Hamiltonian Flows Research Articles

Testing the conservative character of particle simulations: I. Canonical and noncanonical guiding center model in Boozer coordinates

The guiding center (GC) Lagrangian in Boozer coordinates for toroidally confined plasmas can be cast into canonical form by eliminating terms containing the covariant component BΨP of the magnetic field vector with respect to the poloidal flux function ΨP. In an unperturbed plasma, BΨP can be eliminated via exact coordinate transformations, but, in general, one relies on approximations, assuming that the effect of BΨP is small. Here, we are interested in the question whether Hamiltonian conservation laws are still satisfied when BΨP is retained in the presence of fluctuations. Considering fast ions in the presence of a shear Alfvén wave field with fixed amplitude, fixed frequency, and a single toroidal mode number n, we show that simulations using the code ORBIT with and without BΨP yield practically the same resonant and nonresonant GC orbits. The numerical results are consistent with theoretical analyses (presented in the appendix), which show that the unabridged GC Lagrangian with BΨP retained yields equations of motion that possess two key properties of Hamiltonian flows: (i) phase space conservation and (ii) energy conservation. As counter-examples, we also show cases where energy conservation (ii) or both conservation laws (i) and (ii) are broken by omitting certain small terms. When testing the conservative character of the simulation code, it is found to be beneficial to apply perturbations that do not resemble normal (eigen) modes of the plasma. The deviations are enhanced and, thus, more easily spotted when one inspects wave-particle interactions using nonnormal modes.

Global solutions of aggregation equations and other flows with random diffusion

Aggregation equations, such as the parabolic-elliptic Patlak–Keller–Segel model, are known to have an optimal threshold for global existence versus finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. (Int Math Res Not IMRN 23:9370–9385, 2020) showing that, with high probability, the inviscid SQG equation with random diffusion has global classical solutions, we investigate whether suitable random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevrey-type Fourier–Lebesgue spaces with quantifiable high probability.

Viscosity subsolutions of Hamilton–Jacobi equations and invariant sets of contact Hamilton systems

This paper aims at providing a dynamic perspective of viscosity subsolutions to contact Hamiltonian–Jacobi equations H(x, ∂xu, u) = 0 on connected, closed manifold M. Based on implicit variational principles, we focus on the connection between positive invariant sets under Hamiltonian flow and viscosity subsolutions. We apply the connection to give a new necessary and sufficient condition for the existence of viscosity solutions of the stationary equation from the dynamical view. Finally, we discuss the multiplicity of viscosity solutions and give several illustrative examples.

Discrete representations of orbit structures of flows for topological data analysis

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called the flow of finite type, are in one-to-one correspondence with discrete structures such as trees/graphs and sequences of letters. The flow of finite type is an extension of structurally stable Hamiltonian vector fields, which appear in many theoretical and numerical investigations of two-dimensional (2D) incompressible fluid flows. Moreover, it contains compressible 2D vector fields such as the Morse–Smale vector fields and the projection of 3D vector fields onto 2D sections. The discrete representation is not only a simple symbolic identifier for the topological structure of complex flows, but it also gives rise to a new methodology of topological data analysis for flows when applied to data brought by measurements, experiments, and numerical simulations of complex flows. As a proof of concept, we provide some applications of the representation theory to 2D compressible vector fields and a 3D vector field arising in an industrial problem.

A Continuation Multiple Shooting Method for Wasserstein Geodesic Equation

In this paper, we propose a numerical method to solve the classic $L^2$-optimal transport problem. Our algorithm is based on the use of multiple shooting, in combination with a continuation procedure, to solve the boundary value problem associated to the transport problem. Based on the viewpoint of Wasserstein Hamiltonian flow with initial and target densities, our algorithm reflects the Hamiltonian structure of the underlying problem and exploits it in the numerical discretization. Several numerical examples are presented to illustrate the performance of the method.

Higher elastica: geodesics in jet space

Carnot groups are subRiemannian manifolds. As such, they admit geodesic flows, which are left-invariant Hamiltonian flows on their cotangent bundles. Some of these flows are integrable; some are not. The k-jets space of for real-valued functions on the real line forms a Carnot group of dimension k+2. In this study, it is shown that its geodesic flow is integrable and that its geodesics generalize Euler’s elastica, with the case k=2 corresponding to the elastica.

On conformal points of area preserving maps and related topics

In this article we consider area preserving diffeomorphisms of planar domains, and we are interested in their conformal points, i.e., points at which the derivative is a similarity. We present some conditions that guarantee existence of conformal points for the infinitesimal problem of Hamiltonian vector fields as well as for what we call moderate symplectomorphisms of simply connected domains. We also link this problem to the Carathéodory and Loewner conjectures.

Limit Cycles in Discontinuous Piecewise Linear Planar Hamiltonian Systems Without Equilibrium Points

In this paper, we study the limit cycles in the discontinuous piecewise linear planar systems separated by a nonregular line and formed by linear Hamiltonian vector fields without equilibria. Motivated by [Llibre & Teixeira, 2017], where an open problem was posed: Can piecewise linear differential systems without equilibria produce limit cycles? We prove that such systems have at most two limit cycles, and the limit cycles must intersect the nonregular separation line in two or four points. More precisely, the exact upper bound of crossing limit cycles is two, and this upper bound can indeed be reached: either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points. Based on Poincaré map, the stability of various limit cycles is also proved. In addition, we give some concrete examples to illustrate our main results.

Identification of Kuroshio meanderings south of Japan via a topological data analysis for sea surface height

This study proposes an algorithm to identify stable Kuroshio meanderings by extracting topological features from a sea surface height (SSH) gridded dataset in 1993–2020. Based on the mathematical theory of topological classifications for streamline patterns, the algorithm provides a unique symbolic representation and a discrete graph structure, which is referred to as the partially cyclically ordered rooted tree (COT) representation and the Reeb graph, respectively, to structurally stable Hamiltonian vector fields. We have confirmed that the temporal variations in the Kuroshio southernmost position south of the Tokai district captured by the algorithm are well consistent with the existing results by the Japan Meteorological Agency (JMA). The algorithm based on the topology detects five meandering periods: The three of them correspond to large meandering events detected by the JMA, while the two of them are offshore non-large meandering events. The topological data analysis reveals that a large cyclonic eddy inside of the meandering is split into two small eddies near the termination of the most meandering events.

Invariant measures for Hamiltonian flows and diffusion in infinitely dimensional phase spaces

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the group of symplectomorphisms preserving two-dimensional symplectic subspaces are investigated. This construction gives the opportunity to present a random Hamiltonian flow in phase space by means of a random unitary group in the space of functions that are quadratically integrable by invariant measure. The properties of mean values of random shift operators are studied.

The symplectic geometrical formulation of quantum hydrodynamics

Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.

Non-uniqueness of integral curves for autonomous Hamiltonian vector fields

In this work we prove the existence of an autonomous Hamiltonian vector field in $W^{1,r}(\mathbb T^d; \mathbb R^d)$ with $r < d-1$ and $d \geq 4$ for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio's superposition principle [2], we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover, we show that the Hamiltonian is not constant along these integral curves.

Nilpotent Center in a Continuous Piecewise Quadratic Polynomial Hamiltonian Vector Field

In this paper, we study the global dynamics of continuous piecewise quadratic Hamiltonian systems separated by the straight line [Formula: see text], where these kinds of systems have a nilpotent center at [Formula: see text], which comes from the combination of two cusps of both Hamiltonian systems. By the Poincaré compactification we classify the global phase portraits of these systems. We must mention that it is extremely rare to find works studying the center-focus problem in piecewise smooth systems with nonelementary singular points as we did here.

Modeling for Improved Performance of Ultra-Fast Nonvolatile Toggle Spin Torque MRAM Bit-Cell

In the recent past, exploiting nonvolatile memory devices to facilitate in-memory computation benchmarks has demonstrated a considerable capability to address the von-Neumann drawbacks. This process resulted in a tremendous effort towards spintronics memory development. Due to the voltage-controlled magnetic anisotropy (VCMA) effect, the toggle magnetoresistive random-access memory (MRAM) exhibits faster switching and lower power consumption than spin-transfer torque (STT) MRAM. We assessed read/write access time, magnetized time and anisotropic field for crystalline perpendicular magnetic tunnel junctions (CPMTJ) with detailed material study and device parameters in this work. These junctions are affected by the VCMA effect. Various VCMA parameters, including the VCMA coefficient, the time constant and antiparallel (AP) resistance on the saturated magnetization, are measured via the memory cell’s SPICE modeling. The Rashba spin–orbit interaction (SOI) and spin-valve model are developed by MATLAB scripting for evaluating a Rashba and Hamiltonian vector field. The toggle spin torque (TST) in the TST-MRAM cell is generated due to the interconnection between STT and spin–orbit torque (SOT) mechanisms. The TST-MRAM cell’s benefits are energy efficiency, large TMR ratio and superfast read and write operation. This work’s novel part is that the Hamiltonian field can control the net magnetization factor by the Rashba SOI effect. We end this paper with the performance enhancement techniques of MRAM cells due to VCMA and Rashba SOI effect.

The mean-field approximation for higher-dimensional Coulomb flows in the scaling-critical L ∞ space

In the mean-field scaling regime, a first-order system of particles with binary interactions naturally gives rise to a scalar partial differential equation (PDE), which, depending on the nature of the interaction, corresponds to either the Hamiltonian or gradient flow of the effective energy of the system for a large number of particles. The empirical measure of such systems is a weak solution to this limiting mean-field PDE, and one expects that as the number of particles tends to infinity, it converges along its lifespan in the weak-* sense to a more regular solution of the PDE, provided it does so initially. Much effort has been invested over the years in proving and quantifying this convergence under varying regularity assumptions. When the interaction potential is Coulomb, the mean-field PDE has a scaling invariance which leaves the L ∞ norm unchanged; i.e., L ∞ is a critical function space for the equation. Moreover, the L ∞ norm is either conserved or decreasing, and the equation is globally well-posed in this space, making it a natural choice for studying the dynamics. Building on our previous work (Rosenzweig 2022 Arch. Ration. Mech. Anal. 243 1361–431) for point vortices (i.e. d = 2), we prove quantitative convergence of the empirical measure to the L ∞ solution of the mean-field PDE for short times in dimensions d ⩾ 3. To the best of our knowledge, this is the first such work outside of the 2D case. Our proof is based on a combination of the modulated-energy method of Serfaty (2020 Duke Math. J. 169 2887–935) and a novel mollification argument first introduced by the author in Rosenzweig (2022 Arch. Ration. Mech. Anal. 243 1361–431). Compared to our prior work (Rosenzweig 2022 Arch. Ration. Mech. Anal. 243 1361–431), the new challenge is the non-logarithmic nature of the potential.

On the dispersionless Davey–Stewartson hierarchy: Zakharov–Shabat equations, twistor structure, and Lax–Sato formalism

AbstractIn this paper, we continue the study of the Davey–Stewartson system, which is one of the most important (2 + 1)‐dimensional integrable models. As we showed in the previous paper, the dDS (dispersionless Davey–Stewartson) system arises from Hamiltonian vector fields Lax pair. A new hierarchy of compatible partial differential equations (PDEs) defining infinitely many symmetries, which is associated with the dDS system, is defined in this paper. We show that this hierarchy arises from the commutation condition of a particular series of one‐parameter Hamiltonian vector fields.

Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds.

Flexibility of sections of nearly integrable Hamiltonian systems

Given any symplectomorphism on [Formula: see text] which is [Formula: see text] close to the identity, and any completely integrable Hamiltonian system [Formula: see text] in the proper dimension, we construct a [Formula: see text] perturbation of [Formula: see text] such that the resulting Hamiltonian flow contains a “local Poincaré section” that “realizes” the symplectomorphism. As a (motivating) application, we show that there are arbitrarily small perturbations of any completely integrable Hamiltonian system which are entropy non-expansive (and, in particular, exhibit hyperbolic behavior on a set of positive measure). We use some results in Berger–Turaev [On Herman’s positive entropy conjecture, Adv. Math. 349 (2019) 1234—1288], though in higher dimensions we could simply apply a construction from [D. Burago and S. Ivanov, Boundary distance, lens maps and entropy of geodesic ows of Finsler metrics, Geom. &amp; Topol. 20 (2016) 469–490].

Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter kR-spaces and locally convex spaces E such that E×E is a kR-space.

Couplings for Andersen dynamics

Andersen dynamics is a standard method for molecular simulations, and a precursor of the Hamiltonian Monte Carlo algorithm used in MCMC inference. The stochastic process corresponding to Andersen dynamics is a PDMP (piecewise deterministic Markov process) that iterates between Hamiltonian flows and velocity randomizations of randomly selected particles. Both from the viewpoint of molecular dynamics and MCMC inference, a basic question is to understand the convergence to equilibrium of this PDMP particularly in high dimension. Here we present couplings to obtain sharp convergence bounds in the Wasserstein sense that do not require global convexity of the underlying potential energy.