Exact Analytical Expressions Research Articles

Order–disorder transition in a linear system of anti-parallel magnetic dipoles with long-range interactions

In the present study, we investigate the magnetic ordering of a system with a large number of nanomagnetic dipoles, approximated as point dipoles, arranged on a regular ring, and having anti-parallel (anti-ferromagnetic) ordering at zero external magnetic fields. We can impose different restrictions on the degrees of freedom of the dipoles: dipoles along the ring and dipoles perpendicular to the ring. When the dipoles are along the ring, the model is called theϕ-model (the dipoles are along the ϕ-coordinate of the cylindrical coordinate system) or head-to-tail model. When the dipoles are perpendicular to the plane of the ring the model is called the z-model (the dipoles are along the z-axis of the cylindrical coordinate system), and when the dipoles are along the radials of the ring, the model is called the r-model (the radial model). The z-and the r- models are similar to the one-dimensional Ising chain model (spins ‘up’ or ‘down’): the only difference is that in the z-and r-models the interactions are dipolar long-range ones. By a combination of numerical and analytical methods, we have found the exact ground states of the models, which are degenerated. A perfectly ordered system is possible only at zero temperature: by increasing the temperature the perfect order is destroyed. Employing the Monte Carlo simulation, we show that in a system of magnetic dipoles forming a linear chain, similar to the one-dimensional Ising chain model, there are no phase transitions in the thermodynamics limit, when N→∞, except at zero temperature. The system continuously transforms from a perfectly-ordered state at zero temperature to a perfectly-disordered state at very high temperatures. However, the disordered states of the models consist of perfectly ordered domains, where the ordering corresponds to one of the degenerated ground states. With the temperature, the average size of the domains decreases to zero (at very high temperatures), and numerically this size is equal to the correlation length. For finite-sized models the perfectly ordered state is kept to some critical temperature depending on the number of dipoles: T1≈4.8/lnN/2 for the head-to-tail model, Tc≈1.8/lnN/2 for the z- and radial models, Tc is the temperature when the first defect appears in the perfectly-ordered ground state,measured in J=μ0m24πr03 units, where m is the dipole moment of the dipoles, r0 the nearest distance between them. Introducing quasi-dipoles, in the present study we investigate the basic properties of the z-models, such as the internal energy, the heat capacity, the order parameter, the correlation length, etc. Employing Boltzmann’s statistics, we have obtained the exact solution of the z- and 1d Ising models. The zero-temperature state of the z-model is a perfectly ordered configuration of anti-parallel pairs of dipoles, and it is degenerated because inverting the directions of the dipoles does not change the energy of the configuration. When the model consists of N odd number of dipoles, in the ground state there is a natural ‘defect’ – a non-paired dipole, which degenerates additionally N-times the ground state and increases the energy per dipole of the ground state. For any large number of dipoles N∼100 this effect becomes negligible. We have obtained exact analytical expressions for the energy and the magnetic field of the z-model at zero temperature.

Stochastic resetting and first arrival subjected to Gaussian noise and Poisson white noise.

We study the dynamics of an overdamped Brownian particle subjected to Poissonian stochastic resetting in a nonthermal bath, characterized by a Poisson white noise and a Gaussian noise. Applying the renewal theory we find an exact analytical expression for the spatial distribution at the steady state. Unlike the single exponential distribution as observed in the case of a purely thermal bath, the distribution is double exponential. Relaxation of the transient spatial distributions to the stationary one, for the limiting cases of Poissonian rate, is investigated carefully. In addition, we study the first-arrival properties of the system in the presence of a delta-function sink with strength κ, where κ=0 and κ=∞ correspond to fully nonreactive and fully reactive sinks, respectively. We explore the effect of two competitive mechanisms: the diffusive spread in the presence of two noises and the increase in probability density around the initial position due to stochastic resetting. We show that there exists an optimal resetting rate, which minimizes the mean first-arrival time (MFAT) to the sink for a given value of the sink strength. We also explore the effect of the strength of the Poissonian noise on MFAT, in addition to sink strength. Our formalism generalizes the diffusion-limited reaction under resetting in a nonequilibrium bath and provides an efficient search strategy for a reactant to find a target site, relevant in a range of biophysical processes.

Exact expressions for the height of the interatomic step in the exchange-correlation potential from the derivative discontinuity of the energy

Popular approximations to the exchange-correlation (xc) energy of density-functional theory do not yield the spatial ``step'' structures in the exact xc potential which are necessary to describe dissociation and electron excitation with the Kohn-Sham (KS) system. Via the discontinuity in the derivative of the xc energy as a function of electron number I derive exact analytic expressions in terms of the KS single-particle energies for the height of the step in the xc potential between a variety of open- and closed-shell atoms within stretched molecules.

Analytical solutions for the magnon frequencies at high-symmetry points of the Brillouin zone in anisotropic kagome antiferromagnets

The knowledge of the magnon dispersion relations in antiferromagnetic materials with nontrivial spin textures has considerable interest to the understanding of magnonic and spintronic phenomena involving these materials. One particularly interesting nontrivial spin texture existing in several antiferromagnets has spins at an angle of 120° with the in-plane neighbors and arranged in kagome lattices. Here we present a spin-wave calculation for antiferromagnets with kagome spin lattices considering exchange and single-ion anisotropy interactions between the spins. The theory yields exact analytical expressions for the frequencies of magnons at high-symmetry points of the Brillouin zone, that can be readily use to obtain the interaction parameters from experimental data with one-and two-magnon inelastic light scattering. The analytical expressions are used to obtain the field parameters for the kagome lattice antiferromagnet L12-IrMn3 from four experimentally measured frequencies. Both exchange field parameters are in reasonable agreement with the values obtained with ab initio calculations, while the anisotropy field is in very good agreement with the one calculated with atomistic spin models and Monte Carlo simulations.

Stability of one and two-dimensional spatial solitons in a cubic–quintic–septimal nonlinear Schrödinger equation with fourth-order diffraction and [formula omitted]-symmetric potentials

In this paper, the existence and stability of solitons in parity-time (PT)-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with fourth-order diffraction (FOD) coefficient and higher-order nonlinearities have been investigated. For the linear case, we have demonstrated numerically that, the FOD parameter can alter the PT-breaking points. Exact analytical expressions of the localized modes are obtained respectively, in one and two dimensional nonlinear Schrödinger (NLS) equation with both self-focusing and self-defocusing Kerr, and higher-order nonlinearities for nonlinear case. The effects of both FOD and higher-order nonlinearities on the stability/instability structure of these localized modes have also been discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Some stable and unstable solutions have been given and, it has been seen how higher-order self-focusing and self-defocusing nonlinearities can influence the stability/instability of the system.

OPTICAL-ELECTRONIC INSTRUMENT FOR MEASURING THE RADIATION COEFFICIENT AND TEMPERATURE OF THE CONTROLLED OBJECT

The operating principle of optoelectronic instruments for measuring the temperature of the heated products based on the measurement of the radiation flux from the heated product, which depends on the temperature and emissivity of the surface material. The main error of such optocal-electronic devices is the methodological component, which is due to the variability of the radiation coefficient of the surface of the product material. The radiation coefficient of an object depends on the material, the surface state of the material, and the temperature. In the measurement process, it is difficult to take this dependence into account, since there are no exact analytical expressions of these dependencies. In practice, the radiation coefficient of the surface of material the product is determined approximately using reference books. For a more accurate determination of the radiation coefficient, a preliminary study is necessary, which requires more complex equipment than a device for measuring the radiation flux from a heated body. To solve this problem, there are empirical generalized equations of the functional dependences of the radiation coefficient. The article analyzes the errors in determining the radiation coefficient using generalized equations in comparison with experimental data. The analysis indicates that the error in determining the radiation coefficient can reach large values that may not meet the requirements of consumers. To improve the accuracy of measuring the temperature of the object, a device has been developed that implements the method of sample signals. The developed device predetermines the radiation coefficient of the measured product at a certain temperature and introduces a correction when measuring the temperature.

Expected maximum of bridge random walks & Lévy flights

We consider one-dimensional discrete-time random walks (RWs) with arbitrary symmetric and continuous jump distributions f(η), including the case of Lévy flights. We study the expected maximum of bridge RWs, i.e. RWs starting and ending at the origin after n steps. We obtain an exact analytical expression for valid for any n and jump distribution f(η), which we then analyze in the large n limit up to second leading order term. For jump distributions whose Fourier transform behaves, for small k, as with a Lévy index 0 < μ ⩽ 2 and an arbitrary length scale a > 0, we find that, at leading order for large n, . We obtain an explicit expression for the amplitude h 1(μ) and find that it carries the signature of the bridge condition, being different from its counterpart for the free RW. For μ = 2, we find that the second leading order term is a constant, which, quite remarkably, is the same as its counterpart for the free RW. For generic 0 < μ < 2, this second leading order term is a growing function of n, which depends non-trivially on further details of , beyond the Lévy index μ. Finally, we apply our results to compute the mean perimeter of the convex hull of the 2d Rouse polymer chain and of the 2d run-and-tumble particle, as well as to the computation of the survival probability in a bridge version of the well-known ‘lamb–lion’ capture problem.

Exact solution of two-potential system under same range approximation and its implication

In this paper, exact analytical expressions for the Jost solution and Jost function are derived for motion in the nuclear Manning–Rosen plus the Hulthén potential to study both the bound and scattering state observables. The proton-deuteron and alpha-carbon systems are studied to judge the merit of our approach. Our results are found in reasonable agreement with experimental data.

Multifractality and self-averaging at the many-body localization transition

Finite-size effects have been a major and justifiable source of concern for studies of many-body localization, and several works have been dedicated to the subject. In this paper, however, we discuss yet another crucial problem that has received much less attention, that of the lack of self-averaging and the consequent danger of reducing the number of random realizations as the system size increases. By taking this into account and considering ensembles with a large number of samples for all system sizes analyzed, we find that the generalized dimensions of the eigenstates of the disordered Heisenberg spin-1/2 chain close to the transition point to localization are described remarkably well by an exact analytical expression derived for the non-interacting Fibonacci lattice, thus providing an additional tool for studies of many-body localization.

A general analytical framework for the mechanics of heterogeneous hexagonal lattices

The in-plane mechanics of two-dimensional heterogeneous hexagonal lattices are investigated. The heterogeneity originates from two physically realistic considerations: different constituent materials and different wall thicknesses. Through the combination of multi-material and multi-thickness elements, the most general form of 2D heterogeneous hexagonal lattices is proposed in this paper. By exploiting the mechanics of a unit cell with multi-material and multi-thickness characteristics, exact closed-form analytical expressions of equivalent elastic properties of the general heterogeneous lattice have been derived. The equivalent elastic properties of the 2D heterogeneous lattice are Young’s modulli and Poisson’s ratios in both directions and the shear modulus. Two distinct cases, namely lattices with thin and thick constituent members, are considered. Euler–Bernoulli beam theory is employed for the thin-wall case, and Timoshenko beam theory is employed for the thick-wall case. The closed-form expressions are validated by independent finite element simulation results. The generalized expressions can be considered as benchmark solutions for validating future numerical and experimental investigations. The conventional single-material and single-thickness homogeneous lattice appears as a special case of the heterogeneous considered here. By introducing the Material Disparity Ratio (MDR) and Geometric Disparity Ratio (GDR), variability in the equivalent elastic properties has been graphically demonstrated. As opposed to classical homogeneous lattices, heterogeneous lattices significantly expand the design space for 2D lattices. Orders-of-magnitude of variability in the equivalent elastic properties is possible by suitably selecting material and geometric disparities within the lattices. The general closed-form expressions proposed in this paper open up the opportunity to design next-generation heterogeneous lattices with highly tailored effective elastic properties.

Correlations in scattered perfect optical vortices

We study correlations in the speckle patterns generated by the scattering of perfect optical vortex (POV) beams and use them to produce a new class of coherence functions, namely Bessel coherence functions. Higher (zeroth) order Bessel coherence functions have been realized in cross (auto)-correlation between the speckle patterns generated by the scattering of perfect vortex beams of different orders. We have also studied the propagation of produced Bessel coherence functions and characterized their divergence with respect to the radius of their first ring for different orders m = 0–4. We observe that the divergence varies linearly with the order of the coherence function. We provide the exact analytical expression for the auto-correlation, as well as cross-correlation functions for speckle patterns. Our experimental results are in good agreement with the analytical results.

On detour index of cycloparaphenylene and polyphenylene molecular structures

Cycloparaphenylene is a particle that comprises a few benzene rings associated with covalent bonds in the para positions to frame a ring-like structure. Similarly, poly (para-phenylenes) are macromolecules that include benzenoid compounds straightforwardly joined to each other by C–C bonds. Because of their remarkable architectural highlights, these structures have fascinated attention from numerous vantage focuses. Descriptors are among the most fundamental segments of prescient quantitative structure-activity and property relationship (QSAR/QSPR) demonstrating examination. They encode chemical data of particles as quantitative numbers, which are utilized to create a mathematical correlation. The nature of a predictive model relies upon great demonstrating insights, yet additionally on the extraction of compound highlights. To a great extent, Molecular topology has exhibited its adequacy in portraying sub-atomic structures and anticipating their properties. It follows a two-dimensional methodology, just thinking about the interior plan, including molecules. Explicit subsets speak the design of every atom of topological descriptors. When all around picked, these descriptors give a unique method of describing an atomic system that can represent the most significant highlights of the molecular structure. Detour index is one such topological descriptor with much application in chemistry, especially in QSAR/QSPR studies. This article presents an exact analytical expression for the detour index of cycloparaphenylene and poly (para-phenylene).

Hulthén off-shell transition matrix for nuclear systems

Exact analytical expression for the Hulthén off-shell Jost solution is constructed by using the particular integral of the inhomogeneous Schrödinger equation. The Jost solution is exploited to calculate the off-shell physical solution and in turn the off-shell T-matrix. The off-shell T-matrices are computed for the (n-p) and (n-d) systems and are found in order with respect to its half-shell behavior.

Analysis of the spectral properties of free electron lasers in X-RAY and other bands

Analytical formalism for the description of free electron laser (FEL) spectrum is proposed with account for the electron beam size, divergences, beam energy spread, the off-axis deflection and the periodic harmonic and non-periodic constant magnetic field components in undulators. The exact analytical expressions for the spontaneous undulator radiation (UR) are obtained in terms of generalized Bessel and Airy functions, which describe the spectrum line shape and intensities. The obtained analytical formulae distinguish the effects of each installation and beam parameter on the FEL harmonic radiation. The enhanced phenomenological model is employed for the FEL modeling; it describes the harmonic powers evolution along the undulators and gradual saturation around the saturation length with account for the saturated powers oscillations. With this model, we study the generation of FEL harmonics, including even, in several instances of FEL experiments at LCLS, LEUTL and PAL-XFEL. We analytically demonstrate the even harmonics reasons in the LCLS and LEUTL FEL experiments. The spectrum line split due to the beam size is demonstrated analytically and compared with experimentally measured spectral characteristics. The PAL-XFEL experiment is analyzed and compared with the LCLS experiment; the possibility of high harmonic radiation in it is explored. The modeling results fully agree with the measured values in the studied FELs. The analytical formalism shows physical background for each harmonic in every FEL. It allows the analysis of the radiation and beam alignment in projected, built and working FELs and it can help minimizing losses and correcting magnetic fields as well as projecting experiments with FELs.

The eigenbuckling analysis of hexagonal lattices: closed-form solutions

Elastic instability such as the buckling of cellular materials plays a pivotal role in their analysis and design. Despite extensive research, the quantifi- cation of critical stresses leading to elastic instabi- lities remains challenging due to the inherent nonlinearities. We develop an analytical approach considering the spectral decomposition of the elasticity matrix of two-dimensional hexagonal lattice materials. The necessary and sufficient condition for the buckling is established through the zeros of the eigenvalues of the elasticity matrix. Through the analytical solution of the eigenvalues, the conditions involving equivalent elastic properties of the lattice were directly connected to the mathematical requirement of buckling. The equivalent elastic properties are expressed in closed form using geometric properties of the lattice and trigonometric functions of a non-dimensional axial force parameter. The axial force parameter was identified for four different stress cases, namely, compressive stress in the longitudinal and transverse directions separately and together and torsional stress. By solving the resulting nonlinear equations, we derive exact analytical expressions of critical eigenbuckling stresses for these four cases. Crucial parameter combinations leading to minimum buckling stresses are derived analytically. The exact closed-form analytical expressions derived in the paper can be used for quick engineering design calculations and benchmarking related experimental and numerical studies.

Multi-mode oscillations of a mechanical system with higher-order singularity

This article is based on earlier author’s works on the behavior of mechanical systems with a singular configuration space. We obtained exact analytical expressions for the constraints providing multi-mode oscillations of a double mathematical pendulum. It was found that there are infinitely many modes of oscillations determined by the degree of degeneracy at a singular point in the configuration space of the pendulum. The problem of a double pendulum motion with an elliptical constraint, posed by the author earlier, received a complete solution, both theoretical and experimental. For such a constraint, there are exactly two different oscillation modes: symmetric and asymmetric. Each mode can be realized by selecting the system parameters and is completely determined by the smoothness of the trajectory in the configuration space with a singularity. In the study we used only standard tools of classical analytical mechanics.

Polytropic wind solutions via the Complex Plane Strategy

Solar-type stars generate spherical winds, which are pressure driven flows, that start subsonic, reach the sound speed at the sonic point and transition to supersonic flows. The sonic point, mathematically corresponds to a singularity of the system of differential equations describing the flow. In the problem of an isothermal wind, the Parker solution provides an exact analytical expression tuned appropriately so that the singularity does not affect the solution. However, if the wind is polytropic it is not possible to find an analytical solution and a numerical approach needs to be followed. We study solutions of spherical winds that are driven by pressure within a gravitational field. The solutions pass smoothly from the critical point and allow us to study the impact of the changes of the polytropic index to these winds. We explore the properties of these solutions as a function of the polytropic index and the boundary conditions used. We apply the Complex Plane Strategy (CPS) and we obtain numerically solutions of polytropic winds. This allows us to avoid the singularity appearing in the equations through the introduction complex variables and integration on the complex plane. Applying this method, we obtain solutions with physical behaviour at the stellar surface, the sonic point and at large distances from the star. We further explore the role of the polytropic index in the flow and the effect of mass-loss rate and temperature on the solution. We find that the increase of polytropic index as well as the decrease of flow parameter both yield to a smoother velocity profile and lower velocities and shifts the transition point from subsonic to supersonic behaviour further from the star. Finally, we verify that the increasing of coronal temperature yields higher wind velocities and a weaker dependence on polytropic index.

A modified fluctuation test for elucidating drug resistance in microbial and cancer cells

Clonal populations of microbial and cancer cells are often driven into a drug-tolerant persister state in response to drug therapy, and these persisters can subsequently adapt to the new drug environment via genetic and epigenetic mechanisms. Estimating the frequency with which drug-tolerance states arise, and its transition to drug-resistance, is critical for designing efficient treatment schedules. Here we study a stochastic model of cell proliferation where drug-tolerant persister cells transform into a drug-resistant state with a certain adaptation rate, and the resistant cells can then proliferate in the presence of the drug. Assuming a random number of persisters to begin with, we derive an exact analytical expression for the statistical moments and the distribution of the total cell count (i.e., colony size) over time. Interestingly, for Poisson initial conditions the noise in the colony size (as quantified by the Fano factor) becomes independent of the initial condition and only depends on the adaptation rate. Thus, experimentally quantifying the fluctuations in the colony sizes provides an estimate of the adaptation rate, which then can be used to infer the starting persister numbers from the mean colony size. Overall, our analysis introduces a modification of the classical Luria–Delbrück experiment, also called the “Fluctuation Test”, providing a valuable tool to quantify the emergence of drug resistance in cell populations.

A Cartesian grid based tailored finite point method for reaction-diffusion equation on complex domains

This paper presents a Cartesian grid based tailored finite point method (TFPM) for singularly perturbed reaction-diffusion equation on complex domains. The method is incorporated with the kernel-free boundary integral algorithm, where the semi-discrete boundary value problems after time integration are reformulated into corresponding Fredholm boundary integral equations (BIEs) of the second kind, however with no algorithmic dependence on the exact analytical expression for the kernels of integrals. The BIEs are iteratively solved by the GMRES method while integral evaluation during each iteration resorts to solving an equivalent interface problem, which in practice is achieved by a series of manipulations in the framework of TFPM including discretization, correction, solution, and interpolation. The proposed method has second-order accuracy for the reaction-diffusion equation as demonstrated by the numerical examples.

Decoherence suppression with both spontaneous emission and dephasing by non-Hermitian operation

Decoherence suppression from disturbance of the environment is an essential task in quantum information processing. Using the non-Hermitian operation approach, decoherence suppression of a qubit system interacting with a zero-temperature reservoir with both spontaneous emission and dephasing is investigated. We derive the exact analytical expressions of the quantum fidelity and find that quantum coherence can be significantly enhanced by controlling the non-Hermitian operator parameters. In particular, the non-Hermitian is more effective in the case of strong decay rates and show better inhibition effect on dephasing process than the spontaneous emission process.