Large Time Approximation Research Articles

Inconsistencies in, and short pathlength correction to, RAA(pT)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{AA}(p_T)$$\\end{document} in A+A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{A}+\ extrm{A}$$\\end{document} and p+A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{p} + \ extrm{A}$$\\end{document} collisions

We present the first leading hadron suppression predictions in Pb+Pb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pb}+\ extrm{Pb}$$\\end{document} and p+Pb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{p}+\ extrm{Pb}$$\\end{document} collisions from a convolved radiative and collisional energy loss model in which partons propagate through a realistic background and in which the radiative energy loss receives a short pathlength correction. We find that the short pathlength correction is small for D and B meson RAA(pT)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{AA}(p_T)$$\\end{document} in both Pb+Pb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pb}+\ extrm{Pb}$$\\end{document} and p+Pb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{p}+\ extrm{Pb}$$\\end{document} collisions. However the short pathlength correction leads to a surprisingly large reduction in suppression for π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} mesons in p+Pb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{p}+\ extrm{Pb}$$\\end{document} and even Pb+Pb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pb}+\ extrm{Pb}$$\\end{document} collisions. We systematically check the consistency of the assumptions used in the radiative energy loss derivation-such as collinearity, softness, and large formation time-with the final numerical model. While collinearity and softness are self-consistently satisfied in the final numerics, we find that the large formation time approximation breaks down at modest to high momenta pT≳30GeV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_T \\gtrsim 30~\ extrm{GeV}$$\\end{document}. We find that both the size of the small pathlength correction to RAA(pT)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{AA}(p_T)$$\\end{document} and the pT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_T$$\\end{document} at which the large formation time assumption breaks down are acutely sensitive to the chosen distribution of scattering centers in the plasma.

Induced entanglement by three-level laser in an open cavity coupled to thermal reservoir

In this study, we have investigated the squeezing and entanglement characteristics of the light produced by three-level atoms available in an open cavity and pumped to the top level by electron bombardment at constant rate. We have obtained steady-state solutions to the quantum Langevin equations for the cavity mode operators and the equations of evolution for the expectation values of the atomic operators by employing the large-time approximation approach. The squeezing characteristics, entanglement amplification, and normalized second-order correlation function of the cavity radiation are discussed using the resulting steady-state solutions. We have seen that the light generated by the three-level laser is in a squeezed state, and the squeezing occurs in the minus quadrature. It so turns out that the maximum quadrature squeezing of the two-mode cavity light is 37.5% for n̄=0, 36.5% for n̄=0.01, and 35.5% for n̄=0.02 below the vacuum-state level. We have also established that the impact of the thermal reservoir process results in increases in the mean and variance of the photon number. Additionally, it is discovered that quadrature squeezing and entanglement are increased by the existence of spontaneous emission. This indicates that there are actually more photons fluctuating in the cavity. It is found that the squeezing and entanglement in the two-mode light are directly related. As a result, an increase in the degree of squeezing directly leads to an increase in the degree of entanglement and vice versa. This shows that, whenever there is squeezing in the two-mode light; an entanglement exists in the system.

Fermionic and bosonic fluctuation-dissipation theorem from a deformed AdS holographic model at finite temperature and chemical potential

In this work we study fluctuations and dissipation of a string in a deformed anti-de Sitter (AdS) space at finite temperature and density. The deformed AdS space is a charged black hole solution of the Einstein–Maxwell–Dilaton action. In this background we take into account the backreaction on the horizon function from an exponential deformation of the AdS space. From this model we compute the admittance and study the influence of the temperature and the chemical potential on it. We calculate the two-point correlations functions, and the mean square displacement for bosonic and fermionic cases, from which we obtain the short and large time approximations. For the long time, we obtain a sub-diffusive regime sim log t. Combining the results from the admittance and the correlations functions we check the fluctuation-dissipation theorem for bosonic and fermionic systems.

Infinite Series Asymptotic Expansions for Decaying Solutions of Dissipative Differential Equations with Non-smooth Nonlinearity

We study the precise asymptotic behavior of a non-trivial solution that converges to zero, as time tends to infinity, of dissipative systems of nonlinear ordinary differential equations. The nonlinear term of the equations may not possess a Taylor series expansion about the origin. This absence technically cripples previous proofs in establishing an asymptotic expansion, as an infinite series, for such a decaying solution. In the current paper, we overcome this limitation and obtain an infinite series asymptotic expansion, as time goes to infinity. This series expansion provides large time approximations for the solution with the errors decaying exponentially at any given rates. The main idea is to shift the center of the Taylor expansions for the nonlinear term to a non-zero point. Such a point turns out to come from the non-trivial asymptotic behavior of the solution, which we prove by a new and simple method. Our result applies to different classes of non-linear equations that have not been dealt with previously.

Enhancing Steady-State Entanglement Generated by a Nondegenerate Three-Level Laser with Thermal Reservoir

We analyze a nondegenerate three-level cascade laser with an open cavity and coupled to a two-mode thermal reservoir employing the stochastic differential equations for atomic operators associated with the normal ordering. Applying the large-time approximation scheme, we obtain the solutions for the corresponding equations of evolution of the expectation values of atomic operators. Furthermore, employing the resulting solutions, we studied the photon as well as cavity atomic-state entanglement amplification of the cavity radiation.

Large-Time Asymptotics of Fundamental Solutions for Diffusion Equations in Periodic Media and its Application to Averaging-Theory Estimates

The diffusion equation is considered in an infinite 1-periodic medium. We find large-time approximations for its fundamental solution. The approximation precision has pointwise and integral estimates of orders $$ O\left({t}^{-\frac{d+j+1}{2}}\right) $$ and $$ O\left({t}^{-\frac{j+1}{2}}\right) $$ , j = 0, 1, …, respectively. The approximations are constructed on the base of the known fundamental solution of the averaged equation with constant coefficients, its derivatives, and solutions of a family of auxiliary problems on the periodicity cell. The family of problems on the cell is generated recurrently. These results are used to construct approximations of the operator exponential of the diffusion equation with precision estimates in operator norms in Lp-spaces, 1 ≤ p ≤ ∞. For the analogous equation in an e-periodic medium, where e is a small parameter, we obtain approximations of the operator exponential in Lp-operator norms for a fixed time with precision of order O(en), n = 1, 2, …

Three-Level Laser Coupled to Squeezed Vacuum Reservoir

We analyze the quantum properties of the light generated by a three-level laser with an open cavity and coupled to a two-mode squeezed vacuum reservoir via a single-port mirror. The three-level laser consists of three-level atoms available in an open cavity and pumped from the bottom to the top level by means of electron bombardment. Applying the large-time approximation scheme, we have obtained the steady-state solutions of the equations of evolution for the expectation values of the atomic operators and the quantum Langevin equations for the cavity mode operators. Using the resulting steady-state solutions, we have calculated the mean photon number, the variance of the photon number, and the quadrature variance for the two-mode cavity light. We have seen that the light generated by the three-level laser is in a squeezed state and the squeezing occurs in the plus quadrature. It so turns out that the maximum quadrature squeezing of the two-mode cavity light is 45:68% for M0 = 0:59 and N0 = 0:27 below the vacuum-state level. In addition, we have shown that the effect of the squeezed parameter is to increase the mean and variance of the photon number.

Switchback effect of holographic complexity in multiple-horizon black holes

In this paper, we use the “complexity equals action” (CA) conjecture to explore the switchback effect in the strongly-coupled quantum field theories with finite N and finite coupling effects. In the perspective of holography, this is equivalent to evaluating the CA complexity in a Vaidya geometry equipped with a light shockwave for a higher curvature gravitational theory. Based on the Noether charge formalism of Iyer and Wald, we obtain the slope of the complexity of formation in the small- and large-time approximations. By circuit analogy, we show that our results concur with the switchback effect of the quantum system. These results show that the switchback effect is a general feature of the CA complexity in stationary black holes and its existence is independent of the explicit gravitational theory as well as spacetime background. From the viewpoint of AdS/CFT, this also implies that the switchback effect is a general feature of the thermofield double state in the strongly-coupled quantum field systems with finite N and finite coupling effects. Moreover, we also illustrate that unlike the late-time complexity growth rate, the counterterm plays an important role in the study of the switchback effect.

Short-path-length corrections to Djordjevic-Gyulassy-Levai-Vitev energy loss

We compute the correction to the energy loss of a hard parton due to short separation distances between the creation of the particle and the in-medium scattering center that stimulates bremsstrahlung radiation, to first order in opacity. In deriving the result we make full use of the large formation time assumption, which results in a significant reduction of the number of diagrams contributing to the small separation distance correction. An asymptotic analysis of our small separation distance correction term finds that the correction dominates at large $\sim 100$ GeV parent parton energies; scales like $L$ with the size of the system for small $L$, but like $L^0$ at larger $L$; and breaks color triviality. An extensive numerical investigation of the correction term confirms the aforementioned analytic findings, reveals that the correction term does not go to zero for large $L$, finds that the correction is sensitive to the mass of the parent parton, and shows a crucial dependence of the energy loss on a proper treatment of the physics of separation distances on the order of the Debye screening length. However, an examination of the large formation time approximation shows that it is violated for much of the phase space of the emitted radiation, implying a need to investigate the sensitivity of jet quenching results from relaxing this approximation. Our result constitutes an important step toward understanding partonic energy loss in small colliding systems.

Two-Level Atom with Squeezed Light from Optical Parametric Oscillators

The dynamics of a coherently driven two-level atom with parametric amplifier and coupled to a vacuum reservoir is analyzed. The combination of the master equation and the quantum Langevin equation is presented to study the quantum properties of light. By using these equations, we have determined the time evolution of the expectation values of the cavity mode and atomic operators. Moreover, with the aid of these results, the correlation properties of noise operators, and the large-time approximation scheme, we calculate the mean photon number, power spectrum, second-order correlation function, and quadrature variances for the cavity-mode light and fluorescence. It is found that the half-width of the power spectrum for the fluorescent light in the presence of a parametric amplifier increases, while it decreases for the cavity-mode light. Moreover, we have found the probability for the atom to be in the upper level in the presence of a parametric amplifier.

On the large time approximation of the Navier–Stokes equations in $\mathbb{R}^n$ by Stokes flows

On the large time approximation of the Navier–Stokes equations in $\mathbb{R}^n$ by Stokes flows

Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle

We consider the Hamiltonian system consisting of scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subject to an external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. It is assumed that the charge density satisfies the Wiener condition which is a version of the "Fermi Golden Rule". We prove that in the large time approximation any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.

Large Time Approximation for Shearing Motions

Small- and large-amplitude oscillatory shear tests are widely used by experimentalists to measure, respectively, linear and nonlinear properties of viscoelastic materials. These tests are based on the quasi-static approximation according to which the strain varies sinusoidally with time after a number of loading cycles. Despite the extensive use of the quasi-static approximation in solid mechanics, few attempts have been made to justify rigorously such an approximation. The validity of the quasi-static approximation is studied here in the framework of the Mooney--Rivlin Kelvin--Voigt viscoelastic model by solving the equations of motion analytically. For a general nonlinear model, the quasi-static approximation is instead derived by means of a perturbation analysis.

Axisymmetric waves of small amplitude on the interface between two viscous fluids

Axisymmetric wave of small amplitude on the interface between two viscous fluids is studied. The amplitude of wave is solved as a function of time. Two single-fluid cases in which one of the fluids has negligible dynamic effect are considered. The small and large time approximations are obtained and compared with the exact solution. It is found that the small time approximation is valid for small wave numbers or small viscosity; generally the large time approximation is accurate only at large times. The characteristic of vorticity is affected by wave number and viscosity significantly.

Relation between heat kernel method and scattering spectral method

In the present paper, we establish the connection between the heat kernel method and the scattering spectral method in quantum field theory. First, we provide the relation between the heat kernel and the scattering phase shift. Then as applications, we derive a large-time approximation of the heat kernel through the low-energy approximation of the phase shift, and we also derive a high-energy expansion of the phase shift expressed with heat kernel coefficients. Finally, we compare the renormalization schemes in the heat kernel method with that in the scattering spectral method. Concretely, we calculate the first- and second-order contributions to the vacuum energy by heat-kernel and Feynman-diagram methods, respectively, and show the coincidence of the results. Especially, in the heat-kernel framework, we perform both dimensional and zeta-function regularization to calculate the vacuum energy, and the result shows that dimensional renormalization procedure works well in the heat kernel method.

Scattering of solitons for coupled wave-particle equations

We establish a long time soliton asymptotics for a nonlinear system of wave equation coupled to a charged particle. The coupled system has a six-dimensional manifold of soliton solutions. We show that in the large time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution to the free wave equation. It is assumed that the charge density satisfies Wiener condition which is a version of Fermi Golden Rule, and that the momenta of the charge distribution vanish up to the fourth order. The proof is based on a development of the general strategy introduced by Buslaev and Perelman: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.

Error analysis of large-time approximations for decay chains

Consider a radioactive decay chain X 1 → ⋯ → X n → and let N n ( t) be the amount of X n at time t. This paper establishes error bounds for large-time approximations to N n ( t) that include and generalize the transient equilibrium approximations and other known approximations. The error bounds allow one to find the range of t for which these approximations can be used with a given degree of precision.

A dual-well step drawdown method for the estimation of linear and non-linear flow parameters and wellbore skin factor in confined aquifer systems

Summary In this study a method based on dual-well step drawdown test (i.e. a combination of an aquifer and a well performance test) for the determination of hydrodynamic parameters (namely storage coefficient and hydraulic conductivity), mechanical wellbore finite thickness skin factor, non-linear wellbore and non-linear aquifer parameters in an homogeneous confined aquifer is presented in order to put together aquifer and well tests. The interpretation procedure is based on the application of superposition principle to a large time logarithmic approximation of the solution. The advantages of this method, that can be considered an extension of Jacob step-test (1947) and Cooper–Jacob approximation (1946) , are that: (I) it is possible to determine simultaneously aquifer and well properties in a single test; (II) the method is based on a large time approximation and it is therefore independent from wellbore storage; (III) if the well skin is absent, the aquifer parameters (storage coefficient and hydraulic conductivity) can be derived just from a single-well test; (IV) the interpretation procedure is easy to apply and robust and does not require any specific numeric code or software. The same procedure can be easily adapted to gas well testing. It is also shown that, even in the presence of linear and non-linear flow, skin effect and wellbore storage, the hydraulic conductivity (and not the storage coefficient) of the aquifer can be correctly estimated by the Cooper and Jacob (1946) method applied to a single-rate pumping test, using exclusively the large time drawdown data measured at the pumping well.

Approximate Solutions for Pressure Buildup During CO2 Injection in Brine Aquifers

If geo-sequestration of CO 2 is to be employed as a key emissions reduction method in the global effort to mitigate against climate change, simple yet robust screening of the risks of disposal in brine aquifers will be needed. There has been significant development of simple analytical and semi-analytical techniques to support screening analysis and performance assessment for potential carbon sequestration sites. These techniques have generally been used to estimate the size of CO 2 plumes for the purpose of leakage rate estimation. A common assumption is that both the fluids and the geological formation are incompressible. Consequently, calculation of pressure distribution requires the specification of an arbitrary radius of influence. In this article, a new similarity solution is derived using the method of matched asymptotic expansions. A large time approximation of this solution is then extended to account for inertial effects using the Forchheimer equation. By allowing for slight compressibility in the fluids and formation, the solutions improve on previous work by not requiring the specification of an arbitrary radius of influence. The validity of both solutions is explored by comparison with equivalent finite difference solutions, revealing that the new method can provide robust and mathematically rigorous solutions for screening level analysis, where numerical simulations may not be justified or cost effective.

Pressure buildup during CO2 injection in brine aquifers using the Forchheimer equation

AbstractIf geo-sequestration of CO2 is to be employed as a key emissions reduction method in the global effort to mitigate climate change, simple yet robust screening of the risks of disposal in brine aquifers will be needed. There has been significant development of simple analytical and semi-analytical techniques to support screening analysis and performance assessment for potential carbon sequestration sites. These techniques have generally been used to estimate the size of CO2 plumes for the purpose of leakage rate estimation. A common assumption has been that both the fluids and the geological formation are incompressible. Consequently, calculation of pressure distribution requires the specification of an arbitrary radius of influence. In this talk, a new similarity solution is derived using the method of matched asymptotic expansions. By allowing for slight compressibility in the fluids and formation, the solution improves on previous work by not requiring the specification of an arbitrary radius of influence. A large-time approximation of the solution is then extended to account for non-Darcy inertial effects using the Forchheimer equation. Both solutions are verified by comparison with finite difference solutions. The results show that inertial losses will often be comparable, and sometimes greater than, the viscous Darcy-like losses associated with the brine displacement, although this is strongly dependent on formation porosity and permeability.