Energy Scaling Law Research Articles

Energy barriers for boundary nucleation in a two-well model without gauge invariances

We study energy scaling laws for a simplified, singularly perturbed, double-well nucleation problem confined in a half-space, in the absence of gauge invariance and for an inclusion of fixed volume. Motivated by models for boundary nucleation of a single-phase martensite inside a parental phase of austenite, our main focus in this nonlocal isoperimetric problem is how the relationship between the rank-1 direction and the orientation of the half-space influences the energy scaling with respect to the fixed volume of the inclusion. Up to prefactors depending on this relative orientation, the scaling laws coincide with the corresponding ones for bulk nucleation (Knüpfer in Proc R Soc A Math Phys Eng Sci 467(2127): 695-717, 2011) for all rank-1 directions, but the ones normal to the confining hyperplane, where the scaling is as in a three-gradient problem in full space, resulting in a lower energy barrier (Rüland et al. J Nonlinear Sci 33(2): 25 2023).

Mechanics of structural size effect in impact brittle fracture of steels and consequent scaling laws for fracture

A large structural size effect on brittle fracture energy was observed in studies of impact fracture of steels by Stanton, Batson and Docherty, nearly 100 years ago. The size effect was seen in macroscopically brittle fractures of geometrically similar specimens tested over a large size range. These historical findings have great significance in designing large-scale steel structures such as ships and bridges. However, there has been no physical principle or theory yet to explain the size effects observed by Stanton, Batson and Docherty. In the present work, a new approach based on net-section mechanics is used to explain the size effect. It is shown that the change in net-section strain energy predicts well the size effect on fracture energy , under the constant condition of cleavage crack initiation stress at the notch root. The spectacular decrease in specific fracture energy with increased specimen size is explained by noting the three-dimensional nature of stored strain energy and the two-dimensional nature of fracture energy dissipated in the crack layer. By scaling both energies to a reference cubic specimen volume and equating, an expression that describes the inverse size dependence of specific fracture energy has been formulated. The mechanics of the size effect is illustrated by assessing the theory with extensive experimental data. Further, scaling laws for fracture energy have been developed. This work honours the pioneering studies of Stanton, Batson and Docherty, which have been instrumental for the development of the present theory and analysis.

A Transistor Operations Model for Deep Learning Energy Consumption Scaling Law

Deep Neural Networks (DNN) has transformed the automation of a wide range of industries and finds increasing ubiquity in society. The high complexity of DNN models and its widespread adoption has led to global energy consumption doubling every 3-4 months. Current energy consumption measures largely monitor system wide consumption or make linear assumptions of DNN models. The former approach captures other unrelated energy consumption anomalies, whilst the latter does not accurately reflect nonlinear computations. In this paper, we are the first to develop a bottom-up Transistor Operations (TOs) approach to expose the role of non-linear activation functions and neural network structure. As there will be inevitable energy measurement errors at the core level, we statistically model the energy scaling laws as opposed to absolute consumption values. We offer models for both feedforward DNNs and convolution neural networks (CNNs) on a variety of data sets and hardware configurations - achieving a 93.6% - 99.5% precision. This outperforms existing FLOPs-based methods and our TOs method can be further extended to other DNN models.

The role of viscoplastic drop shape in impact

The impact of fluid drops on solid substrates is a cardinal fluid dynamics phenomenon intrinsically related to many fields. Although these impacting objects are very often non-spherical and non-Newtonian, previous studies have mainly focused on spherical Newtonian drops. As a result, both shape and rheological effects on the drop-spreading dynamics remain largely unexplored. In the present work we use a mixed approach combining experiments with multiphase three-dimensional numerical simulations to extend the work reported by Luu & Forterre (J. Fluid Mech., vol. 632, 2009, pp. 301–327) by highlighting the fundamental role of shape in the normal impact of viscoplastic drops. Such complex fluids are highly common in various industrial domains and ideally behave either like a rigid body or a shear-rate-dependent liquid, according to the stress solicitation. Spherical, prolate, cylindrical and prismatic drops are considered. The results show that, under negligible capillary effects, the impacting kinetic energy of the drop is dissipated through viscoplastic effects during the spreading process, giving rise to three flow regimes: (i) inertio-viscous, (ii) inertio-plastic, and (iii) mixed inertio-visco-plastic. These regimes are deeply affected by the drop initial aspect ratio, which in turn reveals the possibility of using drop shape to control spreading. The physical mechanisms driving the considered phenomenon are underlined by energy budget analyses and scaling laws. The results are summarised in a two-dimensional diagram linking the drop maximum spreading, minimum height and final shape with different spreading regimes through a single dimensionless parameter, here called the impact number.

Energy scaling laws for microstructures: from helimagnets to martensites

We consider scalar-valued variational models for pattern formation in helimagnetic compounds and in shape memory alloys. Precisely, we consider a non-convex multi-well bulk energy term on the unit square, which favors four gradients (±α,±β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\pm }\\,\\alpha ,{\\pm }\\,\\beta )$$\\end{document}, regularized by a singular perturbation in terms of the total variation of the second derivative. We derive scaling laws for the minimal energy in the case of incompatible affine boundary conditions in terms of the singular perturbation parameter as well as the ratio α/β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha /\\beta $$\\end{document} and the incompatibility of the boundary condition. We discuss how well-studied models for martensitic microstructures in shape-memory alloys arise as a limiting case, and relations between the different models in terms of scaling laws. In particular, we show that scaling regimes arise in which an interpolation between the rather different branching-type constructions in the spirit of Kohn and Müller (Commun Pure Appl Math 47(4):405–435, 1994) and Ginster and Zwicknagl (J Nonlinear Sci 33:20, 2023), respectively, occurs. A particular technical difficulty in the lower bounds arises from the fact that the energy scalings involve various logarithmic terms that we capture in matching upper and lower scaling bounds.

Quartic soliton dynamics and pattern formations in passive driven fiber resonators with different quartic dispersion regimes

Solitons are particle-like localized structures which can sustain characteristic shape in various nonlinear systems. Very recently, the novel concept of quartic soliton with remarkable energy scaling law arising from the interaction of quartic dispersion and Kerr nonlinearity, provides innovations in nonlinear optics and applications. Here we theoretically investigate the quartic soliton dynamics in passive driven fiber resonators from negative to positive quartic dispersion driving. In the pure negative quartic dispersion regimes, the typical energy-duration scaling relationship is sufficiently demonstrated. Worth to note that, we further investigate the dissipative quartic soliton formation in the positive quartic dispersion regimes, achieving remarkable pulse energy increase. Interestingly, echoing our previous work on the influence of cavity group-velocity dispersion on dissipative Faraday instability formation, pattern formation dynamics in the quartic dispersion driving fiber resonators is also explored. Our investigations might benefit the fundamental understanding of the quartic soliton dynamics.

Energy Scaling Law for a Singularly Perturbed Four-Gradient Problem in Helimagnetism

We study pattern formation in magnetic compounds near the helimagnetic/ferromagnetic transition point in case of Dirichlet boundary conditions on the spin field. The energy functional is a continuum approximation of a J_1-J_3 model and was recently derived in Cicalese et al. (SIAM J Math Anal 51: 4848–4893, 2019). It contains two parameters, one measuring the incompatibility of the boundary conditions and the other measuring the cost of changes between different chiralities. We prove the scaling law of the minimal energy in terms of these two parameters. The constructions from the upper bound indicate that in some regimes branching-type patterns form close to the boundary of the sample.

Control of plasmonic field enhancement by mode-mixing

We demonstrate experimentally that nanoscale control of plasmonic field enhancement becomes available by changing the polarization state of light. This is revealed by photoelectron emission from plasmonic nanorods illuminated with linearly and circularly polarized femtosecond laser pulses. Simulations show that the tunability of the field enhancement originates from the mode-mixing property of circularly polarized illumination, meaning simultaneous excitation of multiple plasmon modes of the nanostructures. Performing trajectory calculations of the photoemitted electrons, we prove that the kinetic energy scaling law remains the same irrespective to the polarization state.

Optimal fine-scale structures in compliance minimization for a uniaxial load in three space dimensions

We consider the shape and topology optimization problem to design a structure that minimizes a weighted sum of material consumption and (linearly) elastic compliance under a fixed given boundary load. As is well-known, this problem is in general not well-posed since its solution typically requires the use of infinitesimally fine microstructure. Therefore we examine the effect of singularly perturbing the problem by adding the structure perimeter to the cost. For a uniaxial and a shear load in two space dimensions, corresponding energy scaling laws were already derived in the literature. This work now derives the scaling law for the case of a uniaxial load in three space dimensions, which can be considered the simplest three-dimensional setting. In essence, it is expected (and confirmed in this article) that for a uniaxial load the compliance behaves almost like the dissipation in a scalar flux problem so that lower bounds from pattern analysis in superconductors can directly be applied. The upper bounds though require nontrivial modifications of the constructions known from superconductors. Those become necessary since in elasticity one has the additional constraint of torque balance.

Worst-case control energy of single-input network systems

The controllability of large-scale network systems has been extensively investigated in the past few years. In spite of the recent advances in this field, there are still a number of unsolved problems which are of critical importance to fully understand the practical limitations arising in the control of large-scale networks. These include the derivation of informative bounds and scaling laws for the control energy of networks controlled by a limited number of nodes. In this paper, we aim to fill this gap by establishing new numerically reliable bounds and asymptotic estimates on the worst-case control energy of continuous-time linear network systems controlled by a single node. Our results rely on a convenient reformulation of the controllability Gramian of a single-input linear system in terms of a Cauchy matrix. We illustrate and validate our theoretical findings through several examples, ranging from structured networks to random ones.

Scaling laws for laser-driven ion acceleration from nanometer-scale ultrathin foils.

Laser-driven ion acceleration has attracted global interest for its potential towards the development of a new generation of compact, low-cost accelerators. Remarkable advances have been seen in recent years with a substantial proton energy increase in experiments, when nanometer-scale ultrathin foil targets and high-contrast intense lasers are applied. However, the exact acceleration dynamics and particularly the ion energy scaling laws in this novel regime are complex and still unclear. Here, we derive a scaling law for the attainable maximum ion energy from such laser-irradiated nanometer-scale foils based on analytical theory and multidimensional particle-in-cell simulations, and further show that this scaling law can be used to accurately describe experimental data over a large range of laser and target parameters on different facilities. This provides crucial references for parameter design and experimentation of the future laser devices towards various potential applications.

Electrostatics, conformation, and rheology of unentangled semidilute polyelectrolyte solutions

Polyelectrolyte (PE) solutions, which are charged polymers in polar solvents, are ubiquitous and essential to life. Due to the electrostatic interactions among the charged monomers and mobile ions, the dependence of the rheological properties on the polymer concentration of PE solutions differs significantly from that of solutions of uncharged macromolecules. In addition, salt in PE solutions, whether added intentionally or intrinsically present, can affect the properties of the solutions. Here, we analyze the ion distribution near a monomer using the nonlinear Poisson–Boltzmann equation for scenarios with nonoverlapping and overlapping electric double layers. Consequently, by incorporating the electrostatic interactions into the blob model and Zimm–Rouse dynamic model, we obtain different scaling laws for the electrostatic energy per monomer, correlation length, end-to-end distance, relaxation time and viscosity η of semidilute, unentangled PE solutions in consecutive regimes of polymer concentration np, and salt concentration ns, which are summarized in tables. With our theory, we anticipate that the empirical Fuoss law η∝np1/2 is expected for solutions prepared with salt-contaminated PE samples, while η∝np0.68 might be found for those with purer PE samples. A new critical charge fraction φc is defined, where we predict that the peak, which is present in the measurements of the reduced viscosity η/np as a function of np, is only expected for weakly charged PEs φ<φc prepared with pure PE samples. Comparisons with the experimental data as well as the classical scaling theories are provided, and the range of applicability of the theory is discussed.

Dimension reduction for thin films prestrained by shallow curvature

We are concerned with the dimension reduction analysis for thin three-dimensional elastic films, prestrained via Riemannian metrics with weak curvatures. For the prestrain inducing the incompatible version of the Föppl–von Kármán equations, we find the Γ -limits of the rescaled energies, identify the optimal energy scaling laws, and display the equivalent conditions for optimality in terms of both the prestrain components and the curvatures of the related Riemannian metrics. When the stretching-inducing prestrain carries no in-plane modes, we discover similarities with the previously described shallow shell models. In higher prestrain regimes, we prove new energy upper bounds by constructing deformations as the Kirchhoff–Love extensions of the highly perturbative, Hölder-regular solutions to the Monge–Ampere equation obtained by means of convex integration.

Raman pure quartic solitons in Kerr microresonators.

Pure quartic solitons (PQSs) are a unique class of solitons advantageous for developing promising applications due to their broad and flat-top spectrum, as well as the distinctive energy scaling law. Here we investigate the characteristics and dynamics of the PQS in the presence of the Raman effect based on Kerr microresonators. Stimulated Raman scattering leads to reduced pulse peak power, self-frequency shifts, and distortion of the symmetry pulse shape of the PQS. Besides, dynamical evolution of the Raman PQS, especially the breathing state, is investigated. We discover the appearance of an intermediate stable state between the existence region of breathers and chaos in the parameter space, which has not been discovered yet in other soliton regimes. A stability analysis is performed to investigate the spatial dynamics in the context of the Raman PQS.

Global Energetics of Solar Flares. XII. Energy Scaling Laws

Abstract In this study we test 18 versions of five fundamental energy scaling laws that operate in large solar flares. We express scaling laws in terms of the magnetic potential field energy E p , the mean potential field strength B p , the free energy E free, the dissipated magnetic flare energy E diss, the magnetic length scale L, the thermal length scale L th, the mean helically twisted flux tube radius R, the sunspot radius r, the emission-measure-weighted flare temperature T e , the electron density n e , and the total emission measure EM, measured from a data set of 173 GOES M- and X-class flare events. The five categories of physical scaling laws include (i) a scaling law of the potential field energy, (ii) a scaling law for helical twisting, (iii) a scaling law for Petschek-type magnetic reconnection, (iv) the Rosner–Tucker–Vaiana scaling law, and (v) the Shibata–Yokoyama scaling law. We test the self-consistency of these theoretical scaling laws with observed parameters by requiring two criteria: a cross-correlation coefficient of CCC > 0.5 between the theoretically predicted scaling laws and observed values, and a linear regression fit with a slope of α ≈ 1 within one standard deviation σ. These two criteria enable us (i) to corroborate some existing (or modified) scaling laws, (ii) to reject other scaling laws that are not consistent with the observations, (iii) to probe the dimensionality of flare geometries, and (iv) to predict various energy parameters based on tested scaling laws.

Branching of twins in shape memory alloys revisited

We study the branching of twins appearing in shape memory alloys at the interface between austenite and martensite. In the framework of three-dimensional non-linear elasticity theory, we propose an explicit, low-energy construction of the branched microstructure, generally applicable to any shape memory material without restrictions on the symmetry class of martensite or on the geometric parameters of the interface. We show that the suggested construction follows the expected energy scaling law, i.e., that (for the surface energy of the twins being sufficiently small) the branching leads to energy reduction. Furthermore, the construction can be modified to capture different features of experimentally observed microstructures without violating this scaling law. By using a numerical procedure, we demonstrate that the proposed construction is able to predict realistically the twin width in a Cu-Al-Ni single crystal and to estimate an upper bound to the number of the branching generations.

All-or-none folding of a flexible polymer chain in cylindrical nanoconfinement.

Geometric confinement of a polymer chain results in a loss of conformational entropy. For a chain that can fold into a compact native state via a first-order-like transition, as is the case for many small proteins, confinement typically provides an entropic stabilization of the folded state, thereby shifting the location of the transition. This allows for the possibility of confinement (entropy) driven folding. Here, we investigate such confinement effects for a flexible square-well-sphere N-mer chain (monomer diameter σ) confined within a long cylindrical pore (diameter D) or a closed cylindrical box (height H = D). We carry out Wang-Landau simulations to construct the density of states, which provides access to the complete thermodynamics of the system. For a wide pore, an entropic stabilization of the folded state is observed. However, as the pore diameter approaches the size of the folded chain (D ∼ N1/3σ), we find a destabilization effect. For pore diameters smaller than the native ground-state, the chain folds into a different, higher energy, ground state ensemble and the T vs D phase diagram displays non-monotonic behavior as the system is forced into different ground states for different ranges of D. In this regime, isothermal reduction of the confinement dimension can induce folding, unfolding, or crystallite restructuring. For the cylindrical box, we find a monotonic stabilization effect with decreasing D. Scaling laws for the confinement free energy in the athermal limit are also investigated.

Universal energy scaling law for optimally excited nonlinear oscillators.

We compute the optimal temporal profile for an external driving force F(t) that can maximize the energy absorption of any driven nonlinear oscillator. The technique is based on constraining the maximum amplitude of the force field such that optimal control theory can provide quasianalytical solutions. We illustrate this computational technique for the undamped Duffing oscillator as well as for a driven quantum mechanical two-level system. We find that under optimal force conditions the asymptotic time-dependence of the maximum amplitude growth is given by a power law X(t)∼t^{2/α}, where the (possibly noninteger) exponent is determined by the highest degree of the oscillator's nonlinearity α. As a universal result, this predicts that the maximal energy absorption of any nonlinear oscillator grows (under an optimized force field) quadratically in time. We also find for the two-level system that-even under optimized excitation conditions-the maximally achievable inversion does not monotonically increase with the force amplitude. It is characterized by an interesting sequence of n-cycle thresholds as well as a self-termination of the growth.

Singular perturbation of an elastic energy with a singular weight

We study the singular perturbation of an elastic energy with a singular weight. The minimization of this energy results in a multi-scale pattern formation. We derive an energy scaling law in terms of the perturbation parameter and prove that, although one cannot expect periodicity of minimizers, the energy of a minimizer is uniformly distributed across the sample. Finally, following the approach developed by Alberti and Müller (2001) we prove that a sequence of minimizers of the perturbed energies converges to a Young measure supported on piecewise-linear periodic functions of slope ±1 whose period depends on the location in the domain and the weights in the energy.

Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys

We consider a singularly-perturbed two-well problem in the context of planar geometrically linear elasticity to model a rectangular martensitic nucleus in an austenitic matrix. We derive the scaling regimes for the minimal energy in terms of the problem parameters, which represent the shape of the nucleus, the quotient of the elastic moduli of the two phases, the surface energy constant, and the volume fraction of the two martensitic variants. We identify several different scaling regimes, which are distinguished either by the exponents in the parameters, or by logarithmic corrections, for which we have matching upper and lower bounds.