Closed-form Analytical Results Research Articles

Analytic thermodynamic properties of the Lieb-Liniger gas

We present a comprehensive review on the state-of-the-art of the approximate analytic approaches describing the finite-temperature thermodynamic quantities of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact repulsive interactions. This paradigmatic model of quantum many-body-theory plays an important role in many areas of physics-thanks to its integrability and possible experimental realization using, e.g., ensembles of ultracold bosonic atoms confined to quasi-1D geometries. The thermodynamics of the uniform Lieb-Liniger gas can be obtained numerically using the exact thermal Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the TBA numerical calculations do not allow for the in-depth understanding of the underlying physical mechanisms that govern the thermodynamic behavior of the Lieb-Liniger gas at finite temperature. Our work is then motivated by the insights that emerge naturally from the transparency of closed-form analytic results, which are derived here in six different regimes of the gas and which exhibit an excellent agreement with the TBA numerics. Our findings can be further adopted for characterising the equilibrium properties of inhomogeneous (e.g., harmonically trapped) 1D Bose gases within the local density approximation and for the development of improved hydrodynamic theories, allowing for the calculation of breathing mode frequencies which depend on the underlying thermodynamic equation of state. Our analytic approaches can be applied to other systems including impurities in a quantum bath, liquid helium-4, and ultracold Bose gas mixtures.

Smoothing in linear multicompartment biological processes subject to stochastic input.

Many physical and biological systems rely on the progression of material through multiple independent stages. In viral replication, for example, virions enter a cell to undergo a complex process comprising several disparate stages before the eventual accumulation and release of replicated virions. While such systems may have some control over the internal dynamics that make up this progression, a challenge for many is to regulate behavior under what are often highly variable external environments acting as system inputs. In this work, we study a simple analog of this problem through a linear multicompartment model subject to a stochastic input in the form of a mean-reverting Ornstein-Uhlenbeck process, a type of Gaussian process. By expressing the system as a multidimensional Gaussian process, we derive several closed-form analytical results relating to the covariances and autocorrelations of the system, quantifying the smoothing effect discrete compartments afford multicompartment systems. Semianalytical results demonstrate that feedback and feedforward loops can enhance system robustness, and simulation results probe the intractable problem of the first passage time distribution, which has specific relevance to eventual cell lysis in the viral replication cycle. Finally, we demonstrate that the smoothing seen in the process is a consequence of the discreteness of the system, and does not manifest in systems with continuous transport. While we make progress through analysis of a simple linear problem, many of our insights are applicable more generally, and our work enables future analysis into multicompartment processes subject to stochastic inputs.

Electrohydrodynamics of diffuse porous colloids.

The present article deals with the electrohydrodynamic motion of diffuse porous particles governed by an applied DC electric field. The spatial distribution of monomers as well as the charge distribution across the particle are considered to follow sigmoidal distribution involving decay length. Such a parameter measures the degree of inhomogeneity of the monomer distribution across the particle. The diffuse porous particles resemble several colloidal entities which are often seen in the environment as well as in biological and pharmaceutical industries. Considering the impact of bulk pH and ion steric effects, we modelled the electrohydrodynamics of such porous particulates based on the modified Boltzmann distribution for the spatial distribution of electrolyte ions and the Poisson equation for electric potential as well as the conservation of mass and momentum principles. We adopt regular perturbation analysis with weak field assumption and the perturbed equations are solved numerically to calculate the electrophoretic mobility and neutralization fraction of the particle charge during its motion as well as fluid collection efficiency. We further deduced the closed form relation between the drag force experienced by the charged porous particle and the fluid collection efficiency. In addition to the numerical results, we further derived the closed form analytical results for all the intrinsic parameters indicated above derived within the Debye-Hückel electrostatic framework and homogeneous distribution of monomers within the particle for which the decay length vanishes. The deduced mathematical results as indicated above will be useful to analyze several electrostatic and hydrodynamic features of a wide class of porous particulate and environmental entities.

Diffusiophoresis of Weakly Charged Fluid Droplets in a General Electrolyte Solution: An Analytical Theory.

Owing to the importance of analytical results for electrokinetics of colloidal entities, we performed a mathematical analysis to determine the closed form analytical results for the diffusiophoretic velocity of a hydrophobic and polarizable fluid droplet. A comprehensive mathematical model is developed for diffusiophoresis, considering the background aqueous medium as general electrolytes (e.g., binary valence-symmetric/asymmetric electrolytes and a mixed solution of binary electrolytes). We performed our analysis under a weak concentration gradient, and the analytical results for diffusiophoretic velocity are calculated within the Debye-Hückel electrostatic framework. The exact form of the diffusiophoretic velocity is further approximated with negligible error, and the approximate form is found to be free from any of the cumbersome exponential integrals and thus very convenient for practical use. The present theory also covers the diffusiophoresis of perfectly dielectric as well as perfectly conducting droplets as its limiting case. In addition, we have further derived a number of closed form expressions for diffusiophoretic velocity pertaining to several particular cases, and we observed that the derived limit correctly recovers the available existing analytical results for diffusiophoretic velocity. Thus, the present analytical theory for diffusiophoresis can be applied to a wide class of fluidic droplets, e.g., hydrophobic and dielectric oil/conducting mercury droplets, air bubbles, nanoemulsions, as well as any polarizable and hydrophobic fluidic droplet suspended in a solution of general electrolytes.

Helical inflation correlators: partial Mellin-Barnes and bootstrap equations

Massive spinning particles acquire helicity-dependent chemical potentials during the inflation from axion-type couplings. Such spinning fields can mediate sizable inflaton correlators which we call the helical inflation correlators. Helical inflaton correlators are approximately scale invariant, dS boost breaking, parity violating, and are promising observables of cosmological collider physics. In this work, we present complete and analytical results for 4-point helical inflation correlators with tree-level exchanges of massive spinning particles, including both the smooth background and the oscillatory signals. We compute the bulk Schwinger-Keldysh integrals in two independent ways, including the partial Mellin-Barnes representation and solving bootstrap equations. We also present new closed-form analytical results for 3-point functions with massive scalar or helical spinning exchanges. The analytical results allow us to concretely and efficiently explore the phenomenological consequences of helicity-dependent chemical potentials. In particular, we show that the chemical potential can exponentially enhance oscillatory signals of both local and nonlocal types, but only affects the background in a rather mild way. Our results extend the de Sitter bootstrap program to include nonperturbative breaking of de Sitter boosts. Our results also explicitly verify the recently proposed cutting rule for cosmological collider signals.

A 10-MHz 85.1-dB SFDR 1.1-mW continuous-time Delta–sigma modulator employing calibration-free SC DAC and passive front-end low-pass filter

This article demonstrates the circuit technique of a highly power-efficient continuous-time (CT) Delta–sigma modulator (DSM) based on a switched-capacitor (SC) feedback digital-to-analog converter (DAC) and a passive front-end low-pass filter (LPF). The technique addresses the clock sensitivity issue of conventional DACs while maintaining the alias rejection. It is validated by a fourth-order CT DSM prototype fabricated in 65-nm CMOS technology. The input stage of the DSM is an active-RC integrator merged with a passive front-end LPF. A resonator follows and realizes the cascade of resonators with feedforward (CRFF) structure. The accurate feedback loop employs a highly linear and calibration-free 5-level SC DAC. Based on the linear periodically time-varying (LPTV) system model, an exact analysis is given to explain the effects of the passive LPF and SC DAC combo on the signal transfer function (STF), and hence, on the alias rejection. The prototype DSM achieves a 70.2-dB signal to noise and distortion ratio (SNDR) and an 85.1-dB spurious-free dynamic range (SFDR) in a 10-MHz bandwidth, while dissipating 1.1 mW with a 1-V supply. The modulator achieves a 65.4-dB alias rejection and confirms the closed-form analytical results.

Broadband multimode piezoelectric spherical shell transducers for underwater sound

Broadband multimode piezoelectric spherical shell transducers are investigated for underwater sound by exciting the zeroth and/or first order (0 + 1) extensional modes of vibration with and without acoustical baffles of different types. Measurements of the frequency response (transmit pressure per volt or TVR), radiation patterns and electromechanical impedance reveal useful tradeoffs and design guidance for many practical applications. Measurements are in good agreement with theoretical closed form analytical results.

Performance Evaluation and Security Analysis of UAV-Based FSO/CV-QKD System Employing DP-QPSK/CD

Unmanned aerial vehicles (UAV) based on free-space optical (FSO) systems combine the benefits of both the high data rate of the FSO and the mobility of UAV. However, cumulative effects of laser beam divergence and turbulence-induced fading on the received irradiance in the FSO link might allow an external eavesdropper located near the authorized receiver to break the transmission under certain conditions. Quantum key distribution (QKD) is a technique of secure communication that employs quantum mechanics-based cryptography principles. A combination of a continuous-variable quantum key distribution (CV-QKD) and Gaussian modulation of quantum coherent states (GMCS) enables a relatively secure communication system against collective attacks over FSO transmission links. However, the performance is restricted and degraded due to the influence of atmospheric turbulence and excess noise. This paper aims to evaluate the performance of a secured UAV-based FSO system utilizing prepare and measure CV-QKD protocol based on GMCS under collective attack in terms of quantum bit error rate (QBER), outage probability, and secret key rate (SKR) considering the impairments of the FSO channel, link budget, deviations in the position and attitude angles of the UAV, channel fluctuating transmittance, and excess noise. The system parameters, including transmit power, receiver’s field-of-view, and beam divergence angle, are optimized to satisfy the system’s design criteria of threshold-QBER <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$=10^{-3}$</tex-math></inline-formula> and threshold outage probability <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$=10^{-2}$</tex-math></inline-formula> . The proposed system tolerates boresight displacement up to 7 cm with the optimized values. The Closed-form expressions and analytical results are confirmed by Monte-Carlo simulations.

Three-Variable Reduction of the Spin-Flip Model for Spin-VCSELs

Recent increased interest in spin-polarised vertical-cavity surface-emitting lasers (spin-VCSELs) as potential high-speed sources has spurred research into the analysis of their dynamics. This has often been explored by combining the spin-flip model (SFM) with numerical methods. However, numerical simulation does not readily expose key dependencies and to date there is a lack of accessible closed-form analytical results for the steady-state solutions and dynamic stability boundaries. Thus in the present contribution we address this and show that, for zero dichroism, the five rate equations of the SFM can be reduced to a set of three. These can be solved in the steady-state in terms of the intensity and polarisation of the pump (optical or electrical), and the ellipticity of the output. Additionally, a small-signal analysis leads to analytic results for the boundaries between stable and unstable operation in the plane of pump ellipticity versus pump intensity. Comparison of the results from these expressions with those from numerical bifurcation and continuation methods shows very good agreement. The accuracy of the reduced set of equations is confirmed by comparing the results with those from the full set of SFM equations. In the limiting case of very high birefringence, as would be required for a potential THz source, a simple algebraic relation is derived for the spin relaxation rate in terms of other parameters. Hence we find that the range of spin relaxation rates to achieve THz oscillations is very limited. The relative simplicity of the present approach thus offers a rapid, intuitive and convenient route to study the dynamics of spin-VCSELs.

Reflectance statistics from a thin weakly disordered optical media: application to detection of structural alterations in cells/tissues

A linearized stochastic model is introduced to describe light-scattering from a thin weakly disordered optical medium with a mean refractive index (RI) different from that of the uniform surrounding medium. The parameters are chosen suitably to study scattering from biologically relevant cell/tissue samples using partial wave spectroscopy technique. Closed-form analytical results are derived for the statistics of the reflected light. We show in detail how they can be directly compared with experiments to extract the sample thickness and mean RI with the aim of determining the structural disorder parameter - changes in which are useful biological markers for disease related abnormal alterations in the intracellular structures. A strong spectral dependence of the auto-correlation function is found arising from the interaction between disorder scattering and thin-film interference leading to a new decorrelation lengthscale.

Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely spaced modes

It is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle oscillations. However, when the intensity of the self-excitation reaches a condition of maximum friction damping, the limit cycle loses stability via a fold bifurcation. The behavior may become even more complicated in the presence of any internal resonance conditions. In this work, we consider a two-degree-of-freedom system with an elastic dry friction element (Jenkins element) having closely spaced natural frequencies. The symmetric in-phase motion is subjected to self-excitation by negative (viscous) damping, while the symmetric out-of-phase motion is positively damped. In a previous work, we showed that the limit cycle loses stability via a secondary Hopf bifurcation, giving rise to quasi-periodic oscillations. A further increase in the self-excitation intensity may lead to chaos and finally divergence, long before reaching the fold bifurcation point of the limit cycle. In this work, we use the method of complexification-averaging to obtain the slow flow in the neighborhood of the limit cycle. This way, we show that chaos is reached via a cascade of period-doubling bifurcations on invariant tori. Using perturbation calculus, we establish analytical conditions for the emergence of the secondary Hopf bifurcation and approximate analytically its location. In particular, we show that non-periodic oscillations are the typical case for prominent nonlinearity, mild coupling (controlling the proximity of the modes), and sufficiently light damping. The range of validity of the analytical results presented herein is thoroughly assessed numerically. To the authors’ knowledge, this is the first work that shows how the challenging Jenkins element can be treated formally within a consistent perturbation approach in order to derive closed-form analytical results for limit cycles and their bifurcations.

Lubrication analysis of peristaltic motion in non-axisymmetric annular tubes

This paper addresses peristaltic flow induced in a non-axisymmetric annular tube by a periodic small-amplitude wave of arbitrary shape propagating axially along its inner surface, assumed to be a circular cylinder. The study is motivated by recent in vivo experimental observations pertaining to the flow of cerebrospinal fluid along the perivascular spaces of cerebral arteries. The analysis employs the lubrication approximation, describing low-Reynolds-number peristaltic flow in the long-wavelength approximation. Closed-form analytic expressions are derived for the average pumping rate in infinitely long tubes and also in tubes of finite length. Consideration is also given to the transverse motion arising in non-axisymmetric tubes. For small-amplitude waves, the solution is reduced to the integration of a parameter-free Stokes-flow problem, which is solved for relevant cross-sectional shapes, with closed-form analytical results derived for thin canals.

A study on multi-GNSS phase-only positioning

The global navigation satellite systems (GNSS) carrier phase measurements form the basis of high-precision satellite positioning. These measurements are often accompanied by their code counterparts to enable one to compute single-epoch ambiguity-resolved positioning solutions. To avoid unwanted code modelling errors, such as code multipath, one may opt for a phase-only solution and take recourse to carrier phase measurements of two successive epochs. In this paper we study the ambiguity resolution performance of a dual-epoch phase-only model, upon which the unknown positioning parameters are assumed to be completely unlinked in time. With the aid of closed-form analytical results, it is investigated how ambiguity resolution performs when dealing with high-rate phase data. It is thereby shown that multi-GNSS integration makes near real-time centimetre-level phase-only positioning possible. Our analytical analysis is supported by means of numerical results.

Endless elastic beam travelling on a moving rough surface with zones of stick and sliding

We study the planar deformation of a beam that travels across a given control domain supported by a moving rough plane, which is a prototype for various technological processes. A sufficiently small misalignment between the guideways at the ends of the domain results in a stationary regime of motion, which features a zone of sticking contact near the entry to the control domain, followed by infinitely many segments of transverse sliding with alternating directions. Self-similarity of this solution of an essentially nonlinear boundary value problem is the primary novel result of the present contribution. Closed-form analytic results are validated against a finite element simulation of the transient evolution process, which demonstrates stability of the obtained solution and provides insights regarding the characteristic time scales of establishing of subsequent zones of sliding.

Precoder and Spatial Compression Filter Designs for Uplink Cloud Radio Access Networks

This work considers an uplink cloud radio access network (C-RAN), and proposes a MIMO spatial compression filtering and which compressed symbols to upload at RRH side as well as MIMO precoding at UE (user equipment) side. The error rate performance of this proposed scheme is analyzed and closed-form analytical results are obtained. Based on the analytical results, we propose solutions for the spatial compression filter, compressed symbols and UE precoder. Fronthaul loading can be significantly reduced by the proposed MIMO spatial compression filter and compressed symbols; meanwhile this proposed scheme can greatly outperform conventional schemes under the same settings of fronthaul loading. Moreover, when the UE has multiple antennas, the performance can further be significantly improved by using the proposed UE precoder. Simulation results are provided to show the accuracy of the analytical results as well as the advantages of the proposed solutions in terms of error rate performance and reduced fronthaul loading.

Advanced User Association in Non-Orthogonal Multiple Access-Based Fog Radio Access Networks

Non-orthogonal multiple access (NOMA) is promising to further improve spectral efficiency (SE) and decrease transmit latency in fog radio access networks (F-RANs) through serving multi-users in the same frequency-time resource block simultaneously, while the complexity of user association is challenging to exploit the corresponding performance gains. In this paper, a performance analysis framework for the user association in NOMA based F-RANs is proposed and the closed-form analytical results are developed by using stochastic geometry tool. In particular, we propose two user association algorithms based on evolutionary game and reinforcement learning, respectively. The performance model jointly considering quality of service, delay cost, and power consumption is formulated as a payoff function, and the corresponding performance expressions are derived for these two user association algorithms. Numerical and simulation results demonstrate that the derived expressions are accurate, and the NOMA based F-RAN can provide over 50% performance gains on SE compared to the orthogonal multiple access scheme. Furthermore, these two proposed user association algorithms work well with high convergence, which can effectively enhance the overall performance and the fairness of users.

Exact Solutions to the Quantum Rabi-Stark Model Within Tunable Coherent States**Supported by the National Science Foundation of China under Grant Nos. 11674285 and 11834005

The quantum Rabi-Stark model, where the linear dipole coupling and the nonlinear Stark-like coupling is present on an equal footing, is studied within the tunable extended coherent states. The eigenvalues and eigenstates are therefore obtained exactly. Surprisingly, the entanglement entropy in the ground-state is found to jump suddenly with the coupling strength. The first-order quantum phase transition can be detected by level crossing of the ground state and the first excited state, which is however lacking in the original linear quantum Rabi model. Performing the first-order approximation in the present theory, we can derive closed-form analytical results for the ground-state. Interestingly, it agrees well with the exact solutions up to the ultra-strong coupling regime in a wide range of model parameters. The spectral collapses when the absolute value of the nonlinear coupling strength approaches to twice the cavity frequency is observed with the help of new solutions in the limits.

Purcell enhancement of the parametric down-conversion in two-dimensional nonlinear materials

Ultracompact nonlinear optical devices utilizing two-dimensional (2D) materials and nanostructures are emerging as important elements of photonic circuits. Integration of the nonlinear material into a subwavelength cavity or waveguide leads to a strong Purcell enhancement of the nonlinear processes and compensates for a small interaction volume. The generic feature of such devices which makes them especially challenging for analysis is strong dissipation of both the nonlinear polarization and highly confined modes of a subwavelength cavity. Here we solve a quantum-electrodynamic problem of the spontaneous and stimulated parametric down-conversion in a nonlinear quasi-2D waveguide or cavity. We develop a rigorous Heisenberg-Langevin approach which includes dissipation and fluctuations in the electron ensemble and in the electromagnetic field of a cavity on equal footing. Within a relatively simple model, we take into account the nonlinear coupling of the quantized cavity modes, their interaction with a dissipative reservoir and the outside world, amplification of thermal noise and zero-point fluctuations of the electromagnetic field, and other relevant effects. We derive closed-form analytic results for relevant quantities such as the spontaneous parametric signal power and the threshold for parametric instability. We find a strong reduction in the parametric instability threshold for 2D nonlinear materials in a subwavelength cavity and provide a comparison with conventional nonlinear photonic devices.

Adaptive Multiuser Scheduling for Simultaneous Wireless Information and Power Transfer in a Multicell Environment

Simultaneous wireless information and power transfer (SWIPT) is a promising technique to transmit information and harvest energy at the same time. In this paper, we investigate multiuser scheduling criteria in a multiuser SWIPT system taking into account co-channel interference in a multicell environment, where each transmitter operate independently. We propose an $\alpha $ -adaptive scheduling scheme, which is able to adjust a scheduling criterion between maximizing achievable rate and maximizing harvested energy of the scheduled device by adjusting $\alpha $ factor. Employing a power splitting receiver, we derive closed-form analytical results including cumulative distribution function (CDF) of the harvested energy, CDF of the achievable rate, average harvested energy, and average achievable rate. We also investigate the effect of adjustable $\alpha $ factor in the proposed scheduling scheme and multiuser diversity in terms of the CDF of the harvested energy, the CDF of the achievable rate, and rate-energy tradeoff, compared with two conventional scheduling schemes: random scheduling and max-SNR scheduling. Our analytical results provide insightful information for designing multiuser scheduling criteria considering co-channel interference in the SWIPT systems.

What is an integrable quench?

Inspired by classical results in integrable boundary quantum field theory, we propose a definition of integrable initial states for quantum quenches in lattice models. They are defined as the states which are annihilated by all local conserved charges that are odd under space reflection. We show that this class includes the states which can be related to integrable boundary conditions in an appropriate rotated channel, in loose analogy with the picture in quantum field theory. Furthermore, we provide an efficient method to test integrability of given initial states. We revisit the recent literature of global quenches in several models and show that, in all of the cases where closed-form analytical results could be obtained, the initial state is integrable according to our definition. In the prototypical example of the XXZ spin-s chains we show that integrable states include two-site product states but also larger families of matrix product states with arbitrary bond dimension. We argue that our results could be practically useful for the study of quantum quenches in generic integrable models.