Leading Order Behavior Research Articles

Wrinkling of a thin circular sheet bonded to a spherical substrate

We consider a disc-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) If the bonding is stiff (but not too stiff), the geometry of the sphere makes the sheet wrinkle to avoid azimuthal compression. The total energy of this system is the elastic energy of the sheet plus a (Winkler-type) substrate energy. Treating the thickness of the sheet h as a small parameter, we determine the leading-order behaviour of the energy as h tends to zero, and we give (almost matching) upper and lower bounds for the next-order correction. Our analysis of the leading-order behaviour determines the macroscopic deformation of the sheet; in particular, it determines the extent of the wrinkled region, and predicts the (non-trivial) radial strain of the sheet. The leading-order behaviour also provides insight about the length scale of the wrinkling, showing that it must be approximately independent of the distance r from the centre of the sheet (so that the number of wrinkles must increase with r). Our results on the next-order correction provide insight about how the wrinkling pattern should vary with r Roughly speaking, they suggest that the length scale of wrinkling should not be exactly constant-rather, it should vary slightly, so that the number of wrinkles at radius r can be approximately piecewise constant in its dependence on r, taking values that are integer multiples of h-a with [Formula: see text]This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications'.

Edge statistics for a class of repulsive particle systems

We study a class of interacting particle systems on $$\mathbb {R}$$ which was recently investigated by Gotze and Venker (Ann Probab 42(6):2207–2242, 2014. doi: 10.1214/13-AOP844 ). These ensembles generalize eigenvalue ensembles of Hermitian random matrices by allowing different interactions between particles. Although these ensembles are not known to be determinantal one can use the stochastic linearization method of Gotze and Venker (Ann Probab 42(6):2207–2242, 2014. doi: 10.1214/13-AOP844 ) to represent them as averages of determinantal ones. Our results describe the transition between universal behavior in the regime of the Tracy–Widom law and non-universal behavior for large deviations of the rightmost particle. Moreover, a detailed analysis of the transition that occurs in the regime of moderate deviations, is provided. We also compare our results with the corresponding ones obtained recently for determinantal ensembles (Eichelsbacher et al. in Symmetry Integr Geom Methods Appl 12:18, 2016. doi: 10.3842/SIGMA.2016.093 ; Schuler in Moderate, large and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles. Ph.D. thesis, University of Bayreuth, 2015). In particular, we discuss how the averaging effects the leading order behavior in the regime of large deviations. In the analysis of the averaging procedure we use detailed asymptotic information on the behavior of Christoffel–Darboux kernels that is uniform for perturbative families of weights. Such results have been provided by K. Schubert, K. Schuler and the authors in Kriecherbauer et al. (Markov Process Relat Fields 21(3, part 2):639–694, 2015).

A note on 2D focusing many-boson systems

We consider a 2D quantum system of $N$ bosons in a trapping potential $|x|^s$, interacting via a pair potential of the form $N^{2\beta-1} w(N^\beta x)$. We show that for all $0 < \beta < (s+1)/(s+2)$, the leading order behavior of ground states of the many-body system is described in the large $N$ limit by the corresponding cubic nonlinear Schrodinger energy functional. Our result covers the focusing case ($w \leq 0$) where even the stability of the many-body system is not obvious. This answers an open question mentioned by X.~Chen and J.~Holmer for harmonic traps ($s=2$). Together with the BBGKY hierarchy approach used by these authors, our result implies the convergence of the many-body quantum dynamics to the focusing NLS equation with harmonic trap for all $0 < \beta < 3/4$.

Bogoliubov correction to the mean-field dynamics of interacting bosons

We consider the dynamics of a large quantum system of $N$ identical bosons in 3D interacting via a two-body potential of the form $N^{3\beta-1} w(N^\beta(x-y))$. For fixed $0\leq \beta <1/3$ and large $N$, we obtain a norm approximation to the many-body evolution in the $N$-particle Hilbert space. The leading order behaviour of the dynamics is determined by Hartree theory while the second order is given by Bogoliubov theory.

Gibbs Phenomenon for Dispersive PDEs on the Line

We investigate the Cauchy problem for linear, constant-coefficient evolution PDEs on the real line with discontinuous initial conditions (ICs) in the small-time limit. The small-time behavior of the solution near discontinuities is expressed in terms of universal, computable special functions. We show that the leading-order behavior of the solution of dispersive PDEs near a discontinuity of the ICs is characterized by Gibbs-type oscillations and gives exactly the Wilbraham--Gibbs constants.

On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials $\pi_n(z)$ with the quartic exponential weight $\exp[-N(\frac 12 z^2+\frac 14 tz^4)]$, where $t\in {\mathbb C}$ and $N\in{\mathbb N}$, $N\to\infty$. Our goal is to describe these asymptotic behaviors globally for $t\in {\mathbb C}$ in different regions. We also describe the "breaking" curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic "phase portrait" of the recurrence coefficients of $\pi_n(z)$, which was studied numerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the $g$-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex $t$-plane to all noncritical values of $t$. We also provide explicit solutions for recurrence coefficients in terms of the Riemann theta functions. The leading order behaviour of the recurrence coefficients in the full scaling neighbourhoods the critical points (double and triple scaling limits) was obtained in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321] and [Asymptotics of complex orthogonal polynomials on the cross with varying quartic weight: critical point behaviour and the second Painlev\'e transcendents, in preparation].

Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail

We prove precise deviations results in the sense of Cram\'er and Petrov for the upper tail of the distribution of the maximal value for a special class of determinantal point processes that play an important role in random matrix theory. Here we cover all three regimes of moderate, large and superlarge deviations for which we determine the leading order description of the tail probabilities. As a corollary of our results we identify the region within the regime of moderate deviations for which the limiting Tracy-Widom law still predicts the correct leading order behavior. Our proofs use that the determinantal point process is given by the Christoffel-Darboux kernel for an associated family of orthogonal polynomials. The necessary asymptotic information on this kernel has mostly been obtained in [Kriecherbauer T., Schubert K., Sch\"uler K., Venker M., Markov Process. Related Fields 21 (2015), 639-694]. In the superlarge regime these results of do not suffice and we put stronger assumptions on the point processes. The results of the present paper and the relevant parts of [Kriecherbauer T., Schubert K., Sch\"uler K., Venker M., Markov Process. Related Fields 21 (2015), 639-694] have been proved in the dissertation [Sch\"uler K., Ph.D. Thesis, Universit\"at Bayreuth, 2015].

Equilibration and GGE in interacting-to-free quantum quenches in dimensions

Ground states ofinteracting QFTs are non-Gaussian states, i.e. their connected n-point correlation functions do not vanish for , in contrast to the free QFT case. We show that, when the ground state of an interacting QFT evolves under a free massive QFT for a long time (a scenario that can be realised by a quantum quench), the connected correlation functions decay and all local physical observables equilibrate to values that are given by a Gaussian density matrix that retains memory only of the two-point initial correlation function. The argument hinges upon the fundamental physical principle of cluster decomposition, which is valid for the ground state of a general QFT. An analogous result was already known to be valid in the case of d = 1 spatial dimensions, where it is a special case of the so-called generalised Gibbs ensemble (GGE) hypothesis, and we now generalise it to higher dimensions. Moreover, in the case of massless free evolution, despite the fact that the evolution may lead not to equilibration but instead to unbounded increase of correlations with time, the GGE gives correctly the leading-order asymptotic behaviour of correlation functions in the thermodynamic and large time limit. The demonstration is performed in the context of a bosonic relativistic QFT, but the arguments apply more generally.

Slender-ribbon theory

Ribbons are long narrow strips possessing three distinct material length scales (thickness, width, and length) which allow them to produce unique shapes unobtainable by wires or filaments. For example, when a ribbon has half a twist and is bent into a circle it produces a Möbius strip. Significant effort has gone into determining the structural shapes of ribbons but less is know about their behavior in viscous fluids. In this paper, we determine, asymptotically, the leading-order hydrodynamic behavior of a slender ribbon in Stokes flows. The derivation, reminiscent of slender-body theory for filaments, assumes that the length of the ribbon is much larger than its width, which itself is much larger than its thickness. The final result is an integral equation for the force density on a mathematical ruled surface, termed as the ribbon plane, located inside the ribbon. A numerical implementation of our derivation shows good agreement with the known hydrodynamics of long flat ellipsoids and successfully captures the swimming behavior of artificial microscopic swimmers recently explored experimentally. We also study the asymptotic behavior of a ribbon bent into a helix, that of a twisted ellipsoid, and we investigate how accurately the hydrodynamics of a ribbon can be effectively captured by that of a slender filament. Our asymptotic results provide the fundamental framework necessary to predict the behavior of slender ribbons at low Reynolds numbers in a variety of biological and engineering problems.

Asymptotic analysis of cold sandwich rolling

An analytical model for the cold rolling of sandwich sheets is proposed, assuming a rigid-plastic flow rule and Coulomb friction. Asymptotic analysis is used to solve the equations based on the assumptions of small aspect ratio and friction coefficient. This model relaxes the assumption, crucial to slab methods, that the stresses are uniformly distributed through the thickness of each material layer. Thus, our model is able to predict the through-thickness velocity and stress distributions. The leading-order behaviour is shown to be consistent with the slab method, and the predictions are compared with finite element simulations. Computation times are orders of magnitude smaller than finite element calculations.

Axisymmetric flow of ideal fluid moving in a narrow domain: A study of the axisymmetric hydrostatic Euler equations

In this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain. After providing a formal derivation of the model, we will prove the well-posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition. Finally, a blowup result regarding this model will be discussed as well.

The singular values of the GUE (less is more)

Some properties that nominally involve the eigenvalues of the Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable given the well-known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large [Formula: see text] limits of the GUE are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quarter-circle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter [Formula: see text]. Similarly, we write the absolute value of the determinant of the [Formula: see text] GUE as a product [Formula: see text] independent random variables to gain new insight into its asymptotic log-normality. The decomposition also provides a description of the distribution of the smallest singular value of the GUE, which in turn permits the study of the leading order behavior of the condition number of GUE matrices. The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.

On the black hole limit of rotating discs of charged dust

Investigating the rigidly rotating disc of dust with constant specific charge, we find that it leads to an extreme Kerr–Newman black hole in the ultra-relativistic limit. A necessary and sufficient condition for a black hole limit is, that the electric potential in the co-rotating frame is constant on the disc. In that case certain other relations follow. These relations are reviewed with a highly accurate post-Newtonian expansion. Remarkably it is possible to survey the leading order behaviour close to the black hole limit with the post-Newtonian expansion. We find that the disc solution close to that limit can be approximated very well by a ‘hyperextreme’ Kerr–Newman solution with the same gravitational mass, angular momentum and charge.

Comment on “Painlevé integrability of two sets of nonlinear evolution equations with nonlinear dispersions” [Phys. Lett. A 262 (1999) 344

Abstract In the Letter [1] , Lou and Wu studied the Rosenau–Hyman equation and presented various integrable cases. However, some of their results are not correct, due to missing leading order behaviors in the Painleve test, or to the vanishing of a denominator in the application of the hodograph transformations. Moreover, an integrable case is missing in [1] . In this Comment, we summarize the results of [1] and point out the mistakes and incompletions. For a better understanding, we briefly outline the Painelve analysis and the application of the hodograph transformations.

Explicit solutions in evolutionary genetics and applications

We show that the solution to a nonlocal reaction–diffusion equation, present in evolutionary genetics, can be related explicitly to the solution of the heat equation with the same initial data. As a consequence, we show different possible scenario for the solution: it can be either well-defined for all time, or become extinct in finite time, or even be defined for no positive time. In the former case, we give the leading-order asymptotic behavior of the solution for large time, which is universal.

The Transition to a Point Constraint in a Mixed Biharmonic Eigenvalue Problem

The mixed-order eigenvalue problem $-\delta \Delta^2 u + \Delta u + \lambda u = 0$ with $\delta>0$, modeling small amplitude vibrations of a thin plate, is analyzed in a bounded two-dimensional domain $\Omega$ that contains a single small hole of radius $\varepsilon$ centered at some $x_0\in \Omega$. Clamped conditions are imposed on the boundary of $\Omega$ and on the boundary of the small hole. In the limit $\varepsilon\to 0$, and for $\delta={\mathcal O}(1)$, the limiting problem for $u$ must satisfy the additional point constraint $u(x_0)=0$. To determine how the eigenvalues of the Laplacian in a domain with a small hole are perturbed by adding the small fourth-order term $-\delta \Delta^2 u$, together with an additional boundary condition on $\partial\Omega$ and on the hole boundary, the asymptotic behavior of the eigenvalues of the mixed-order eigenvalue problem are studied in the dual limit $\varepsilon\to 0$ and $\delta\to 0$. Leading-order behaviors of eigenvalues are determined for three ranges ...

Damage dynamics, rate laws, and failure statistics via Hamilton’s principle

We present a new model for studying the coupled-field nonlinear dynamics of systems with evolving distributed damage, focusing on the case of high-cycle fatigue. A 1D continuum model is developed using Hamilton’s principle together with Griffith energy arguments. It captures the interaction between a damage field variable, representing the density of microcracks, and macroscopic vibrational displacements. We use the perturbation method of averaging to show that the nonautonomous coupled-field model yields an autonomous Paris-Erdogan rate law as a limiting case. Finite element simulations reveal a brittle limit for which the life cycle dynamics is dominated by leading-order power-law behavior. Space-time failure statistics are explored using large ensembles of simulations starting from random initial conditions. We display typical probability distributions for failure locations and times, as well as “\(F{-}N\)” curves relating load to the number of cycles to failure. A universal time scale is identified for which the failure time statistics are independent of the applied load and the damage rate constant. We show that the evolution of the macro-displacement frequency response function, as well as the statistical variability of failure times, can vary substantially with changes in system parameters, both of which have significant implications for the design of failure diagnostic and prognostic systems.

Holographic entanglement entropy of nonlocal field theories

We study holographic entanglement entropy of non-local field theories both at extremality and finite temperature. The gravity duals, constructed in arXiv:1208.3469 [hep-th], are characterized by a parameter $w$. Both the zero temperature backgrounds and the finite temperature counterparts are exact solutions of Einstein-Maxwell-dilaton theory. For the extremal case we consider the examples with the entangling regions being a strip and a sphere. We find that the leading order behavior of the entanglement entropy always exhibits a volume law when the size of the entangling region is sufficiently small. We also clarify the condition under which the next-to-leading order result is universal. For the finite temperature case we obtain the analytic expressions both in the high temperature limit and in the low temperature limit. In the former case the leading order result approaches the thermal entropy, while the finite contribution to the entanglement entropy at extremality can be extracted by taking the zero temperature limit in the latter case. Moreover, we observe some peculiar properties of the holographic entanglement entropy when $w=1$.

Towards a double scaling limit for tensor models: probing sub-dominant orders

The definition of a double scaling limit represents an important goal in the development of tensor models. We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the colored tensor models. We show that the radius of convergence of the nlo series coincides with that of the leading order melonic sector. Meanwhile, the value of the susceptibility exponent, γnlo = 3/2, signals a departure from the leading order behavior. Both pieces of information provide clues for a non-trivial double scaling limit, for which we put forward some precise conjecture.

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

We study the leading order behaviour of positive solutions of the equation -\Delta u +\epsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\mathbb R^N, where N\ge 3 , q&gt;p&gt;2 and when \epsilon &gt;0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p , q and N . The behavior of solutions depends sensitively on whether p is less, equal or bigger than the critical Sobolev exponent 2^\ast=\frac{2N}{N-2} . For p&lt;2^\ast the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p&gt;2^\ast the solution asymptotically coincides with the solution of the equation with \epsilon =0 . In the most delicate case p=2^\ast the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden–Fowler equation, whose choice depends on \epsilon in a nontrivial way.