Nonintegrable Potentials Research Articles

Number-rigidity and β-circular Riesz gas

For an inverse temperature β>0, we define the β-circular Riesz gas on Rd as any microscopic thermodynamic limit of Gibbs particle systems on the torus interacting via the Riesz potential g(x)=‖x‖−s. We focus on the nonintegrable case d−1<s<d. Our main result ensures, for any dimension d≥1 and inverse temperature β>0, the existence of a β-circular Riesz gas which is not number-rigid. Recall that a point process is said number rigid if the number of points in a bounded Borel set Δ is a function of the point configuration outside Δ. It is the first time that the nonnumber-rigidity is proved for a Gibbs point process interacting via a nonintegrable potential. We follow a statistical physics approach based on the canonical DLR equations. It is inspired by the recent paper (Comm. Pure Appl. Math. 74 (2021) 172–222) where the authors prove the number-rigidity of the Sineβ process.

Chaotic instability in the BFSS matrix model

Chaotic scattering is a manifestation of transient chaos realized by the scattering with non-integrable potential. When the initial position is taken in the potential, a particle initially exhibits chaotic motion, but escapes outside after a certain period of time. The time to stay inside the potential can be seen as lifetime and this escape process may be regarded as a kind of instability. The process of this type exists in the Banks-Fischler-Shenker-Susskind (BFSS) matrix model in which the potential has flat directions. We discuss this chaotic instability by reducing the system with an ansatz to a simple dynamical system and present the associated fractal structure. We also show the singular behavior of the time delay function and compute the fractal dimension. This chaotic instability is the basic mechanism by which membranes are unstable, which is also common to supermembranes at quantum level.

$3D$-flows generated by the curl of a vector potential &amp; Maurer-Cartan equations

We examine $3D$ flows $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ admitting vector identity $M\mathbf{v} = \nabla \times \mathbf{A}$ for a multiplier $M$ and a potential field $\mathbf{A}$. It is established that, for those systems, one can complete the vector field $\mathbf{v}$ into a basis fitting an $\mathfrak{sl}(2)$-algebra. Accordingly, in terms of covariant quantities, the structure equations determine a set of equations in Maurer-Cartan form. This realization permits one to obtain the potential field as well as to investigate the (bi-)Hamiltonian character of the system. The latter occurs if the system has a time-independent first integral. In order to exhibit the theoretical results on some concrete cases, three examples are provided, namely the Gulliot system, a system with a nonintegrable potential, and the Darboux-Halphen system in symmetric polynomials.

The Discreteness-driven Relaxation of Collisionless Gravitating Systems: Entropy Evolution in External Potentials, N-dependence, and the Role of Chaos

Abstract We investigate the old problem of the fast relaxation of collisionless N-body systems that are collapsing or perturbed, emphasizing the importance of (noncollisional) discreteness effects. We integrate orbit ensembles in fixed potentials, estimating the entropy to analyze the time evolution of the distribution function. These estimates capture the correct physical behavior expected from the second law of thermodynamics, without any spurious entropy production. For self-consistent (i.e., stationary) samples, the entropy is conserved, while for non-self-consistent samples, it increases within a few dynamical times, stabilizing at a maximum (even in integrable potentials). Our results make transparent that the main ingredient for this fast collisionless relaxation is the discreteness (finite N) of gravitational systems in any potential. Additionally, in nonintegrable potentials, the presence of chaotic orbits accelerates the entropy production. Contrary to the traditional violent relaxation scenario, our results indicate that a time-dependent potential is not necessary for this relaxation. For the first time, in connection with the Nyquist–Shannon theorem, we derive the typical timescale for this discreteness-driven relaxation, with slightly weaker N-dependencies for nonintegrable potentials with substantial fractions of chaotic orbits. This timescale is much smaller than the collisional relaxation time even for small-N systems such as open clusters and represents an upper limit for the relaxation time of real N-body collisionless systems. Additionally, our results reinforce the conclusion of Beraldo e Silva et al. that the Vlasov equation does not provide an adequate kinetic description of the fast relaxation of collapsing collisionless N-body systems.

Bose-condensed atomic systems with nonlocal interaction potentials

The general approach for describing systems with a Bose–Einstein condensate, where atoms interact through nonlocal pair potentials, is presented. Special attention is paid to nonintegrable potentials, such as the dipolar interaction potential. The potentials that are not absolutely integrable can have not well-defined Fourier transforms. Using formally these not defined Fourier transforms leads to unphysical conclusions. To make the Fourier transform well defined, the interaction potential has to be regularised. This is illustrated by the example of dipolar interactions.

Semiclassical treatment of symmetry breaking and bifurcations in a non-integrable potential

We have derived an analytical trace formula for the level density of the Hénon–Heiles potential using the improved stationary phase method, based on extensions of Gutzwiller's semiclassical path integral approach. This trace formula has the correct limit to the standard Gutzwiller trace formula for the isolated periodic orbits far from all (critical) symmetry-breaking points. It continuously joins all critical points at which an enhancement of the semiclassical amplitudes occurs. We found a good agreement between the semiclassical and the quantum oscillating level densities for the gross shell structures and for the energy shell corrections, solving the symmetry breaking problem at small energies.

Closed-Orbit Theory of Spatial Density Oscillations in Finite Fermion Systems

We investigate the particle and kinetic-energy densities for N noninteracting fermions confined in a local potential. Using Gutzwiller's semiclassical Green function, we describe the oscillating parts of the densities in terms of closed nonperiodic classical orbits. We derive universal relations between the oscillating parts of the densities for potentials with spherical symmetry in arbitrary dimensions and a "local virial theorem" valid also for arbitrary nonintegrable potentials. We give simple analytical formulas for the density oscillations in a one-dimensional potential.

On the fundamental paradox of stellar dynamics

Various ways of solving the relaxation paradox for stellar systems are discussed. They include scattering on massive wave or material objects (giant molecular clouds, transient spiral arms and halo black holes), a revision of the Jeans–Chandrasekhar relaxation theory (resonant relaxation, relaxation in the smoothed field and collective relaxation in a steady non-integrable potential and in a non-steady (oscillating) mean field (violent relaxation)).

Chaos and the continuum limit in the gravitational N-body problem. II. Nonintegrable potentials.

This paper continues a numerical investigation of the statistical properties of "frozen-N orbits," i.e., orbits evolved in frozen, time-independent N-body realizations of smooth density distributions rho corresponding to both integrable and nonintegrable potentials, allowing for 10(2.5)</=N</=10(5.5). The focus is on distinguishing between, and quantifying, the effects of graininess on initial conditions corresponding, in the continuum limit, to regular and chaotic orbits. Ordinary Lyapunov exponents chi do not provide a useful diagnostic for distinguishing between regular and chaotic behavior. Frozen-N orbits corresponding in the continuum limit to both regular and chaotic characteristics have large positive chi even though, for large N, the "regular" frozen-N orbits closely resemble regular characteristics in the smooth potential. Alternatively, viewed macroscopically, both regular and "chaotic" frozen-N orbits diverge as a power law in time from smooth orbits with the same initial condition. However, convergence towards the continuum limit is much slower for chaotic orbits. For regular orbits, the time scale associated with this divergence t(G) approximately N(1/2)t(D), with t(D) a characteristic dynamical, or crossing, time; for chaotic orbits t(G) approximately (ln N)t(D). For N>10(3)-10(4), clear distinctions exist between the phase mixing of initially localized ensembles, which, in the continuum limit, exhibit regular versus chaotic behavior. Regular ensembles evolved in a frozen-N density distribution diverge as a power law in time, albeit more rapidly than ensembles evolved in the smooth distribution. Chaotic ensembles diverge in a fashion that is roughly exponential, albeit at a larger rate than that associated with the exponential divergence of the same ensemble evolved in smooth rho. For both regular and chaotic ensembles, finite-N effects are well mimicked, both qualitatively and quantitatively, by energy-conserving white noise with amplitude eta proportional, variant 1/N. This suggests strongly that earlier investigations of the effects of low amplitude noise on phase space transport in smooth potentials are directly relevant to real physical systems.

Boundary Value Problems for a Class of Sturm–Liouville Operators with Nonintegrable Potential

Boundary Value Problems for a Class of Sturm–Liouville Operators with Nonintegrable Potential

Collapse in systems with attractive nonintegrable potentials.

Collapse, or a gravitational-like phase transition, is found in microcanonical ensembles of particles with an attractive 1/r(alpha) potential not only for alpha = 1 but for all 0<alpha<3. The phase behavior of the system is complex: If an effective sufficiently short-range cutoff is applied, the density of the collapsed phase is finite everywhere; if not, the collapse results in a density singularity. Also, with increasing effective cutoff range, the gravitational phase transition will cross over to a normal first-order phase transition.

Invariant distributions and collisionless equilibria

This paper discusses the possibility of constructing time-independent solutions to the collisionless Boltzmann equation which depend on quantities other than global isolating integrals such as energy and angular momentum. The key point is that, at least in principle, a self-consistent equilibrium can be constructed from any set of time-independent phase-space building blocks which, when combined, generate the mass distribution associated with an assumed time-independent potential. This approach provides a way to justify Schwarzschild’s method for the numerical construction of self-consistent equilibria with arbitrary time-independent potentials, generalizing thereby an approach developed by Vandervoort for integrable potentials. As a simple illustration, Schwarzschild’s method is reformulated to allow for a straightforward computation of equilibria which depend only on one or two global integrals and no other quantities, as is reasonable, for example, for modelling axisymmetric configurations characterized by a non-integrable potential.

Excitation of a quantum gas of independent particles under periodic perturbation in integrable or nonintegrable potentials

The excitation of a quantum gas of 112 independent fermions in the time-dependent potential well, periodically oscillating around the spherical shape, was followed over ten oscillation cycles. Five different oscillation frequencies are considered for each of the five types of deformations: spheroidal and Legendre polynomial ripples P{sub 3}, P{sub 4}, P{sub 5}, and P{sub 6}. The excitation rate of the gas in the deforming hard-walled cavities substantially decreases after initial one or two cycles and the final excitation energy is a few times smaller than the wall formula predictions. Qualitatively similar results are obtained for the diffused Woods-Saxon well. The details and possible origins of this behavior are discussed as well as the consequences for the one-body dissipation model of nuclear dynamics. {copyright} {ital 1997} {ital The American Physical Society}

Resonant driving of chaotic orbits.

Finite time segments of chaotic orbits in strongly nonintegrable potentials often exhibit complicated power spectra, which, despite being broadband, are dominated by frequencies \ensuremath{\omega} appropriate for ``nearby'' regular orbits. This implies that even low amplitude periodic driving can trigger complicated resonant couplings, evidencing a sensitive dependence on the driving frequency \ensuremath{\Omega}. Numerical experiments involving individual chaotic orbits indicate that the response to a low amplitude time-periodic perturbation, as measured, e.g., by the maximum excursion in energy arising within a given time interval, can exhibit a sensitive dependence on \ensuremath{\Omega}, with substantial structure even on scales \ensuremath{\delta}\ensuremath{\Omega}${10}^{\mathrm{\ensuremath{-}}4}$ times a typical natural frequency \ensuremath{\omega}. Ensembles of chaotic initial conditions driven with a frequency \ensuremath{\Omega} comparable to the natural frequencies of the unperturbed orbits typically display diffusive behavior: The distribution of energy changes, N(\ensuremath{\delta}E(t)), at any given time t is Gaussian and the rms value of the change in energy \ensuremath{\delta}${\mathit{E}}_{\mathrm{rms}}$=A(\ensuremath{\Omega},E)\ensuremath{\alpha}${\mathit{t}}^{1/2}$, where \ensuremath{\alpha} denotes the driving amplitude. For fixed energy E, the proportionality constant A is independent of the detailed choice of initial conditions, but can exhibit a complicated dependence on \ensuremath{\Omega}. Potential implications for galactic dynamics are discussed.

Inverse eigenvalue problems for a singular Sturm-Liouville operator on (0,1)

In this paper we investigate two boundary value problems on the finite interval with potentials that may have non-integrable singularity at the origin. The first problem allows unique determination of the potential by only one spectrum for certain values of the parameter in one of the boundary conditions. The second problem deals with the Dirichlet problem for potentials which are the sum of Bessel's main part with constant bounded from below and non-integrable potential. Uniqueness results are proved for both problems.

Hamiltonian perturbation theory for numerically constructed phase-space tori

The methods of McGill & Binney and Kaasalainen & Binney are employed in creating invariant phase-space tori in non-integrable potentials supporting minor-orbit families. Following Binney & Kumar, these tori are used to define an integrable Hamiltonian H0 closely approximating the original one, and Hamiltonian perturbation theory is then used to demonstrate that a minor-orbit family can be treated as one made up of orbits trapped by a resonance of H0. The main objectives of this paper are (i) to develop a model for perturbative calculations, resembling the pendulum analysis of standard secular perturbation theory, that takes into account the special properties of H0, and (ii) to show how H0 may be constructed in the presence of powerful resonances, which render ineffective the approach of Binney & Kumar. As an illustration of the method, the ‘banana’ and ‘fish’ minor-orbit families in the planar logarithmic potential are explicitly derived by perturbing constructed H0s.

Angle variables for numerically fitted orbital tori

Angle variables are constructed for orbital tori least-squares fitted to general potentials by the method of McGill & Binney . These angle variables enable one to determine the densities ⍴J(x) associated with the orbit that has given actions J. They also make it possible to treat any non-integrable potential as a perturbation on a nearby integrable one. As an illustration of this approach, Hamiltonian perturbation theory is used to derive the width of the 1:1 resonant-orbit family in a realistic model of the potential of a disk galaxy.