arXiv subjects

Naoki Sato

Publications and source records attributed to Naoki Sato.

At least 37 records · Page 2Linked to original sources

arXiv2024

Generalizing Hamiltonian Mechanics with Closed Differential Forms

Classical Hamiltonian mechanics, characterized by a single conserved Hamiltonian (energy) and symplectic geometry, `hides' other invariants into symmetries of the Hamiltonian or into the kernel of the Poisson tensor. Nambu mechanics aims to generalize classical Hamiltonian mechanics to ideal dynamical systems bearing two Hamiltonians, but its connection to a suitable geometric framework has remained elusive. This work establishes a novel correspondence between generalized Hamiltonian mechanics, defined for systems with a phase space conservation law (invariance of a closed form) and a matter conservation law (invariance of multiple Hamiltonians), and multisymplectic geometry. The key lies in the invertibility of differential forms of degree higher than 2. We demonstrate that the cornerstone theorems of classical Hamiltonian mechanics (Lie-Darboux and Liouville) require reinterpretation within this new framework, reflecting the unique properties of invertibility in multisymplectic geometry. Furthermore, we present two key theorems that solidify the connection: i) any classical Hamiltonian system with two or more invariants is also a generalized Hamiltonian system and ii) given a generalized Hamiltonian system with two or more invariants, there exists a corresponding classical Hamiltonian system on the level set of all but one invariant, with the remaining invariant playing the role of the Hamiltonian function.

math.DG↗

arXiv2024

Revealing Noncanonical Hamiltonian Structures in Relativistic Fluid Dynamics

We present the noncanonical Hamiltonian structure of the relativistic Euler equations for a perfect fluid in Minkowski spacetime. By identifying the system's noncanonical Poisson bracket and Hamiltonian, we show that relativistic fluid flows preserve helicity and enstrophy as conserved quantities in three-dimensional and two-dimensional cases, respectively. This holds when the fluid follows a relativistic $γ$-barotropic equation of state, which generalizes the classical barotropic condition. Furthermore, we demonstrate that these conserved quantities are Casimir invariants associated with the noncanonical Poisson structure. These findings open new avenues for applying Hamiltonian theory to the study of astrophysical fluids and relativistic plasmas.

math-ph↗

arXiv2024

Quantum-Fluid Correspondence in Relativistic Fluids with Spin: From Madelung Form to Gravitational Coupling

This paper explores the quantum-fluid correspondence in a charged relativistic fluid with intrinsic spin. We begin by examining the nonrelativistic case, showing that the inclusion of spin introduces a quantum correction to the classical fluid energy. This correction, coupled with Maxwell's equations, naturally leads to the Schrödinger equation in Madelung form. Building on this foundation, we extend the formalism to a relativistic perfect fluid, identifying the system's stress-energy-momentum tensor. Our analysis reveals that the trace of the quantum correction to this tensor corresponds to the energy density of an oscillator, with its frequency determined by the vorticity of the spin motion. We then use the stress-energy-momentum tensor to establish the relationship between the Ricci scalar curvature, as dictated by the Einstein field equations, and the fluid density in a static, spherically symmetric configuration. Lastly, we generalize the Madelung transformation to compressible Navier-Stokes flows with vorticity and viscosity by developing a tailored Clebsch representation of the velocity field. This theoretical framework offers potential applications for studying fluid-like systems with internal rotational degrees of freedom, commonly encountered in astrophysical settings.

gr-qc↗

arXiv2024

Steady Compressible 3D Euler Flows in Toroidal Volumes without Continuous Euclidean Isometries

We demonstrate the existence of smooth three-dimensional vector fields where the cross product between the vector field and its curl is balanced by the gradient of a smooth function, with toroidal level sets that are not invariant under continuous Euclidean isometries. This finding indicates the existence of steady compressible Euler flows, either influenced by an external potential energy or maintained by a density source in the continuity equation, that are foliated by asymmetric nested toroidal surfaces. Our analysis suggests that the primary obstacle in resolving Grad's conjecture regarding the existence of nontrivial magnetohydrodynamic equilibria arises from the incompressibility constraint imposed on the magnetic field.

math.AP↗

arXiv2024

Thermoelectric properties of marcasite-type compounds MSb$_2$ (M = Ta, Nb): A combined experimental and computational study

Here, we investigate the thermoelectric properties of the marcasite-type compounds MSb$_2$ (M = Ta, Nb) in the temperature range of 310-730 K. These compounds were synthesized by a solid-state reaction followed by the spark plasma sintering process. The Rietveld refinement method confirms the monoclinic phase with space group C2/m for both compounds. The observed values of Seebeck coefficients exhibit non-monotonic behaviour in the studied temperature range, with the maximum magnitude of -14.4 and -22.7 $μ$V K$^{-1}$ for TaSb$_2$ and NbSb$_2$, respectively at ~444 K. The negative sign of S in the full temperature window signifies the n-type behaviour of these compounds. Both electrical and thermal conductivities show an increasing trend with temperature. The experimentally observed thermoelectric properties are understood through the first-principles DFT and Boltzmann transport equation. A pseudogap in the density of states around the Fermi level characterizes the semimetallic behaviour of these compounds. The multi-band electron and hole pockets were found to be mainly responsible for the temperature dependence of transport properties. The experimental power factors are found to be ~0.09 and ~0.42 mW m$^{-1}$ K$^{-2}$ at 310 K for TaSb2 and NbSb2, respectively. From the DFT-based calculations, the maximum possible power factors for p-type conduction are predicted as ~1.14 and ~1.74 mW m$^{-1}$ K$^{-2}$, while these values are found to be ~1.16 and ~1.80 mW m$^{-1}$ K$^{-2}$ for n-type TaSb$_2$ and NbSb$_2$, respectively at 300 K with the corresponding doping concentrations. The present study suggests that the combined DFT and Boltzmann transport theory are found to be reasonably good at explaining the experimental transport properties, and moderate power factors are predicted.

cond-mat.mtrl-sci↗

arXiv2024

Generalization of Hamiltonian Mechanics to a Three Dimensional Phase Space

Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The properties of the Poisson bracket are encapsulated in the symplectic 2-form, a closed second order differential form. Due to closure, the symplectic 2-form is preserved by the Hamiltonian flow, and it assigns an invariant (Liouville) measure on the phase space through the Lie-Darboux theorem. In this paper we propose a generalization of classical Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson bracket is replaced with a generalized Poisson bracket acting on a pair of Hamiltonian functions, while the symplectic 2-form is replaced by a symplectic 3-form. We show that, using the closure of the symplectic 3-form, a result analogous to the classical Lie-Darboux theorem holds: locally, there exist smooth coordinates such that the components of the symplectic 3-form are constants, and the phase space is endowed with a preserved volume element. Furthermore, as in the classical theory, the Jacobi identity for the generalized Poisson bracket mathematically expresses the closure of the associated symplectic form. As a consequence, constant skew-symmetric third order contravariant tensors always define generalized Poisson brackets. This is in contrast with generalizations of Hamiltonian mechanics postulating the fundamental identity as replacement for the Jacobi identity. In particular, we find that the fundamental identity represents a stronger requirement than the closure of the symplectic 3-form.

math-ph↗

arXiv2024

A Collision Operator for Describing Dissipation in Noncanonical Phase Space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in an equilibrium described by a Maxwell-Boltzmann distribution. However, the time scale required to reach Maxwell-Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell-Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

cond-mat.stat-mech↗

arXiv2023

A Reduced Ideal MHD System for Nonlinear Magnetic Field Turbulence in Plasmas with Approximate Flux Surfaces

This paper studies the nonlinear evolution of magnetic field turbulence in proximity of steady ideal MHD configurations characterized by a small electric current, a small plasma flow, and approximate flux surfaces, a physical setting that is relevant for plasma confinement in stellarators. The aim is to gather insight on magnetic field dynamics, to elucidate accessibility and stability of three-dimensional MHD equilibria, as well as to formulate practical methods to compute them. Starting from the ideal MHD equations, a reduced dynamical system of two coupled nonlinear PDEs for the flux function and the angle variable associated with the Clebsch representation of the magnetic field is obtained. It is shown that under suitable boundary and gauge conditions such reduced system preserves magnetic energy, magnetic helicity, and total magnetic flux. The noncanonical Hamiltonian structure of the reduced system is identified, and used to show the nonlinear stability of steady solutions against perturbations involving only one Clebsch potential. The Hamiltonian structure is also applied to construct a dissipative dynamical system through the method of double brackets. This dissipative system enables the computation of MHD equilibria by minimizing energy until a critical point of the Hamiltonian is reached. Finally, an iterative scheme based on the alternate solution of the two steady equations in the reduced system is proposed as a further method to compute MHD equilibria. A theorem is proven which states that the iterative scheme converges to a nontrivial MHD equilbrium as long as solutions exist at each step of the iteration.

physics.plasm-ph↗

arXiv2023

Existence and Estimation of Critical Batch Size for Training Generative Adversarial Networks with Two Time-Scale Update Rule

Previous results have shown that a two time-scale update rule (TTUR) using different learning rates, such as different constant rates or different decaying rates, is useful for training generative adversarial networks (GANs) in theory and in practice. Moreover, not only the learning rate but also the batch size is important for training GANs with TTURs and they both affect the number of steps needed for training. This paper studies the relationship between batch size and the number of steps needed for training GANs with TTURs based on constant learning rates. We theoretically show that, for a TTUR with constant learning rates, the number of steps needed to find stationary points of the loss functions of both the discriminator and generator decreases as the batch size increases and that there exists a critical batch size minimizing the stochastic first-order oracle (SFO) complexity. Then, we use the Fr'echet inception distance (FID) as the performance measure for training and provide numerical results indicating that the number of steps needed to achieve a low FID score decreases as the batch size increases and that the SFO complexity increases once the batch size exceeds the measured critical batch size. Moreover, we show that measured critical batch sizes are close to the sizes estimated from our theoretical results.

cs.LG↗

arXiv2023

Numerical simulation of two-dimensional incompressible Navier-Stokes turbulence by Clebsch potentials

The Clebsch representation of a velocity field represents an effective tool for the analysis of physical properties of fluid flows. Indeed, a suitable choice of Clebsch potentials can be used to extract structural features that would otherwise be hidden within the complexity of fluid patterns and their evolution. In this work, we report the solution of the two-dimensional incompressible Navier-Stokes equations via Clebsch potentials. The results are in agreement with the solution of the vorticity equation for the stream function. Furthermore, we numerically demonstrate that the Shannon information entropy associated with each Clebsch potential is a growing function of time, and that it evolves at a slower rate than the rate of change in energy and enstrophy, as predicted by theory. These results pave the way for an alternative approach in the numerical study of fluid flows.

physics.flu-dyn↗

arXiv2023

Guiding Center Derivation of the Generalized Hasegawa-Mima Equation for Drift Wave Turbulence in Curved Magnetic Fields

Recently, a generalized Hasegawa-Mima (gHM) equation describing drift wave turbulence in curved magnetic fields has been derived in [N. Sato and M. Yamada, J. Plasma Phys. (2022), vol. 88, 905880319] for an ion-electron plasma modeled as a two-fluid system. In this work, we show that a mathematically equivalent GHM equation can be obtained within the kinetic framework of guiding center motion, and that the relevant drift wave turbulence ordering can be further relaxed, effectively generalizing the applicability of the equation to any magnetic field geometry and electron spatial density, in the sense that no ordering requirements involve spatial derivatives of the magnetic field or the electron spatial density.

physics.plasm-ph↗

arXiv2023

Nested invariant tori foliating a vector field and its curl: toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries

This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface, and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.

math.AP↗

arXiv2022

Maximum entropy states of collisionless positron-electron plasma in a dipole magnetic field

We are developing a positron-electron plasma trap based on a dipole magnetic field generated by a levitated superconducting magnet to investigate the physics of magnetized plasmas with mass symmetry as well as antimatter components. Such laboratory magnetosphere is deemed essential for the understanding of pair plasmas in astrophysical environments, such as magnetars and blackholes, and represents a novel technology with potential applications in antimatter confinement and development of coherent gamma-ray lasers. The design of the device requires a preemptive analysis of the achievable self-organized steady states. In this study, we construct a theoretical model describing maximum entropy states of a collisionless positron-electron plasma confined by a dipole magnetic field, and demonstrate efficient confinement of both species under a wide range of physical parameters by analysing the effect of the three adiabatic invariants on the phase space distribution function. The theory is verified by numerical evaluation of spatial density, electrostatic potential, and toroidal rotation velocity for each species in correspondence of the maximum entropy state.

physics.plasm-ph↗

arXiv2022

Existence of weakly quasisymmetric magnetic fields in asymmetric toroidal domains with non-tangential quasisymmetry

A quasisymmetry is a special symmetry that enhances the ability of a magnetic field to trap charged particles. Quasisymmetric magnetic fields may allow the realization of next generation fusion reactors (stellarators) with superior performance when compared with classical (tokamak) designs. Nevertheless, the existence of such magnetic configurations lacks mathematical proof due to the complexity of the governing equations. Here, we prove the existence of weakly quasisymmetric magnetic fields by constructing explicit examples. This result is achieved by a tailored parametrization of both magnetic field and hosting toroidal domain, which are optimized to fulfill quasisymmetry. The obtained solutions hold in a toroidal volume, are smooth, possess nested flux surfaces, are not invariant under continuous Euclidean isometries, have a non-vanishing current, exhibit a direction of quasisymmetry that is not tangential to the toroidal boundary, and fit within the framework of anisotropic magnetohydrodynamics.

physics.plasm-ph↗

arXiv2022

A Generalized Hasegawa-Mima Equation in Curved Magnetic Fields

We derive a model equation describing electrostatic plasma turbulence in general (inhomogeneous and curved) magnetic fields by analysing the effect of curved geometry on the ion fluid polarization drift velocity. The derived nonlinear equation generalizes the Hasegawa-Mima equation governing drift wave turbulence in a straight homogeneous magnetic field, and may serve as a toy model for the description of turbulent systems such as the core of H-mode plasmas. The equation is most appropriate for configurations with a small ExB drift velocity divergence, or a mild spatial change in ExB drift velocity. We identify the conserved energy of the system, and obtain conditions on magnetic field topology for conservation of generalized enstrophy. Through numerical examples, we further show how the curvature of the magnetic field reshapes self-organized steady turbulent states.

physics.plasm-ph↗

arXiv2021

Vorticity equation on surfaces with arbitrary topology

We derive the vorticity equation for an incompressible fluid on a 2-dimensional surface with arbitrary topology embedded in 3-dimensional Euclidean space by using a tailored Clebsch parametrization of the flow. In the inviscid limit, we identify conserved surface energy and enstrophy, and obtain the corresponding noncanonical Hamiltonian structure. We then discuss the formulation of the diffusion operator on the surface by examining two alternatives. In the first case, we follow the standard approach for the Navier-Stokes equations on a Riemannian manifold and calculate the diffusion operator by requiring that flows corresponding to Killing fields of the Riemannian metric are not subject to dissipation. For an embedded surface, this leads to a diffusion operator including derivatives of the stream function across the surface. In the second case, using an analogy with the Poisson equation for the Newtonian gravitational potential in general relativity, we construct a diffusion operator taking into account the Ricci scalar curvature of the surface. The resulting vorticity equation is 2-dimensional, and the corresponding diffusive equilibria minimize dissipation under the constraint of curvature energy.

math-ph↗

arXiv2021

Quasisymmetric magnetic fields in asymmetric toroidal domains

We explore the existence of quasisymmetric magnetic fields in asymmetric toroidal domains. These vector fields can be identified with a class of magnetohydrodynamic equilibria in the presence of pressure anisotropy. First, using Clebsch potentials, we derive a system of two coupled nonlinear first order partial differential equations expressing a family of quasisymmetric magnetic fields in bounded domains. In regions where flux surfaces and surfaces of constant field strength are not tangential, this system can be further reduced to a single degenerate nonlinear second order partial differential equation with externally assigned initial data. Then, we exhibit regular quasisymmetric vector fields which correspond to local solutions of anisotropic magnetohydrodynamics in asymmetric toroidal domains such that tangential boundary conditions are fulfilled on a portion of the bounding surface. The problems of boundary shape and locality are also discussed. We find that symmetric magnetic fields can be fitted into asymmetric domains, and that the mathematical difficulty encountered in the derivation of global quasisymmetric magnetic fields lies in the topological obstruction toward global extension affecting local solutions of the governing nonlinear first order partial differential equations.

math.AP↗

arXiv2021

The Effect of Spacetime Curvature on Statistical Distributions

The Boltzmann distribution of an ideal gas is determined by the Hamiltonian function generating single particle dynamics. Systems with higher complexity often exhibit topological constraints, which are independent of the Hamiltonian and may affect the shape of the distribution function as well. Here, we study a further source of heterogeneity, the curvature of spacetime arising from the general theory of relativity. The present construction relies on three assumptions: first, the statistical ensemble is made of particles obeying geodesic equations, which define the phase space of the system. Next, the metric coefficients are time-symmetric, implying that, if thermodynamic equilibrium is achieved, all physical observables are independent of coordinate time. Finally, ergodicity is enforced with respect to proper time, so that ambiguity in the choice of a time variable for the statistical ensemble is removed. Under these hypothesis, we derive the distribution function of thermodynamic equilibrium, and verify that it reduces to the Boltzmann distribution in the classical limit. We further show that spacetime curvature affects physical observables, even far from the source of the metric. Two examples are analyzed: an ideal gas in Schwarzschild spacetime and a charged gas in Kerr-Newman spacetime. In the Schwarzschild case, conservation of macroscopic constraints, such as angular momentum, combined with relativistic distortion of the distribution function can produce configurations with decreasing density and growing azimuthal rotation velocity far from the event horizon of the central mass. In the Kerr-Newman case, it is found that kinetic energy associated with azimuthal rotations is an increasing function of the radial coordinate, and it eventually approaches a constant value corresponding to classical equipartition, even though spatial particle density decreases.

gr-qc↗