Infinitesimal Perturbation Research Articles

Nonlinear evolutions of streaky structures in viscoelastic pipe flows

The dynamics of streaky structures in a viscoelastic pipe flow of FENE-P fluids is studied numerically. The values of Weissenberg number ( W i , polymer relaxation time over flow turn-over time) and viscosity ratios are varied to evaluate the effects of viscoelasticity in the pipe. For Reynolds numbers 3000 ∼ 3500 (based on pipe radius and maximum laminar velocity) near the transition boundary, we select finite-amplitude two-dimensional rolls as initial conditions and add infinitesimal three-dimensional perturbations to trigger the transition. The 2D streamwise rolls are highly unstable to small 3D disturbances, which helps to better understand the evolution of large scale structures. The stronger temporal intermittency in viscoelastic flows makes it more difficult to identify a successful transition; we found that quantities directly related to viscoelasticity can serve as better indicators of transition to turbulence. The role of polymer stress is studied through the kinetic energy budget and it is found that polymer torque opposes the vorticity and become more dominant for higher W i , consistent with previous studies. The results show that the secondary instability of the streaky structures can be enhanced by viscoelasticity at low W i , while at high W i viscoelasticity delays the transition to turbulence, sometimes leading to relaminarization. • Direct numerical simulations of viscoelastic pipe flow of FENE-P fluids are conducted. • Viscoelasticity changes the evolution of streaks and causes strong intermittency. • Polymer stress is studied via turbulent kinetic energy budget in streak breakdown. • Non-monotonic influence of Weissenberg number on threshold boundary of transitions.

Spatial decay of Kubo’s canonical correlation functions

Kubo’s canonical correlation functions (canonical correlators) describe the static response of a system in equilibrium to infinitesimal local perturbations. Knowing their decay properties with respect to spatial distance is important for many theoretical and experimental applications. For a thermal state of a system with short-range interactions, we prove that any knowledge of the decay rate of ordinary correlators readily translates into that of canonical correlators. As the former have been extensively studied, our result can lead to many new results on the latter. Throughout the paper, we use the framework of infinite-volume quantum lattice model. However, the method we use is adaptable for other physical scenarios not considered in this paper.

Photon emission from inside the innermost stable circular orbit

We consider a situation where a light source orbiting the innermost stable circular orbit (ISCO) of the Kerr black hole is gently falling from the marginally stable orbit due to an infinitesimal perturbation. Assuming that the light source emits photons isotropically, we show that the last radius at which more than 50\% of emitted photons can escape to infinity is approximately halfway between the ISCO radius and the event horizon radius. To evaluate them, we determine emitter orbits from the vicinity of the ISCO, which are uniquely specified for each black hole spin, and identify the conditions for a photon to escape from any point on the equatorial plane of the Kerr spacetime to infinity by specifying regions in the two-dimensional photon impact parameter space completely. We further show that the proper motion of the emitter affects the photon escape probability and blueshifts the energy of emitted photons.

Thermalization and prethermalization in periodically kicked quantum spin chains

We study the dynamics of periodically-kicked many-body systems away from the high-frequency regime, and discuss a family of Floquet systems where the notion of prethermalization can be naturally extended to intermediate and low driving frequencies. We investigate numerically the dynamics of both integrable and nonintegrable systems, and report on the formation of a long-lived prethermal plateau, akin to the high-frequency limit, where the system thermalizes with respect to an effective Hamiltonian captured by the inverse-frequency expansion (IFE). Unlike the high-frequency regime, we find that the relevant heating times are model dependent: we analyze the stability of the prethermal plateau to small perturbations in the drive period, and show that, in a spin chain whose IFE is intractable, the plateau duration is insensitive to the perturbation strength, in contrast to a chain where the IFE admits the resummation of an entire subseries. Infinitesimal perturbations are enough to restore the ergodic properties of the system, and decrease residual finite-size effects. Although the regime where the Floquet system leaves the prethermal plateau and starts heating up to infinite temperature is not captured by the IFE, we provide evidence that the evolved subsystem is described well by a thermal state w.r.t.~the IFE Hamiltonian, with a gradually changing temperature, in accord with the Eigenstate Thermalization Hypothesis.

Non-Bloch band theory in bosonic Bogoliubov–de Gennes systems

In recent research, it has been shown that non-Hermitian systems exhibit sensitivity to boundaries, and it is caused by the non-Hermitian skin effect. In this work, we construct the non-Bloch band theory in bosonic Bogoliubov--de Gennes (BdG) systems. From our theory, we can calculate the generalized Brillouin zone and the energy spectrum in such systems with open boundary conditions in the thermodynamic limit, and we can thus discuss its non-Hermitian nature, despite Hermiticity of an original Hamiltonian. In fact, we find that the bosonic Kitaev-Majorana chain exhibits rich aspects of the non-Hermitian skin effect, such as instability against infinitesimal perturbations and reentrant behavior, in terms of the non-Bloch band theory. This result indicates that our theory is powerful tool for studying non-Hermitian nature in bosonic BdG systems.

Buffer allocation design for unreliable production lines using genetic algorithm and finite perturbation analysis

The buffer allocation problem in production lines is an NP-hard combinatorial optimisation problem. This paper proposes a new hybrid optimisation approach (using simulation) relying on genetic algorithm (GA) and finite perturbation analysis (FPA). Unlike the infinitesimal perturbation analysis, which deals with small (infinitesimal variation) perturbations for estimating gradients of the performance measure, FPA deals with larger (finite) or more lasting perturbations. It is an extension specifically dedicated to discrete decision variables and applicable to most discrete-event dynamic systems. The proposed method allows a global search using GA, with refinement in specific solution-space regions using FPA. The main objective is to maximise the average production rate of a production line with unreliable machines, by allocating the total buffer capacity in locations between machines. Extensive numerical experiments show that: (1) the proposed hybrid GA-FPA method clearly outperforms the state-of-the-art methods from the literature; (2) combining FPA and GA is beneficial when compared to employing GA or FPA independently.

The validation of new phase-dependent gait stability measures: a modelling approach.

Identification of individuals at risk of falling is important when designing fall prevention methods. Current measures that estimate gait stability and robustness appear limited in predicting falls in older adults. Inspired by recent findings on changes in phase-dependent local stability within a gait cycle, we devised several phase-dependent stability measures and tested for their usefulness to predict gait robustness in compass walker models. These measures are closely related to the often-employed maximum finite-time Lyapunov exponent and maximum Floquet multiplier that both assess a system's response to infinitesimal perturbations. As such, they entail linearizing the system, but this is realized in a rotating hypersurface orthogonal to the period-one solution followed by estimating the trajectory-normal divergence rate of the swing phases and the foot strikes. We correlated the measures with gait robustness, i.e. the largest perturbation a walker can handle, in two compass walker models with either point or circular feet to estimate their prediction accuracy. To also test for the dependence of the measures under state space transform, we represented the point feet walker in both Euler–Lagrange and Hamiltonian canonical form. Our simulations revealed that for most of the measures their correlation with gait robustness differs between models and between different state space forms. In particular, the latter may jeopardize many stability measures' predictive capacity for gait robustness. The only exception that consistently displayed strong correlations is the divergence of foot strike. Our results admit challenges of using phase-dependent stability measures as objective means to estimate the risk of falling.

AN IMPROVEMENT OF MARKOVIAN INTEGRATION BY PARTS FORMULA AND APPLICATION TO SENSITIVITY COMPUTATION

This paper establishes a new version of integration by parts formula of Markov chains for sensitivity computation, under much lower restrictions than the existing researches. Our approach is more fundamental and applicable without using Girsanov theorem or Malliavin calculus as did by past papers. Numerically, we apply this formula to compute sensitivity regarding the transition rate matrix and compare with a recent research by an IPA (infinitesimal perturbation analysis) method and other approaches.

Variance Comparison between Infinitesimal Perturbation Analysis and Likelihood Ratio Estimators to Stochastic Gradient

We theoretically compare variances between the Infinitesimal Perturbation Analysis (IPA) estimator and the Likelihood Ratio (LR) estimator to Monte Carlo gradient for stochastic systems. The conditions proposed in [Cui et al., 2020] when the IPA estimator has a smaller variance can yield sharper inequalities or be further relaxed. We also prove a practically interesting result that the IPA estimators to European vanilla and arithmetic Asian options' Delta, respectively, have smaller variance when the underlying asset's return process is independent of the initial price.

The Price of Decentralization: Event-Driven Optimization for Multiagent Persistent Monitoring Tasks

In persistent monitoring tasks, the objective is to control the movements of cooperating agents in order to minimize an uncertainty metric associated with a finite number of targets. We formulate an optimal control problem and show that the optimal solution can be reduced to or approximated by parametric agent trajectory families. The behavior of agents and targets under optimal control can be described by a hybrid system. This enables the use of infinitesimal perturbation analysis to obtain an online centralized solution through a gradient-based algorithm. We identify conditions under which this centralized solution to the parametric optimization problems can be recovered in a decentralized and event-driven manner. In the decentralized scheme, each agent optimizes its performance based on local information, except for one type of nonlocal event requiring communication from a nonneighbor agent, giving rise to a quantifiable “price of decentralization.” Simulation examples are included to illustrate the effectiveness of this “almost decentralized” optimization algorithm and compare it to its fully decentralized counterpart where the aforementioned nonlocal event is ignored.

A pseudo-spectrum based characterization of the robust strong H-infinity norm of time-delay systems with real-valued and structured uncertainties

Abstract This paper provides a mathematical characterization of the robust (strong) H-infinity norm of an uncertain linear time-invariant system with discrete delays in terms of the robust distance to instability of an associated characteristic matrix. The considered class of uncertainties consists of real-valued, structured, Frobenius norm-bounded matrix uncertainties that act on the coefficient matrices. The robust H-infinity norm, defined as the worst-case value of the H-infinity norm over all admissible uncertainty values, is an important measure of robust performance to quantify the worst-case disturbance rejection of an uncertain dynamical system. For the considered system class, this robust H-infinity norm is however a fragile measure, as for a particular instance of the uncertainties, the H-infinity norm might be sensitive to arbitrarily small perturbations on the delays. Therefore, we introduce the robust strong H-infinity norm, inspired by the notion of strong stability of delay differential equations of neutral type, which takes into account both the uncertainties on the system matrices and infinitesimal delay perturbations. This quantity is a continuous function of both the elements of the system matrices and the delays. The main contribution of this work is the introduction of a relation between this robust strong H-infinity norm and the robust distance to instability of an associated characteristic matrix. This relation is subsequently employed in a novel algorithm for computing the robust strong H-infinity norm of uncertain time-delay systems.

Weak Integrability Breaking: Chaos with Integrability Signature in Coherent Diffusion.

We study how perturbations affect dynamics of integrable many-body quantum systems, causing transition from integrability to chaos. Looking at spin transport in the Heisenberg chain with impurities we find that in the thermodynamic limit transport gets diffusive already at an infinitesimal perturbation. Small extensive perturbations therefore cause an immediate transition from integrability to chaos. Nevertheless, there is a remnant of integrability encoded in the dependence of the diffusion constant on the impurity density, namely, at small densities it is proportional to the square root of the inverse density, instead of to the inverse density as would follow from Matthiessen's rule. We show that Matthiessen's rule has to be modified in nonballistic systems. Results also highlight a nontrivial role of interacting scattering on a single impurity, and that there is a regime where adding more impurities can actually increase transport.

Adiabatic Eigenstate Deformations as a Sensitive Probe for Quantum Chaos

In the past decades, it was recognized that quantum chaos, which is essential for the emergence of statistical mechanics and thermodynamics, manifests itself in the effective description of the eigenstates of chaotic Hamiltonians through random matrix ensembles and the eigenstate thermalization hypothesis. Standard measures of chaos in quantum many-body systems are level statistics and the spectral form factor. In this work, we show that the norm of the adiabatic gauge potential, the generator of adiabatic deformations between eigenstates, serves as a much more sensitive measure of quantum chaos. We are able to detect transitions from non-ergodic to ergodic behavior at perturbation strengths orders of magnitude smaller than those required for standard measures. Using this alternative probe in two generic classes of spin chains, we show that the chaotic threshold decreases exponentially with system size and that one can immediately detect integrability-breaking (chaotic) perturbations by analyzing infinitesimal perturbations even at the integrable point. In some cases, small integrability-breaking is shown to lead to anomalously slow relaxation of the system, exponentially long in system size.

Defining a well-ordered Floquet basis by the average energy

At the moment, the most efficient method to compute the state of a periodically driven quantum system is using Floquet theory and the Floquet eigenbasis. The wide application of this basis set method is limited by: a lack of unique ordering of the Floquet eigenfunctions, an ambiguity in their definition at resonance, and an instability against infinitesimal perturbation at resonance. We address these problems by redefining the eigenbasis using a revised definition of the average energy as a quantum number. As a result of this redefinition, we also obtain a Floquet-Ritz variational principle, and justify the truncation of the Hilbert space.

Does elasticity stabilize a magnetic neutron star?

ABSTRACT The configuration of the magnetic field in the interior of a neutron star is mostly unknown from observations. Theoretical models of the interior magnetic field geometry tend to be oversimplified to avoid mathematical complexity and tend to be based on axisymmetric barotropic fluid systems. These static magnetic equilibrium configurations have been shown to be unstable on a short time-scale against an infinitesimal perturbation. Given this instability, it is relevant to consider how more realistic neutron star physics affects the outcome. In particular, it makes sense to ask if elasticity, which provides an additional restoring force on the perturbations, may stabilize the system. It is well known that the matter in the neutron star crust forms an ionic crystal. The interactions between the crystallized nuclei can generate shear stress against any applied strain. To incorporate the effect of the crust on the dynamical evolution of the perturbed equilibrium structure, we study the effect of elasticity on the instability of an axisymmetric magnetic star. In particular, we determine the critical shear modulus required to prevent magnetic instability and consider the corresponding astrophysical consequences.

The Limit Point of the Pentagram Map and Infinitesimal Monodromy

Abstract The pentagram map takes a planar polygon $P$ to a polygon $P^{\prime }$ whose vertices are the intersection points of the consecutive shortest diagonals of $P$. The orbit of a convex polygon under this map is a sequence of polygons that converges exponentially to a point. Furthermore, as recently proved by Glick, coordinates of that limit point can be computed as an eigenvector of a certain operator associated with the polygon. In the present paper, we show that Glick’s operator can be interpreted as the infinitesimal monodromy of the polygon. Namely, there exists a certain natural infinitesimal perturbation of a polygon, which is again a polygon but in general not closed; what Glick’s operator measures is the extent to which this perturbed polygon does not close up.

The dynamics of stratified horizontal shear flows at low Péclet number

Abstract

Stability of Poiseuille flow in an anisotropic porous layer with oblique principal axes: More accurate solution

AbstractThe stability of fluid flow in an anisotropic porous medium of Brinkman type is investigated. Anisotropy in the permeability is considered such that its longitudinal principal axis is oriented arbitrarily with the horizontal, while transversely it is isotropic. A fourth‐order eigenvalue problem obtained by performing a linear stability analysis is solved numerically using the Chebyshev collocation and the compound matrix methods. The critical Reynolds number and the critical wave number are computed for different values of porous parameter, orientation angle of the principal axis and the mechanical anisotropy parameter. The porous and the mechanical anisotropy parameters disclose contrasting contributions to the stability of fluid flow. The orientation angle instills either stabilizing or destabilizing effects on the fluid flow depending on the value of anisotropy parameter. Besides, the streamlines of the perturbation modes are presented for various values of orientation angle and anisotropy parameter. For the Darcy case, a simple analytical method is employed and established that the flow is stable for all infinitesimal perturbations.

Butterfly effect and a self-modulating El Niño response to global warming.

El Niño and La Niña, collectively referred to as the El Niño-Southern Oscillation (ENSO), are not only highly consequential1-6 but also strongly nonlinear7-14. For example, the maximum warm anomalies of El Niño, which occur in the equatorial eastern Pacific Ocean, are larger than the maximum cold anomalies of La Niña, which are centred in the equatorial central Pacific Ocean7-9. The associated atmospheric nonlinear thermal damping cools the equatorial Pacific during El Niño but warms it during La Niña15,16. Under greenhouse warming, climate models project an increase in the frequency of strong El Niño and La Niña events, but the change differs vastly across models17, which is partially attributed to internal variability18-23. Here we show that like a butterfly effect24, an infinitesimal random perturbation to identical initial conditions induces vastly different initial ENSO variability, which systematically affects its response to greenhouse warming a century later. In experiments with higher initial variability, a greater cumulative oceanic heat loss from ENSO thermal damping reduces stratification of the upper equatorial Pacific Ocean, leading to a smaller increase in ENSO variability under subsquentgreenhouse warming. This self-modulating mechanism operates in two large ensembles generated using two different models, each commencing from identical initial conditions but with a butterfly perturbation24,25; it also operates in a large ensemble generated with another model commencing from different initial conditions25,26 and across climate models participating in the Coupled Model Intercomparison Project27,28. Thus, if the greenhouse-warming-induced increase in ENSO variability29 is initially suppressed by internal variability, future ENSO variability is likely to be enhanced, and vice versa. This self-modulation linking ENSO variability across time presents a different perspective for understanding the dynamics of ENSO variability on multiple timescales in a changing climate.

Limiting stiffness coefficients analysis of texture foil journal bearing

The purpose of this study is to explore the limiting stiffness coefficients of a foil journal bearing with the texture bump profile of top foil. The limiting stiffness coefficients are evaluated based on simplified compressible Reynolds equation for large bearing numbers with high speeds of foil journal bearing. A limiting pressure gradient solution for a texture bump of top foil bearing is analyzed. The analytical model accounts for the top foil texture bump profile and bottom foil bump compliance. Results of linearized nondimensional stiffness coefficients obtained using infinitesimal perturbation method are compared for various top foil texture bump profiles. The influence of top foil texture bump extent and height on the limiting stiffness coefficients of a foil journal bearing are investigated.