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Narrow escape theory deals with the first passage of a particle diffusing in a cavity with small circular windows on the cavity wall to one of the windows. Assuming that (i) the cavity has no size anisotropy and (ii) all windows are sufficiently far away from each other, the theory provides an analytical expression for the particle mean first-passage time (MFPT) to one of the windows. This expression shows that the MFPT depends on the only global parameter of the cavity, its volume, independent of the cavity shape, and is inversely proportional to the product of the particle diffusivity and the sum of the window radii. Amazing simplicity and universality of this result raises the question of the range of its applicability. To shed some light on this issue, we study the narrow escape problem in a cylindrical cavity of arbitrary size anisotropy with two small windows arbitrarily located on the cavity side wall. We derive an approximate analytical solution for the MFPT, which smoothly goes from the conventional narrow escape solution in an isotropic cavity when the windows are sufficiently far away from each other to a qualitatively different solution in a long cylindrical cavity (the cavity length significantly exceeds its radius). Our solution demonstrates the mutual influence of the windows on the MFPT and shows how it depends on the inter-window distance. A key step in finding the solution is an approximate replacement of the initial three-dimensional problem by an equivalent one-dimensional one, where the particle diffuses along the cavity axis and the small absorbing windows are modeled by delta-function sinks. Brownian dynamics simulations are used to establish the range of applicability of our approximate approach and to learn what it means that the two windows are far away from each other. [ABSTRACT FROM AUTHOR]

We consider the spectral radius of a large random matrix X with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of X in Cipolloni et al., Ann. Probab. 51(6), 2192–2242 (2023). To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of X − z for different complex shift parameters z using the Dyson Brownian Motion. [ABSTRACT FROM AUTHOR]

Transfer entropy (TE) has proven to be an effective tool for determining the causal connection between two processes. For example, TE has been used to classify leader and follower agents in collective dynamics in the Vicsek model (VM). However, previous results have limited interaction radii, which are precisely the same among all agents, which is not realistic in practice. Here, we propose a modified version of the VM where the domains in which an agent can be influenced by others vary from agent-to-agent, which matches more closely with a real-life setting where not all agents have the same physical traits. We demonstrate that the TE vs cut-off technique is robust and efficacious in determining the maximum distance at which two interacting agents can transfer information in the system. We find that for two agents with different interaction domains, the derivative of the average inward TE can determine the individual agent's interaction domain. In a system with numerous agents, the TE vs cut-off technique is shown to be effective in predicting the average interaction domain of all agents, where the interaction domain of each agent was randomly selected from a Gaussian distribution. [ABSTRACT FROM AUTHOR]

The interface tends to behave as a circular shape under the effect of surface tension in the two-dimensional two-phase fluid flow. In this study, based on the coupled volume-of-fluid and level set (VOSET) method, an accurate circle-based interface reconstruction (CIR) algorithm for structured meshes is proposed, which features a straightforward implementation procedure. A portion of the standard circle is employed to approximate the curve interface, without the need for complex classification, coordinate transformation, and equation transformation. The radius of this circle is computed simply by the curvature and signed distance at the mixed cell center, and the center of this circle is determined using a straightforward dichotomy method under the mass conservation constraints. Results indicate that the coupled VOSET and CIR (VOSET-CIR) method maintains superior computational accuracy in the signed distance, interface curvature, and dynamic interface reconstruction compared to the coupled VOSET and piecewise linear interface calculation (VOSET-PLIC) method, with accuracy improvements ranging from 34% to 1490% across different test cases. Furthermore, the VOSET-CIR method outperforms the efficient least squares volume-of-fluid interface reconstruction (ELVIRA) and PLIC algorithms in reconstructing random circles and surpasses the quadratic spline based interface reconstruction (QUASI) algorithm in reconstructing random squares. In terms of computational efficiency, except for the initial PLIC algorithm, the time costs of the VOSET-CIR method are substantially lower than those of the ELVIRA and QUASI algorithms, as well as the VOSET-PLIC method. [ABSTRACT FROM AUTHOR]

To explore the curvature dependence of solid–fluid interfacial thermodynamics, we calculate, using Grand Canonical Monte Carlo simulation, the surface free energy for a 2d hard-disk fluid confined in a circular hard container of radius R as a function of the bulk packing fraction η and wall curvature C ̄ = − 1 / R. (The curvature is negative because the surface is concave.) Combining this with our previous data [Martin et al., J. Phys. Chem. B 124, 7938–7947 (2020)] for the positive curvature case (a hard-disk fluid at a circular wall, C ̄ = + 1 / R), we obtain a complete picture of surface thermodynamics in this system over the full range of positive and negative wall curvatures. Our results show that γ is linear in C ̄ with a slope that is the same for both positive and negative wall curvatures, with deviations seen only at high negative curvatures (strong confinement) and high density. This observation indicates that the surface thermodynamics of this system is consistent with the predictions of so-called morphometric thermodynamics at both positive and negative curvatures. In addition, we show that classical density functional theory and a generalized scaled particle theory can be constructed that give excellent agreement with the simulation data over most of the range of curvatures and densities. For extremely high curvatures, where only one or two disks can occupy the container at maximum packing, it is possible to calculate γ exactly. In this limit, the simulations and density functional theory calculations are in remarkable agreement with the exact results. [ABSTRACT FROM AUTHOR]

We determine, for hard spheres, the Helmholtz free energy of a liquid that contains a solid cluster as a function of the size of the solid cluster by means of the formalism of the thermodynamics of curved interfaces. This is done at the constant total number of particles, volume, and temperature. We show that under certain conditions, one may have several local minima in the free energy profile, one for the homogeneous liquid and others for the spherical, cylindrical, and planar solid clusters surrounded by liquid. The variation of the interfacial free energy with the radius of the solid cluster and the distance between equimolar and tension surfaces are inputs from simulation results of nucleation studies. This is possible because stable solid clusters in the canonical ensemble become critical in the isothermal–isobaric ensemble. At each local minimum, we find no difference in chemical potential between the phases. At local maxima, we also find equal chemical potential, albeit in this case the nucleus is unstable. Moreover, the theory allows us to describe the stable solid clusters found in simulations. Therefore, we can use it for any combination of the total number of particles, volume, and global density as long as a minimum in the Helmholtz free energy occurs. We also study under which conditions the absolute minimum in the free energy corresponds to a homogeneous liquid or to a heterogeneous system having either spherical, cylindrical, or planar geometry. This work shows that the thermodynamics of curved interfaces at equilibrium can be used to describe nucleation. [ABSTRACT FROM AUTHOR]

The breakup process of the inviscid liquid bridge sandwiched between two coaxial and equal-sized rods is investigated by tracking its profile. Here, the focus is on the quasi-static profile of the liquid bridge close to rupture and its influence on the subsequent dynamic breakup behaviors. With the increasing distance between the two rods, the profile of the liquid bridge close to rupture undergoes a transition from symmetry to asymmetry. We found there exists a critical slenderness above which the liquid bridge will be asymmetric and present a profile that can be well fitted by one cycle of the sine wave. It is demonstrated both experimentally and theoretically that the ratio of the length of the bridge to its equivalent radius, defined as geometric mean of the radii at the peak and trough of the bridge, is always 2 π for the asymmetric bridge close to rupture. Different with the symmetric evolution of the short bridge, the long asymmetric bridge pinches off first from the side near the bigger sessile drop and then from the other side, which endows the satellite droplet with a lateral momentum, resulting in the satellite re-collected by the sessile drop. The influence of the slenderness on the time interval among the asymmetric pinch-off, velocity, destination, and size of the satellite was investigated. A scaling law was proposed to describe the relationship between the lateral momentum of the satellite and the time interval between two pinch-off. This work is expected to benefit the utilizing or suppressing the satellite in practice. [ABSTRACT FROM AUTHOR]

First principles based beyond Born–Oppenheimer theory has been implemented on the F + H2 system for constructing multistate global diabatic Potential Energy Surfaces (PESs) through the incorporation of Nonadiabatic Coupling Terms (NACTs) explicitly. The spin–orbit (SO) coupling effect on the collision process of the F + H2 reaction has been included as a perturbation to the non-relativistic electronic Hamiltonian. Adiabatic PESs and NACTs for the lowest three electronic states (12A′, 22A′, and 12A″) are determined in hyperspherical coordinates as functions of hyperangles for a grid of fixed values of the hyperradius. Jahn–Teller (JT) type conical intersections between the two A′ states translate along C2 v and linear geometries in F + H2. In addition, A′ and A″ states undergo Renner–Teller (RT) interaction at collinear configurations of this system. Both JT and RT couplings are validated by integrating NACTs along properly chosen contours. Subsequently, we have solved adiabatic-to-diabatic transformation (ADT) equations to evaluate the ADT angles for constructing the diabatic potential matrix of F + H2, including the SO coupling terms. The newly calculated diabatic PESs are found to be smooth, single-valued, continuous, and symmetric and can be invoked for performing accurate scattering calculations on the F + H2 system. [ABSTRACT FROM AUTHOR]

In the presence of advection at a constant flow rate in a rectilinear geometry, the properties of planar A + B → C reaction fronts feature the same temporal scalings as in the pure reaction–diffusion case. In a radial injection geometry where A is injected into B radially at a constant flow rate Q, temporal scalings are conserved, but the related coefficients depend on the injection flow rate Q and on the ratio γ of initial concentrations of the reactants. We show here that this dependence of the front properties on the radial velocity allows us to tune the amount of product obtained in the course of time by varying the flow rate. We compare theoretically the efficiency of the rectilinear and radial geometries by computing the amount of product C generated in the course of time or per volume of reactant injected. We show that a curve γc(Q) can be defined in the parameter space (γ, Q) below which, for similar experimental conditions, the total amount of C is larger in the radial case. In addition, another curve γ*(Q) < γc(Q) can be defined such that for γ < γ*, the total amount of C produced is larger in the radial geometry, even if the production of C per unit area of the contact interface between the two reactants is larger in the rectilinear case. This comes from the fact that the length of the contact zone increases with the radius in the radial case, which allows us to produce in fine more product C for a same injected volume of reactant or in reactors of a same volume than in the rectilinear case. These results pave the way to the geometrical optimization of the properties of chemical fronts. [ABSTRACT FROM AUTHOR]

The vibration and stability of an axially functionally graded (AFG) cylinder with whirl motion in the annular liquid environment are investigated. The model of the performed system is given by the spinning Rayleigh beam assumptions with the rotary inertia and the gyroscopic effects. The fluid forces exerted on the cylinder, as a result of the external fluid, are calculated analytically. The coupled governing equation of motion for the system is developed via Hamilton's principle. The exact and approximate whirl frequency equations are presented for vibration and stability analysis of the AFG cylinder. The validity of the proposed model is confirmed by comparing it with the numerical solutions available in the literature. Detailed parameter discussions are conducted to evaluate the effects of the density ratio, outer-to-inner radius ratio, hollowness ratio, and slenderness ratio on the whirl characteristics and stability of the system. The results show that the whirl characteristics and instability of the AFG cylinder are strongly dependent on the external fluid. [ABSTRACT FROM AUTHOR]

An improved algorithm for evaluation of the minimum circumscribed circle and maximum inscribed circle is proposed in this paper. The candidate points for the minimum circumscribed circle and maximum inscribed circle satisfying the acute triangle are determined based on the local minimax radius in different regions. The more suitable region division method for the maximum inscribed circle is proposed. These simple algorithms are verified to be useful for determining the minimum circumscribed circle and maximum inscribed circle. [ABSTRACT FROM AUTHOR]

Measuring ear-canal absorbance and compensating for effects of the ear-canal acoustics on otoacoustic-emission measurements using an ear probe rely on accurately determining its acoustic source parameters. Using pressure measurements made in several rigid waveguides and models of their input impedances, a conventional calibration method estimates the ear-probe Thévenin-equivalent source parameters via a least-squares fit to an over-determined system of equations. Such a calibration procedure involves critical considerations on the geometry and number of utilized calibration waveguides. This paper studies the effects of calibration-waveguide geometry on achieving accurate ear-probe calibrations and measurements by systematically varying the lengths, length ratios, radii, and number of waveguides. For calibration-waveguide lengths in the range of 10–60 mm, accurate calibrations were generally obtained with absorbance measurement errors of approximately 0.02. Longer waveguides resulted in calibration errors, mainly due to coincident resonance frequencies among waveguides in the presence of mismatches between their assumed and actual geometries. The accuracy of calibrations was independent of the calibration-waveguide radius, except for an increased sensitivity of wider waveguides to noise. Finally, it is demonstrated how reformulating the over-determined system of equations to return the least-squares reflectance source parameters substantially reduces calibration and measurement errors. [ABSTRACT FROM AUTHOR]

Physical cluster formation in a fluid system causes a characteristic dependence of the pair correlation function on the distance r between a specific pair of particles at the liquid–vapor critical point. Its dependence deviates from that expressed by the product of a negative power r−1 and a specific function provided by a Taylor series with respect to positive powers of r. The effects of the physical cluster formation can be estimated by representing the pair correlation function as a sum of two correlation functions; these two correlation functions can be provided as exact solutions for two differential equations that result from a system of two integral equations equivalent to the Ornstein–Zernike equation. [ABSTRACT FROM AUTHOR]

Based on the Vlasov equation, a set of two-dimensional gyrokinetic integral eigenmode equations for drift instabilities in tokamak plasmas is derived. All kinetic effects of ions, including transience parallel to the magnetic field, drifts induced by the curvature and gradient of the magnetic field, and drifts induced by the equilibrium electric potential, and the effects of finite Larmor radius are taken into account. Not only the radial structure but also the poloidal structure of the modes as well as coupling of neighboring poloidal modes induced by toroidicity of the magnetic field can be obtained from these equations. This is an alternative to ballooning mode representation treatment, which can give mode structures along magnetic field lines only. The equations provide a theoretical base for relevant numerical simulation investigations. [ABSTRACT FROM AUTHOR]

When nonlinear measures are estimated from sampled temporal signals with finite-length, a radius parameter must be carefully selected to avoid a poor estimation. These measures are generally derived from the correlation integral, which quantifies the probability of finding neighbors, i.e., pair of points spaced by less than the radius parameter. While each nonlinear measure comes with several specific empirical rules to select a radius value, we provide a systematic selection method. We show that the optimal radius for nonlinear measures can be approximated by the optimal bandwidth of a Kernel Density Estimator (KDE) related to the correlation sum. The KDE framework provides non-parametric tools to approximate a density function from finite samples (e.g., histograms) and optimal methods to select a smoothing parameter, the bandwidth (e.g., bin width in histograms). We use results from KDE to derive a closed-form expression for the optimal radius. The latter is used to compute the correlation dimension and to construct recurrence plots yielding an estimate of Kolmogorov–Sinai entropy. We assess our method through numerical experiments on signals generated by nonlinear systems and experimental electroencephalographic time series. [ABSTRACT FROM AUTHOR]

We propose a novel integral model describing the motion of both flexible and rigid slender fibers in viscous flow and develop a numerical method for simulating dynamics of curved rigid fibers. The model is derived from nonlocal slender body theory (SBT), which approximates flow near the fiber using singular solutions of the Stokes equations integrated along the fiber centerline. In contrast to other models based on (singular) SBT, our model yields a smooth integral kernel which incorporates the (possibly varying) fiber radius naturally. The integral operator is provably negative definite in a nonphysical idealized geometry, as expected from the partial differential equation theory. This is numerically verified in physically relevant geometries. We discuss the convergence and stability of a numerical method for solving the integral equation. The accuracy of the model and method is verified against known models for ellipsoids. Finally, we develop an algorithm for computing dynamics of rigid fibers with complex geometries in the case where the fiber density is much greater than that of the fluid, for example, in turbulent gas-fiber suspensions. [ABSTRACT FROM AUTHOR]

The effects of cavity shapes and rounded corners on polymer filling and mold stress distributions of nanoimprint lithography are investigated using a numerical simulation approach. Three types of mold cavities including a rectangular cavity with vertical sidewalls, a trapezoidal cavity with inclined sidewalls, and a semicircular cavity with curved sidewalls are used to study the polymer flow and the filling situation in the embossing stage. Stress distributions of three types of molds are compared to evaluate the mold durability under different imprint pressures. To further optimize filling ratios and the stress concentration, three molds with round corners are proposed. Simulation results show that the mold with a semicircular cavity can achieve complete filling under the lowest pressure, but the maximum von Mises stress is greater than that of others. The filling ratio of the trapezoidal cavity is better than that of the rectangular cavity, and the maximum von Mises stress is the smallest among three types of molds. With the increase in the radius of round corners for three molds, filling ratios are improved and the maximum stresses decrease apparently, which not only improve the pattern transferring fidelity but also prolong the service life of the mold effectively. [ABSTRACT FROM AUTHOR]

Bridges might have time-varying sunlit–shaded areas due to the rotation and revolution of Earth. This work aims at studying the effects of time-varying sunlit–shaded areas on the temperature field of small radius curved concrete box girder bridges. Based on the law of Earth's rotation and revolution, the solar position is first obtained by considering the relative position between the Sun and the location of the bridge site on Earth. Then, the sunlit–shaded areas of small radius curved concrete bridges are determined using the ray tracing algorithm. With the consideration of various environmental conditions, including the time-varying solar radiation and wind speed, the three-dimensional finite element model of a small radius curved concrete bridge is created in ANSYS and the temperature field of the bridge could be solved. With a 30 m long curved concrete bridge taken as an example, the temperature distribution profiles of the bridge are investigated. The results and analysis provide a good reference for the thermal analysis and design of small radius curved concrete bridges. [ABSTRACT FROM AUTHOR]

Evaluation conditions of the minimum circumscribed circle and maximum inscribed circle in Cartesian coordinates are transformed into those in polar coordinates. According to the radius of the measured points, the candidate points for evaluating the minimum circumscribed circle and maximum inscribed circle are divided into three cases which are solved with the corresponding algorithms. Several examples have been used to validate the validity of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The dynamical formation of a Z-pinch in the strong-shock limit is studied in this paper using one-dimensional (1D) simulations of a two-temperature magnetohydrodynamic model. The classic 1D picture consists of three stages: run-in, reflected-shock, and expansion. The special case of a constant current I and uniform gas fill, which are approximate conditions of the pinch-formation stage in a dense plasma focus, is examined in detail. Time-profiles for the shock-front and piston positions during the run-in stage are compared with some of the commonly used 0D models from the literature. Some practical improvements to these models are presented here and it is shown that this model gives the best agreement with results from the simulations. Maximum compression of the plasma is achieved when the reflected shock from the axis meets the incoming current layer. The ratio of the plasma radius at this time with respect to its initial radius is found from the simulations to be r p / R ≈ 1 / 8 using 5/3 for the adiabatic coefficient γ. The pressure and temperature of the compressed plasma are found to peak a short time after maximum compression due to the inability of the reflected shock to completely stagnate the incoming plasma driven by the converging current layer. The variation of the results with a finite dI/dt and for different values of γ is presented. [ABSTRACT FROM AUTHOR]

A non-coaxial waveguide composed of a cylindrical resonator of radius R and cylindrical waveguides with the radii r1 and r2, respectively, is considered. The radii satisfy the inequality r 1 < r 2 < R. The conversion from the channel with zero orbital angular momentum (OAM) into the channels with non-zero OAM is achieved by shifting the center lines of the waveguides relative to the center line of the cylindrical resonator. The center lines of input and output waveguides are shifted relative to each other by the angle Δ ϕ in order to twist the output acoustic wave. The conversion efficiency of the input wave with zero OAM into the output wave with non-zero OAM as dependent on the frequency, length of the resonator, and Δ ϕ is considered, and the domains where the efficiency can reach almost 100% are found. [ABSTRACT FROM AUTHOR]

It is very important to select the initial four candidate points of the minimum zone circles to improve the computational efficiency. According to the radius of the measured points, three models of error evaluation are proposed and corresponding algorithms are given for evaluating the minimum zone circles based on the lower envelope lines, four-point intersection principle, and bisection method. The points constructing the lower envelope lines can be used for the evaluation of the maximum inscribed circle. Several examples have been carried out to validate the validity of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper studies the transport critical current density in a superconducting cylinder with a hole and a superconducting elliptic cylinder. Based on the Maxwell's equation, the critical current densities are obtained with numerical method for different cross sections. In order to determine the main features of different cross sections, the average critical current density and magnetic field are considered. It is shown that the average critical current density is affected by the geometry and size of the cross section. The average critical current density is a monotonic function of the radius in the superconducting cylinder with a hole. However, the average critical current density in the superconducting elliptic cylinder decreases with major axis first and then increases. [ABSTRACT FROM AUTHOR]

Using molecular dynamics simulations, an embedded-atom model potential, and the mechanistic route, we have computed the pressure tensor and the surface tension γ of Ag-Au liquid alloys. Although the model generally underestimates γ for pure metals, calculations for a bulk planar slab exhibit nonlinear variations of γ with increasing gold concentration, which agree with experiments and can be accounted for by a perfect solution model. Calculations for various nanoscale droplets containing between 100 and 3200 atoms show a systematic decrease of γ with increasing droplet radius R. The positive Tolman length of the alloy determined from these size variations is estimated to vary slightly with gold concentration. The effects of temperature in the range 1300-1700 K are discussed. [ABSTRACT FROM AUTHOR]