Approximate Analytical Theory Research Articles

Motility-Induced Pinning in Flocking System with Discrete Symmetry.

We report a motility-induced pinning transition in the active Ising model for a self-propelled particle system with discrete symmetry. This model was known to exhibit a liquid-gas type flocking phase transition, but a recent study reveals that the polar order is metastable due to droplet excitation. Using extensive MonteCarlo simulations, we demonstrate that, for an intermediate alignment interaction strength, the steady state is characterized by traveling local domains, which renders the polar order short-ranged in both space and time. We further demonstrate that interfaces between colliding domains become pinned as the alignment interaction strength increases. A resonating back-and-forth motion of individual self-propelled particles across interfaces is identified as a mechanism for the pinning. We present a numerical phase diagram for the motility-induced pinning transition, and an approximate analytic theory for the growth and shrink dynamics of pinned interfaces. Our results show that pinned interfaces grow to a macroscopic size preventing the polar order in the regime where the particle diffusion rate is sufficiently smaller than the self-propulsion rate. The growth behavior in the opposite regime and its implications on the polar order remain unresolved and require further investigation.

Active motion of tangentially driven polymers in periodic array of obstacles

One key question about transport of active polymers within crowded environments is how spatial order of obstacles influences their conformation and dynamics when compared to disordered media. To this end, we computationally investigate the active transport of tangentially driven polymers with varying degrees of flexibility and activity in two-dimensional square lattices of obstacles. Tight periodic confinement induces notable conformational changes and distinct modes of transport for flexible and stiff active filaments. It leads to caging of low activity flexible polymers inside the inter-obstacle pores while promoting more elongated conformations and enhanced diffusion for stiff polymers at low to moderate activity levels. The migration of flexible active polymers occurs via hopping events, where they unfold to move from one cage to another, similar to their transport in disordered media. However, in ordered media, polymers are more compact and their long-time dynamics is significantly slower. In contrast, stiff chains travel mainly in straight paths within periodic inter-obstacle channels while occasionally changing their direction of motion. This mode of transport is unique to periodic environment and leads to more extended conformation and substantially enhanced long-time dynamics of stiff filaments with low to moderate activity levels compared to disordered media. At high active forces, polymers overcome confinement effects and move through inter-obstacle pores just as swiftly as in open spaces, regardless of the spatial arrangement of obstacles. We explain the center of mass dynamics of semiflexible polymers in terms of active force and obstacle packing fraction by developing an approximate analytical theory.

Mechanisms of like-charge attraction in many-body systems

The physics of like-charge attraction in a collinear binary system of N particles has been investigated via a polarizable-ion model. It was found that like-charge attraction between an array of (N−1) type-A particles on the left and a single type-B particle on the right is conditional upon a sufficient degree of dissimilarities between the two types of particles. By assuming that (1) the particles of one type are strongly polarizing (but weakly polarizable) such that the Coulomb fields they produce are dominant all over other electric fields, and that (2) the particles of the other type are strongly polarizable (but weakly polarizing) such that their Coulomb-field-induced dipoles are the only polarization effects worth considering, a good agreement between our approximate analytic theory and exact numerical solutions has been obtained. Our analytic results show that the attractive components of the electrostatic force between the two types of particles could be thought of as arising from imaginary charges induced in the polarized particle(s) by the polarizing ones, resulting in a screening of the real charge(s) in the polarized particle(s). Like-charge attraction occurs if the field contributions from the imaginary charges collectively outweigh the field contribution(s) from the real charge(s).

Towards the theory of how a constitutional supercooling layer appears ahead of the planar crystallization front

This study, the effect of constitutional supercooling appearing ahead of the crystallization front and leading to the mushy layer origination is considered. An approximate analytical theory determining the time of mushy layer initiation is constructed. Theoretical predictions are in good agreement with numerical simulations carried out in previous studies.

Mechanical behaviour of rubber blocks

This study investigates the behaviour of rubber blocks bonded between two plates under combined compression and shear loading, using experimental and numerical analyses, and also approximate analytical theories. First, experimental data from a series of compression and shear tests of rubber blocks with different aspect ratios are presented. Next, numerical simulations are carried out with three-dimensional finite element (FE) models, allowing insight to be gained into the stress and strain fields within the blocks. Existing analytical theories for blocks under compression and combined compressive and shear loading are then reviewed, and their accuracy is evaluated against test and numerical results. The study shows that those theories accounting for the effect of the axial shortening of the blocks provide a better description of the combined compression and shear behaviour, compared to theories, developed for laminated structural bearings with many thin rubber layers, that ignore this effect. An improved theory is also proposed, which better describes the effects of the bulging of the compressed blocks on their shear and flexural parameters and provides a better fit to experimental and numerical results.

Surface crossing and energy flow in many-dimensional quantum systems

Energy flow in molecules, like the dynamics of other many-dimensional finite systems, involves quantum transport across a dense network of near-resonant states. For molecules in their electronic ground state, the network is ordinarily provided by anharmonic vibrational Fermi resonances. Surface crossing between different electronic states provides another route to chaotic motion and energy redistribution. We show that nonadiabatic coupling between electronic energy surfaces facilitates vibrational energy flow and, conversely, anharmonic vibrational couplings facilitate nonadiabatic electronic state mixing. A generalization of the Logan-Wolynes theory of quantum energy flow in many-dimensional Fermi resonance systems to the two-surface case gives a phase diagram describing the boundary between localized quantum dynamics and global energy flow. We explore these predictions and test them using a model inspired by the problem of electronic excitation energy transfer in the photosynthetic reaction center. Using an explicit numerical solution of the time-dependent Schrödinger equation for this ten-dimensional model, we find quite good agreement with the expectations from the approximate analytical theory.

Equilibrium flight theory-based orbit entry capability evaluation and guidance method of launch vehicles with thrust faults

<p indent="0mm">Considering the nonfatal engine faults (e.g., abnormal thrust) of a launch vehicle in the outer atmosphere flight phase, a method based on equilibrium flight theory for the evaluation of the vehicle’s orbit entry capability and continuous thrust guidance is proposed. First, the mechanism of mission failure caused by abnormal thrust fault is analyzed from the perspective of flight dynamics; on this basis, the simplified model of launch vehicle flight dynamics is established in the LVLH coordinate system, and the approximate analytical theory of equilibrium flight is provided. The conditions of an “equilibrium flight” and a “quasi-equilibrium flight” are further analyzed, and the differences between their guidance laws are compared. Next, the method and process of the online evaluation and decision-making of launch vehicles with an abnormal thrust are listed. Finally, numerical simulation is performed to reveal the relationship between thrust drop and orbit entry capability, as well as verify the correctness and practicability of the analytical theory of an equilibrium flight comprehensively. This simple and effective method integrates online capability evaluation and optimal orbit guidance, which can provide theoretical and technical support for rapid autonomous fault handling of launch vehicles with thrust faults.

Analytic theory for the tangential YORP produced by the asteroid regolith

Context. The tangential YORP effect is a radiation pressure torque produced by asymmetric thermal emission by structures on the asteroid surface. Previous works considered these structures to be boulders of different shapes lying on the surface of the asteroid. Aims. We study the tangential YORP produced by the rough interface of the asteroid's regolith. Methods. We created an approximate analytic theory of heat conduction on a slightly non-flat sinusoidal surface. We analyzed the published data on the small-scale shape of the asteroid (162173) Ryugu and estimated its tangential YORP due to the surface roughness. Results. We derive an analytic formula that expresses the tangential YORP of a sinusoidal surface in terms of its geometric and thermal properties. The tangential YORP is highest at the thermal parameter on the order of unity and for shape irregularities on the order of the thermal wavelength. Application of this equation to Ryugu predicts a tangential YORP that is 5-70 times greater than its normal YORP effect. Conclusions. The contribution of the small-scale regolith roughness to the YORP effect of the asteroid can be comparable to the normal YORP and the tangential YORP produced by boulders. The same theory can describe the roughness of the asteroid boulders, thus adding a new term to the previously considered the tangential YORP created by boulders.

Approximate calculation of the static analysis of a lifted stay cable in super-long span cable-stayed bridges

The sag effect of long stay cables is one of the key factors restricting further increase in the span of cable-stayed bridges. Based on the formerly proposed concept of long stay cables lifted by an auxiliary suspension cable in cross-strait cable-stayed bridges, corresponding static approximate calculations and analytical theory based on catenary and parabolic cable configurations are established. Taking a main span 1400 m cable-stayed bridge as the research object, three typical lifting conditions and the whole process of auxiliary cable lifting are analyzed and discussed. The results show that the sag effect is effectively reduced. The support efficiency is only improved when the cables are lifted above the original cable chord. Reduction of the horizontal component force of the cable is limited. The equivalent elastic modulus and the vertical support stiffness of the lifted cables are significantly increased with increased horizontal projection length and not sensitive to the change of the lifting point position. The scheme of lifting the cable to the chord midpoint is more economical because of the less steel required for the auxiliary suspension cable, but its effect on improving the vertical support efficiency is limited. The support efficiency is better when the cable is lifted to the cable end tangential to the original cable chord, but the lifting force and the cross-sectional area of the auxiliary suspension cable are doubled. The approximate calculation results of the lifted cables are very close to the numerical analysis results, which verifies the applicability of the approximation method proposed in this study. The results of parabolic approximation calculations are approximately equal to that of catenary cable geometry. As the parabolic approximation analysis theory of lifted cables is more convenient in mathematical processing, it is feasible to use parabolic approximation analysis theory as the analytical method for the conceptual design of lifted cables of super-long span cable-stayed bridges.

The nonlinear propagation of cylindrical vector beams in an optical fiber

We simulate the evolution of cylindrical vector beams (CVBs) in a Kerr nonlinear medium by the full-vector finite-element beam propagation method (VFE-BPM). This beams show poor transverse stability which contrasts with the strong transverse stability of fundamental modes. The approximate analytic theory has been used to qualitative analyze the variations of the numerically calculated instability growth rates. We present a reasonable concentric circle model to avoid the influence of triangular grids during simulation.

Critical threshold for microtubule amplification through templated severing

The cortical microtubule array of dark-grown hypocotyl cells of plant seedlings undergoes a striking, and developmentally significant, reorientation on exposure to light. This process is driven by the exponential amplification of a population of longitudinal microtubules, created by severing events localized at crossovers with the microtubules of the pre-existing transverse array. We present a dynamic one-dimensional model for microtubule amplification through this type of templated severing. We focus on the role of the probability of immediate stabilization-after-severing of the newly created lagging microtubule, observed to be a characteristic feature of the reorientation process. Employing stochastic simulations, we show that in the dynamic regime of unbounded microtubule growth, a finite value of this probability is not required for amplification to occur but does strongly influence the degree of amplification and hence the speed of the reorientation process. In contrast, in the regime of bounded microtubule growth, we show that amplification only occurs above a critical threshold. We construct an approximate analytical theory, based on a priori limiting the number of crossover events considered, which allows us to predict the observed critical value of the stabilization-after-severing probability with reasonable accuracy.

Second harmonic generation in graphene dressed by a strong terahertz field

We observe enhanced second-harmonic generation in monolayer graphene in the presence of an ultra-strong terahertz field pulse with a peak amplitude of 250 kV/cm. This is a strongly nonperturbative regime of light-matter interaction in which particles get accelerated to energies exceeding the initial Fermi energy of 0.2 eV over a timescale of a few femtoseconds. The second-harmonic current is generated as electrons drift through the region of momenta corresponding to interband transition resonance at an optical frequency. The resulting strongly asymmetric distortion of carrier distribution in momentum space gives rise to an enhanced electric-dipole nonlinear response at the second harmonic. We develop an approximate analytic theory of this effect which accurately predicts observed intensity and polarization of the second-harmonic signal.

Computational analyses of fully nonlinear interaction of an internal solitary wave and a free surface wave

This paper is concerned with the interaction of an internal solitary wave (ISW) at the interface of two-layer fluid and the free surface wave on top of the upper layer. It is based on the potential flow theory since internal waves are associated with large Reynolds numbers. The potential flows in the upper layer and lower layer are modeled using a multi-domain boundary element method (MDBEM). The computational model is validated with the experimental results for the profile and speed of the internal wave. The MDBEM is suitable for the simulation of ISW in both small and large density jump stratified fluid system. The wave velocity is compared with the approximate analytical theory for various ratios of the fluid densities of the two layers. In addition, the amplitude, velocity and profile of the surface wave induced by ISWs are investigated. The free surface displacement is opposite to that of the interface, and the amplitude of the surface wave increases with the amplitude and density jump. The surface wave induced by an ISW can be soliton-like wave, propagating with the constant speed of the ISW and maintaining its profile.

Generalized Langevin Equation as a Model for Barrier Crossing Dynamics in Biomolecular Folding.

Conformational memory in single-molecule dynamics has attracted recent attention and, in particular, has been invoked as a possible explanation of some of the intriguing properties of transition paths observed in single-molecule force spectroscopy (SMFS) studies. Here we study one candidate for a non-Markovian model that can account for conformational memory, the generalized Langevin equation with a friction force that depends not only on the instantaneous velocity but also on the velocities in the past. The memory in this model is determined by a time-dependent friction memory kernel. We propose a method for extracting this kernel directly from an experimental signal and illustrate its feasibility by applying it to a generalized Rouse model of a SMFS experiment, where the memory kernel is known exactly. Using the same model, we further study how memory affects various statistical properties of transition paths observed in SMFS experiments and evaluate the performance of recent approximate analytical theories of non-Markovian dynamics of barrier crossing. We argue that the same type of analysis can be applied to recent single-molecule observations of transition paths in protein and DNA folding.

Generalized Langevin Equation as a Model for Barrier Crossing Dynamics in Biomolecular Folding.

Conformational memory in single-molecule dynamics has attracted recent attention and, in particular, has been invoked as a possible explanation of some of the intriguing properties of transition paths observed in single-molecule force spectroscopy (SMFS) studies. Here we study one candidate for a non-Markovian model that can account for conformational memory, the generalized Langevin equation with a friction force that depends not only on the instantaneous velocity but also on the velocities in the past. The memory in this model is determined by a time-dependent friction memory kernel. We propose a method for extracting this kernel directly from an experimental signal and illustrate its feasibility by applying it to a generalized Rouse model of a SMFS experiment, where the memory kernel is known exactly. Using the same model, we further study how memory affects various statistical properties of transition paths observed in SMFS experiments and evaluate the performance of recent approximate analytical theories of non-Markovian dynamics of barrier crossing. We argue that the same type of analysis can be applied to recent single-molecule observations of transition paths in protein and DNA folding.

Self-consistent theory of spatial distribution of an assembly of charged drops in an electrodynamic balance

Levitation of several charged drops in the air by applying oscillating quadrupole electric fields in an electrodynamic (ED) balance offers opportunities for improving our understanding of the role played by inter-particle forces on pattern formation. In these systems, stable structures arise as a balance between the time-averaged quadrupole force, the drag force and the collective force of mutual repulsion acting on individual drops. In order to characterize these structures, we formulate a general theory for the spatial arrangement of the levitated charged drops via a many-body Mathieu-Poisson equation. Although this system is not amenable to an exact analytical treatment, we develop an approximate analytical theory by invoking a jellium picture to predict the charge density profiles of the 2D planar configurations attained by the droplets. This approach reduces the system of equations into a set of dual integral equations, which is solved self-consistently via Hankel transform techniques. The theory predicts functional forms for the droplet density profiles and the size of the planar cluster as a function of the number of charged drops and the system parameters. As a part of validating the theory, numerical integration of the exact equations of motion of the interacting many-drop system has been carried out. Additionally, experiments were also conducted by levitating a large number of drops in an ED balance previously developed in our laboratory. A comparison of the predictions of the theory with those of the numerical simulations shows excellent agreement between the two. The experimental data on the spatial density distribution also seems to be consistent with the predictions of the theory. The results are further discussed.

Three-body correlations and conditional forces in suspensions of active hard disks.

Self-propelled Brownian particles show rich out-of-equilibrium physics, for instance, the motility-induced phase separation (MIPS). While decades of studying the structure of liquids have established a deep understanding of passive systems, not much is known about correlations in active suspensions. In this work we derive an approximate analytic theory for three-body correlations and forces in systems of active Brownian disks starting from the many-body Smoluchowski equation. We use our theory to predict the conditional forces that act on a tagged particle and their dependence on the propulsion speed of self-propelled disks. We identify preferred directions of these forces in relation to the direction of propulsion and the positions of the surrounding particles. We further relate our theory to the effective swimming speed of the active disks, which is relevant for the physics of MIPS. To test and validate our theory, we additionally run particle-resolved computer simulations, for which we explicitly calculate the three-body forces. In this context, we discuss the modeling of active Brownian swimmers with nearly hard interaction potentials. We find very good agreement between our simulations and numerical solutions of our theory, especially for the nonequilibrium pair-distribution function. For our analytical results, we carefully discuss their range of validity in the context of the different levels of approximation we applied. This discussion allows us to study the individual contribution of particles to three-body forces and to the emerging structure. Thus, our work sheds light on the collective behavior, provides the basis for further studies of correlations in active suspensions, and makes a step towards an emerging liquid state theory.

Generalized Nernst layer model for convective-diffusion transport. Numerical solution for bromide ion electroreduction on inactive rotating disk electrode under steady state conditions

The process of electroreduction of bromate anion BrO3 - from aqueous solutions on catalytically inactive (e.g., carbon) electrodes is theoretically described in the framework of the generalized Nernst layer model in which the Nernst-layer thickness is chosen independently for each system’s component according to the Levich formula. For this system, the numerical algorithm is developed for solving the system diffusion- kinetic equations for the case of excessive content of protons in solution and one-dimensional transport (corresponding to RDE) under stationary conditions. The results are compared with conclusions of the approximate analytical theory proposed for the same system in our recent study (J. Electroanal. Chem., 2016, vol. 779, p. 146). The closeness of the numerical and analytical data makes it possible to conclude that both approaches can be used for solving this problem. Deviations are observed only when the approximations lying in the basis of the corresponding analytical relationships are violated.

Analytical and numerical investigation of formation of a vortex ring in the process of rising of a thermal in the atmosphere

Formation of a vortex ring in the process of rising a heated light volume of air medium in the Earth’s gravity field is investigated. The results of calculations carried out on various computational grids are compared both with each other and with solutions of the approximate analytical theory of formation and motion of a buoyant vortex ring approved by the available experimental data. A good agreement between all the data is obtained.

Structure and lubrication of solvent-free dendron brushes

Using analytical and numerical self-consistent field (SCF) methods we analyze equilibrium structure of dry (solvent-free) brush of branched polymers in wide range of chain grafting densities. A detailed analysis is performed for starlike polymers end-tethered to planar surface by terminal monomer of one of their branches. At moderate grafting densities ensuring Gaussian elasticity of tethered stars the brush structure is described by strong stretching SS-SCF theory employing parabolic molecular potential. At high grafting densities an approximate analytical AA-SCF theory is used to describe stratification of dry brush. Analytical predictions are supplemented by results of numerical Scheutjens-Fleer SF-SCF model. Interpenetration between dry brushes is analyzed as a function of grafting density and degree of branching of tethered chains. Comparison of interpenetration and friction between dry brushes with varied chain architectures pointed at optimal lubrication by macrocycles tethered to the surface by branching point. It is demonstrated that shear stress in sliding brushes of starlike macromolecules sharply decreases at high grafting densities corresponding to almost full stretching of the longest elastic path in the polymers.