Perturbation Solution Research Articles

Application of the frozen-modes approximation to classical harmonic oscillator systems

The problems of open classical systems usually correspond to a motion of a test particle that interacts with a large number of bath oscillators. Often, the test particle itself can be considered a harmonic oscillator. For such composite systems, exact numerical solutions are available, but they can become increasingly costly for a large number of bath oscillators. Here we take inspiration from the recent work on open quantum systems and investigate the applicability of the frozen-modes approximation to such classical systems. This approach assumes that some part of the low-frequency bath modes are frozen, thus only their initial values need to be considered. We show that by applying the frozen-modes approximation one can significantly increase the accuracy of the perturbative multiple-scales solution, especially for slow baths. This approach provides a good accuracy even for strong system–bath couplings, a regime that is not accessible to straightforward applications of the perturbation theory. We also suggest a rule for the splitting of spectral density to the fast and slow bath modes. We find that our approach gives excellent results for the ohmic spectral density, but it could be applied for other similar spectral densities as well.

Energy-Preserving Perturbation Method for Thermally Induced Dynamics in Aerospace Tensegrity Structures

Tensegrity structures are emerging as pivotal components in the assembly of ultralarge modular spacecraft in orbit, primarily due to their exceptional volume-to-mass ratio and robust controllability. During on-orbit operations, given the low damping and high flexibility of tensegrity, its vibration-induced deformations affect the magnitude of the heat flux, presenting a typical force–shape coupling problem. This paper introduces an energy-preserving matrix perturbation solution designed to efficiently and accurately address the thermally induced vibration challenges in tensegrity. This method effectively mitigates the energy dissipation and numerical instability issues. Furthermore, the analysis incorporates the impact of the axial component of heat flux on structural vibration, uncovering the influence mechanism of this often-overlooked slow variable on the dynamic behavior of tensegrity. The proposed method demonstrates a deviation of less than 1% between the thermal response and the results obtained from finite element analysis, while achieving a 70% reduction in computation time. The varying axial thermal load leads to a rapid decrease in system frequency, resulting in divergent thermal vibrations. Additionally, the significant dynamic stress can potentially surpass the structural stress limits. These findings underscore the critical importance of considering such effects in the design and calculation of tensegrity structures for aerospace applications.

Rosetta stone for the population dynamics of spiking neuron networks.

Populations of spiking neuron models have densities of their microscopic variables (e.g., single-cell membrane potentials) whose evolution fully capture the collective dynamics of biological networks, even outside equilibrium. Despite its general applicability, the Fokker-Planck equationgoverning such evolution is mainly studied within the borders of the linear response theory, although alternative spectral expansion approaches offer some advantages in the study of the out-of-equilibrium dynamics. This is mainly due to the difficulty in computing the state-dependent coefficients of the expanded system of differential equations. Here, we address this issue by deriving analytic expressions for such coefficients by pairing perturbative solutions of the Fokker-Planck approach with their counterparts from the spectral expansion. A tight relationship emerges between several of these coefficients and the Laplace transform of the interspike interval density (i.e., the distribution of first-passage times). "Coefficients" like the current-to-rate gain function, the eigenvalues of the Fokker-Planck operator and its eigenfunctions at the boundaries are derived without resorting to integral expressions. For the leaky integrate-and-fire neurons, the coupling terms between stationary and nonstationary modes are also worked out paving the way to accurately characterize the critical points and the relaxation timescales in networks of interacting populations.

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

A novel hybrid boundary element for polygonal holes with rounded corners in two-dimensional anisotropic elastic solids

A novel hybrid boundary element is developed for polygonal holes in finite anisotropic elastic plates based on two different special fundamental solutions for holes. Since these special fundamental solutions satisfy traction-free condition along the hole's boundary, there is no mesh required on the boundary of polygonal holes. Various types of polygonal holes with rounded corners, such as triangles, rhombuses, ovals, pentagons, are considered by adding proper perturbation to an elliptical hole. The developed hybrid element is a mixture of two special boundary elements: one is based on the special fundamental solution derived through nonconformal mapping and the other is based on the solution derived through perturbation technique with conformal mapping. The special boundary element methods are combined through submodeling technique. First, the global model is solved using the perturbation solution. Then, using the displacements obtained from global model, an auxiliary submodel is set up and the results are evaluated with the nonconformal solution. The present method is compared and validated with conventional boundary element method and finite element method. The effect of hole curvature, material anisotropy, and loading condition on the stress distribution around the hole is presented.

Machine Learning Guided AQFEP: A Fast and Efficient Absolute Free Energy Perturbation Solution for Virtual Screening.

Structure-based methods in drug discovery have become an integral part of the modern drug discovery process. The power of virtual screening lies in its ability to rapidly and cost-effectively explore enormous chemical spaces to select promising ligands for further experimental investigation. Relative free energy perturbation (RFEP) and similar methods are the gold standard for binding affinity prediction in drug discovery hit-to-lead and lead optimization phases, but have high computational cost and the requirement of a structural analog with a known activity. Without a reference molecule requirement, absolute FEP (AFEP) has, in theory, better accuracy for hit ID, but in practice, the slow throughput is not compatible with VS, where fast docking and unreliable scoring functions are still the standard. Here, we present an integrated workflow to virtually screen large and diverse chemical libraries efficiently, combining active learning with a physics-based scoring function based on a fast absolute free energy perturbation method. We validated the performance of the approach in the ranking of structurally related ligands, virtual screening hit rate enrichment, and active learning chemical space exploration; disclosing the largest reported collection of free energy simulations to date.

Calculation of Green's functions for heat conduction in curved anisotropic media

AbstractSolving the inverse problems of heat conduction often requires performing a very large number of Green's function evaluations. This paper addresses the calculation of Green's functions in anisotropic curved media, such as composite lamina, where the principal axes of thermal conductivity follow the curved surface. We start with established exact series solutions for cylindrical convex and concave shapes. These solutions are extended to accommodate geometries of real‐world composite materials that do not form a closed surface like a cylinder. Unfortunately, the exact solutions have the form of an infinite series of sums or integrals and are computationally infeasible for the inverse problems of interest, especially for large radii of curvature. This motivates a perturbation solution that is accurate and computationally efficient at large radii of curvature. In addition, it motivates a phenomenological approximation that is extremely computationally efficient over a broad range of curvatures, but with some sacrifice in accuracy. These solutions are compared with exact solutions, with each other, and with numerical finite difference calculation. We identify regions in parameter space where the different approaches are preferable and where they lead to the same numerical result.

Parameter Regression for Porous Electrodes Employed in Lithium-Ion Batteries and Application to Ni0.89Co0.05Mn0.05Al0.01O2

We develop a parameter regression scheme that can be used with battery models of interest to the battery-analysis community. We show that the recent reduced order model (ROM1, 2022 J. Electrochem. 169 070520, DOI: 10.1149/1945–7111/ac7c93), which is based on a perturbation solution, can be used in place of the full system of nonlinear partial differential equations with minimal loss of accuracy for the conditions of this work, which are relevant for electric vehicle applications. The use of the computationally efficient ROM1, cast in the Python programming language, along with a routine native to Python for the nonlinear regression of model parameters through the minimization of the squared differences between experimental results and model calculations, provides a fast method for the overall endeavor. We apply the procedure to examine Ni0.89Co0.05Mn0.05Al0.01O2, a high-capacity material that is of increasing interest with respect to electric vehicles and other products that rely on batteries of high energy density. Difficulties encountered in this work include the large number of parameters governing the battery model, parameter sensitivity in the regression analyses, and the potential for multiple solutions. We close this publication with a discussion of these challenges and open questions with respect to parameter identification.

Revisiting flat rotation curves in Chern-Simons modified gravity

We revisit slow rotating black hole (BH) solutions in Chern-Simons modified gravity (CSMG) by considering perturbative solution about Schwarzschild BH. In particular, the case when nondynamical CSMG with noncanonical CS scalar is considered. We provide a new solution different from the previously obtained one [1] which we refer to as KMT model. The present solution accounts for frame dragging effect which includes not only radial dependence as in the KMT. Nevertheless, it reduces to KMT as a particular case. We show that the tidal gravitational force (Kretschmann invariant) associated to the present solution contains a term of order 1/r3 additional to Schwarzschild but absent from directional divergence, unlike KMT model which diverges along the axis of symmetry. We derive the corresponding circular velocity of a massive test particle in which the KMT velocity is recovered in addition to an extra term ∝r. We investigate possible constraints on KMT and the present solutions from the observed rotation curve of UGC11455 galaxy as an example. We show that perturbation solutions cannot physically explain the flattening of galactic rotation curves.

Application of Perturbation Method to Analytical H2 Optimization for a Damped Structure with Inerter-Based Control

Inherent damping exists in all real engineering systems and it affects the dynamic behavior of the structures in vibration reduction. However, it is generally neglected in existing analytical H2 optimization for inerter-based vibration control. Hence, an analytical H2 optimization of a tuned inerter damper (TID) based on the perturbation method is presented to control the vibration of a damped structure subjected to stochastic acceleration ground excitations. Since relative displacement and velocity are crucial to structural damage and absolute acceleration is related to vibration comfort, three sets of closed-form solutions for the optimal stiffness ratio and damping ratio are derived to suppress the corresponding responses. The expressions of these optimal parameters are functions of the inherent damping and inerter-mass ratio. The accuracy of these analytical solutions is demonstrated by comparison with the results provided by the numerical optimization procedure. The relevant parameters affecting the vibration control performance and the robustness of the TID in a damped structure are analyzed in frequency domain. Finally, time–history response and energy dissipation of TID are also explored by numerical examples under real and artificial seismic waves. The results highlight that more than 10% reduction ratio of the mean square value can be further achieved for a structure with a low inherent damping ratio ([Formula: see text]) if the present multi-order perturbation solutions are used for optimization, instead of the existing method which neglects the inherent damping. This superiority of the present perturbation method will become more prominent with the increase of the inherent damping ratio. Therefore, the present perturbation method has a great significance for designing a TID to achieve the best control performance.

Asymptotic matching the self-consistent expansion to approximate the modified Bessel functions of the second kind

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem—too much freedom! The SCE can be used to generate an enormous number of approximations but distinguishing the superb approximations from the deficient ones can only be achieved after the fact by comparison to experimental or numerical results. Here, we propose a method of using the SCE to a priori obtain uniform approximations, namely asymptotic matching. If the asymptotic behaviour of a problem can be identified, then the approximations generated by the SCE can be tuned to asymptotically match the desired behaviour and this can be used to obtain uniform approximations over the entire domain of consideration, without needing to resort to empirical comparisons. We demonstrate this method by applying it to the task of obtaining uniform approximations of the modified Bessel functions of the second kind, Kα(x) .

Perturbation-based nonperturbative method

This paper presents a nonperturbative method for solving eigenproblems. This method applies to almost all potentials and provides nonperturbative approximations for any energy level. The method converts an eigenproblem into a perturbation problem, obtains perturbation solutions through standard perturbation theory, and then analytically continues the perturbative solution into a nonperturbative solution. Concretely, we follow three main steps: (1) Introduce an auxiliary potential that can be solved exactly and treat the potential to be solved as a perturbation on this auxiliary system. (2) Use perturbation theory to obtain an approximate polynomial of the eigenproblem. (3) Use a rational approximation to analytically continue this approximate polynomial into the nonperturbative region.

Open-closed string duality, branes, and topological recursion

We consider matrix models exhibiting open-closed string duality in two-dimensional string theories with various amounts of supersymmetry. In particular, a relationship between matrix models in the β = 2 Wigner-Dyson class and models in the (1 + 2Γ, 2) Altland-Zirnbauer class relates the perturbative solutions of the two systems’ string equations. Point-like operator insertions in the closed string theory are mapped to the topological expansion of the free energy in the open string theory. We compute correlation functions of macroscopic loop operators and FZZT branes in a general topological gravity background. The relationship between the topological recursion of moduli space volumes and branes is discussed by analyzing the Virasoro conditions in the matrix models.

Global well-posedness of three-dimensional magneto-micropolar fluid equations with partial dissipation

In this paper, the three-dimensional magneto-micropolar fluid equations with partial dissipation is studied on R 3 . The main purpose of this paper is to establish global solutions with a small initial value. More specifically, we demonstrate the global stability of the perturbation solutions near the background magnetic field. The proof is mainly based on the energy estimate, the tricky interpolation techniques and bootstrapping argument.

Quantifying macrostructures in viscoelastic sub-diffusive flows

We present a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically consistent decomposition of the polymer conformation tensor, into the so-called structure tensor. Our approach bypasses an inherent problem in the standard arithmetic decomposition, namely, the fluctuating conformation tensor fields may not be positive definite and hence, do not retain their physical meaning. Using well-established results in matrix analysis, the space of positive definite matrices is transformed into a Riemannian manifold by defining and constructing a geodesic via the inner product on its tangent space. This geodesic is utilized to define three scalar invariants of the structure tensor, which do not suffer from the caveats of the regular invariants (such as trace and determinant) of the polymer conformation tensor. First, we consider the problem of formulating perturbative expansions of the structure tensor using the geodesic, which is consistent with the Riemannian manifold geometry. A constraint on the maximum time, during which the evolution of the perturbative solution can be well approximated by linear theory along the Euclidean manifold, is found. A comparison between the linear and the nonlinear dynamics, identifies the role of nonlinearities in initiating the symmetry breaking of the flow variables about the centerline. Finally, fully nonlinear simulations of the viscoelastic sub-diffusive channel flows, underscore the advantage of using these invariants in effectively quantifying the macrostructures.

Local clustering of relic neutrinos: comparison of kinetic field theory and the Vlasov equation

Gravitational clustering in our cosmic vicinity is expected to lead to an enhancement of the local density of relic neutrinos. We derive expressions for the neutrino density, using a perturbative approach to kinetic field theory and perturbative solutions of the Vlasov equation up to second order. Our work reveals that both formalisms give exactly the same results and can thus be considered equivalent. Numerical evaluation of the local relic neutrino density at first and second order provides some fundamental insights into the frequently applied approach of linear response to neutrino clustering (also known as the Gilbert equation). Against the naive expectation, including the second-order contribution does not lead to an improvement of the prediction for the local relic neutrino density but to a dramatic overestimation. This is because perturbation theory breaks down in a momentum-dependent fashion and in particular for densities well below unity.

Perturbation Solution to Modified Nonlinear Schr¨odinger Equation Based on Slowly Varying Envelope Approximation

Perturbation Solution to Modified Nonlinear Schr¨odinger Equation Based on Slowly Varying Envelope Approximation

Theoretical study on multi-physics coupling problems of functionally-graded lead-free piezoelectric rectangular thin plates

This paper carried out a theoretical research on multi-physics coupling problems of functionally-graded lead-free piezoelectric rectangular thin plates. Firstly, the perturbation theory was used to study the bending problem of functionally-graded lead-free piezoelectric rectangular thin plates, and the nonlinear governing equations for this bending problem were established. Then, the three piezoelectric coefficients reflecting the piezoelectric property were taken as the perturbation parameters, and the nonlinear governing equations were transformed into multiple sets of linear equations by the multi-parameter perturbation method. Finally, the solutions of these linear equations were solved by the semi-inverse solving method. Based on the finite element simulations, the effectiveness of the perturbation solutions obtained here was verified. The research results showed that it was reasonable to choose three piezoelectric coefficients as perturbation parameters, which was accord with the basic idea of perturbation theory. The analysis results indicated that there was a cross transfer relationship between external loads and perturbation solutions of every order, which could be used to simplify the perturbation expansion. Moreover, the deformation of the functionally-graded lead-free piezoelectric thin plate was less than that of the functionally-graded thin plate, that is, piezoelectric properties in lead-free piezoelectric materials had a piezoelectric enhancement effect.

Green's functions for anisotropic elastic plates containing polygonal holes

Two sets of Green's functions for anisotropic elastic media containing a polygonal hole with rounded corners are presented. Analyzing non-elliptical holes poses a formidable mathematical challenge for anisotropic materials because existing mapping functions generally fail to meet certain criteria. The nonconformal mapping scheme and the perturbation method with conformal mapping are applied to derive two sets of solutions, respectively. They are solved using the method of analytical continuation. For the nonconformal approach, the hole is mapped onto a unit circle with nonconformal mapping functions, and the solution form is identical to elliptical holes only with a different mapping function. The perturbation solution is expressed in a series form based on the conformal mapping functions and solutions for elliptical holes. Proper adjustments of branch cuts of complex logarithm functions are introduced for both solutions. These adjustments are essential in numerical computation, as they ensure the continuity of an elastic body across branch cuts. Also, the two solutions presented in this paper are compared in-depth. Based upon their pros and cons, a combined use of these two solutions is suggested for application in boundary element method.

Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry

We compute the first twenty moments of three convergent quartic bi-tracial 2-matrix ensembles in the large N limit. These ensembles are toy models for Euclidean quantum gravity originally proposed by John Barrett and collaborators. A perturbative solution is found for the first twenty moments using the Schwinger-Dyson equations and properties of certain bi-colored unstable maps associated to the model. We then apply a result of Guionnet et al. to show that the perturbative and convergent solution coincide for a small neighbourhood of the coupling constants. For each model we compute an explicit expression for the free energy, critical points, and critical exponents in the large N limit. In particular, the string susceptibility is found to be γ = 1/2, hinting that the associated universality class of the model is the continuous random tree.