eigenspace for $λ_0$, we give the Jordan block decomposition for the resolvent of matrix and characterize the pseudospectrum dynamics. Numerical study of the systems perturbed by Gaussian random matrices supports the validity of the present analysis.

math-ph↗

arXiv2025

Extinction and Metastability of Pheromone-Roads in Stochastic Models for Foraging Walks of Ants

Macroscopic changes of group behavior of eusocial insects are studied from the viewpoint of non-equilibrium phase transitions. Recent combined study of experiments and mathematical modeling by the group led by the third author suggests that a species of garden ant switches the individual foraging walk from pheromone-mediated to visual-cues-mediated depending on situation. If an initial pheromone-road between the nest and food sources is a detour, ants using visual cues can pioneer shorter paths. These shorter paths are reinforced by pheromone secreted by following ants, and then the detour ceases to exist. Once the old pheromone-road extincts, there will be almost no chance to reconstruct it. Hence the extinction of pheromone-road is expected to be regarded as a phase transition to an absorbing state. We propose a discrete-time model on a square lattice consisting of switching random walks interacting though time-dependent pheromone field. The numerical study shows that the critical phenomena of the present extinction transitions of pheromone-roads do not seem to belong to the directed percolation universality class associated with the usual absorbing-state transition. The new aspects are cased by the coexistence and competition with newly creating pheromone-roads. In a regime in the extinction phase, the annihilating road shows metastability and takes long time-period to be replaced by a new road.

cond-mat.stat-mech↗

arXiv2025

Interacting Particle Systems Modeling Self-Propelled Motions

In non-equilibrium statistical physics, active matters in both living and non-living systems have been extensively studied. In particular, self-propelled particle systems provide challenging research subjects in experimental and theoretical physics, since individual and collective behaviors of units performing persistent motions can not be described by usual fluctuation theory for equilibrium systems. A typical example of man-made self-propelled systems which can be easily handled in small-sized experiments is a system of camphor floats put on the surface of water. Based on the experimental and theoretical studied by Nishimori et al. (J. Phys. Soc. Jpn. 86 (2017) 101012), we propose a new type of mathematical models for complex motions of camphor disks on the surface of water. In the previous mathematical models introduced by Nishimori et al. are coupled systems of the equations of motion for camphor disks described by ordinary differential equations and the partial differential equation for the concentration field of camphor molecules in water. Here we consider coupled systems of equation of motions of camphor disks and random walks representing individual camphor molecules in water. In other words, we take into account non-equilibrium fluctuations by introducing stochastic processes into the deterministic models. Numerical simulation shows that our models can represent self-propelled motions of individual camphor disk as well as repulsive interactions among them. We focus on the one-dimensional models in which viscosity is dominant, and derive a dynamical system of a camphor disk by taking the average of random variables of our stochastic system. By studying both of stochastic models and dynamical systems, we clarify the transitions between three phases of motions for a camphor disk depending on parameters.

cond-mat.stat-mech↗

arXiv2024

Switching particle systems for foraging ants showing phase transitions in path selections

Switching interacting particle systems studied in probability theory are the stochastic processes of hopping particles on a lattice made up of slow and fast particles, where the switching between these types of particles occurs randomly at a given transition rate. This paper explores how such stochastic processes involving multiple particles can model group behaviors of ants. Recent experimental research by the last author's group has investigated how ants switch between two types of primarily relied cues to select foraging paths based on the current situation. Here, we propose a discrete-time interacting random walk model on a square lattice, incorporating two types of hopping rules. Numerical simulation results demonstrate global changes in selected homing paths, transitioning from trailing paths of the `pheromone road' to nearly optimal paths depending on the switching parameters. By introducing two types of order parameters characterizing the dependence of homing duration distributions on switching parameters, we discuss these global changes as phase transitions in ant path selections. We also study critical phenomena associated with continuous phase transitions.

cond-mat.stat-mech↗