Liouville Equation Research Articles

Spectral parameter power series method for Kurzweil–Henstock integrable functions

<p>In this paper, the convergence of the spectral parameter power series method, proposed by Kravchenko, is performed for the Sturm–Liouville equation with Kurzweil–Henstock integrable coefficients. Numerical simulations of some examples are also presented to validate the performance of the method.</p>

Control of electronic and exchange coupling by bridge substituents in donor acceptor triads with triptycene bridges.

A series of triads, consisting of a triarylamine electron donor and a perylene diimide electron acceptor which were attached to two different wings of a triptycene bridging unit, was investigated concerning the dynamics of photoinduced charge separation and charge recombination processes with a particular focus on the involved spin-chemical aspects. Attaching electron-donating or electron-withdrawing substituents to the third wing of the triptycene bridge allowed tuning the electron transfer processes. These processes were investigated via fs-transient absorption spectroscopy and ns-transient absorption spectroscopy in an external magnetic field. The resulting magnetic field-dependent decay dynamics were analysed and modelled using the stochastic Liouville equation which yielded rate constants for the charge recombination and the exchange energy. In combination with a diabatic rate theory and Anderson's perturbative treatment of the exchange energy, these data gave a complete set of rate constants for charge separation and charge recombination from which the diverse electronic couplings between the involved states were derived. These couplings depend linearly on the inverse energy of virtual triptycene bridge states which allows tuning the electron transfer dynamics by modifying the triptycene bridge.

Neumann series of Bessel functions for inverse coefficient problems

Consider the Sturm–Liouville equation with a real‐valued potential . Let be its solution satisfying certain initial conditions for a number of , where , and are some complex numbers. Denote , where . We propose a method for solving the inverse problem of the approximate recovery of the potential and number from the following data . In general, the problem is ill‐posed; however, it finds numerous practical applications. Such inverse problems as the recovery of the potential from a Weyl function or the inverse two‐spectra Sturm–Liouville problem are its special cases. Moreover, the inverse problem of determining the shape of a human vocal tract also reduces to the considered inverse problem. The proposed method is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid the problem is reduced to the classical inverse Sturm–Liouville problem of recovering from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples.

Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions

The inverse spectral problem method is used to integrate the nonlinear Liouville equation in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution of the nonlinear Liouville equation is introduced. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and using the first trace formula satisfies the Liouville equation.

Quantum simulation for partial differential equations with physical boundary or interface conditions

This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretization of such problems does not necessarily yield Hamiltonian dynamics and even alters the Hamiltonian structure of the dynamics when boundary and interface conditions are included. This seemingly intractable issue can be resolved by using a recently introduced Schrödingerisation method [1,2] – it converts any linear PDEs and ODEs with non-Hermitian dynamics to a system of Schrödinger equations, via the so-called warped phase transformation that maps the equation into one higher dimension. We implement this method for several typical problems, including the linear convection equation with inflow boundary conditions and the heat equation with Dirichlet and Neumann boundary conditions. For interface problems we study the (parabolic) Stefan problem, linear convection, and linear Liouville equations with discontinuous and even measure-valued coefficients. We perform numerical experiments to demonstrate the validity of this approach, which helps to bridge the gap between available quantum algorithms and computational models for classical and quantum dynamics with boundary and interface conditions.

Ensemble fluid simulations on quantum computers

We discuss the viability of ensemble simulations of fluid flows on quantum computers. The basic idea is to formulate a functional Liouville equation for the probability distribution of the flow field configuration and recognize that, due to its linearity, such an equation is in principle more amenable to quantum computing than the dynamic equations of fluid motion. After suitable marginalization and associated closure, the Liouville approach is shown to require several hundreds of logical qubits, hence calling for a major thrust in current noise correction and mitigation techniques.

Integration of the Modified Korteweg–de Vries–Liouville Equation in the Class of Periodic Infinite-Gap Functions

Integration of the Modified Korteweg–de Vries–Liouville Equation in the Class of Periodic Infinite-Gap Functions

A general unsteady Fourier solution for orthotropic heat transfer in 2D functionally graded cylinders

In the present research, an analytical solution is presented for unsteady heat transfer in a two‐dimensional finite cylinder made of functionally graded Materials (FGM). The governing equation, as an unsteady partial differential equation (PDE), is solved with general boundary conditions (BCs). The thermal conductivity coefficients are considered as a power function of the radius, both in the longitudinal and radial directions of the FGM cylinder. The presence of non‐constant coefficients and general BCs will confront us with the challenge of solving an unsteady problem with high complexity. A combination of Laplace transform, Fourier transform, and meromorphic functions are utilized to deal with this complex PDE. The Laplace transform was applied to transform to the frequency domain, the Sturm–Liouville equation was used to extract the orthogonal basis Fourier series, and the meromorphic function was deployed to transform back from the frequency domain to the time domain. Results were well verified against previous research works. Results showed that the proposed analytical solution could well predict the temperature distribution. The current analytical solution is useful to better understand unsteady heat transfer in FGMs.

A hybrid semi-Lagrangian DG and ADER-DG solver on a moving mesh for Liouville's equation of geometrical optics

Liouville's equation describes light propagation through an optical system. It governs the evolution of an energy distribution, the basic luminance, on phase space. The basic luminance is discontinuous across optical interfaces, as is described by non-local boundary conditions at each optical interface. Curved optical interfaces manifest themselves as moving boundaries on phase space. A common situation is that an optical system is described by a piecewise constant refractive index field. Away from optical interfaces, the characteristics of Liouville's equation reduce to straight lines. This property is exploited in the novel solver developed in this paper by employing a semi-Lagrangian discontinuous Galerkin (SLDG) scheme away from optical interfaces. Close to optical interfaces we apply an ADER discontinuous Galerkin (ADER-DG) method on a moving mesh. The ADER-DG method is a fully discrete explicit scheme which must obey a CFL condition that restricts the stepsize, whereas the SLDG scheme can be CFL-free. A moving mesh is used to align optical interfaces with the mesh. Very small elements cannot always be avoided, even when applying mesh refinement. Local time stepping is introduced in the solver to ensure these very small elements only have a local impact on the stepsize. By construction we allow elements of SLDG type to run at a stepsize independent of these small elements. The proposed SLDG scheme uses the exact evolution of the solution, as is described by the characteristics, to update the numerical solution. We impose the condition that no characteristic emanating from an SLDG element can cross an optical interface. In the novel hybrid SLDG and ADER-DG solver this condition is used to naturally divide the spatio-temporal domain into different regions, describing where the SLDG scheme and where the ADER-DG scheme need to be used. An intermediate element is introduced to efficiently couple an SLDG region with an ADER-DG region. Numerical experiments validate that the hybrid solver obeys energy conservation up to machine precision and numerical convergence results show the expected order of convergence. The performance of the hybrid solver is compared to a pure ADER-DG scheme with global time stepping to show the efficiency of the hybrid solver. In particular, a speed-up of 1.6 to 10, in favour of the hybrid solver, for computation times up to 4 minutes was seen in an example. Finally, the hybrid solver and the pure ADER-DG scheme are compared to quasi-Monte Carlo ray tracing. In the examples considered, amongst the three methods the hybrid solver is shown to converge the fastest to high accuracies.

Ricci solitons on anisotropic fluid spacetime in f(R, T)-gravity

In this paper we study the behaviour of anisotropic fluid spacetimes in f(R, T)-gravity for admitting Ricci Soliton. We obtain the form of tangential pressure of the fluid for steady, shrinking and expanding Ricci soliton when Ricci soliton vector field becomes a Killing vector field. Modified Poisson and Liouville equations are derived from the Ricci soliton equation in f(R, T)-gravity having anisotropic fluid. Also harmonic significance of Ricci soliton on anisotropic fluid spacetime in f(R, T)-gravity with a harmonic potential function is derived. We derive field equations for perfect fluid plane symmetric static spacetime and obtain their solutions. For each solution we then obtain Ricci soliton vector fields. We observe that such spacetimes admit expanding, shrinking or steady Ricci solitons. All the obtained solitons are trivial in nature. When these Ricci soliton vector fields become Killing vector fields then our results are in full agreement with the theorems deduced from our study.

Precise eigenvalues in the solutions of generalized Sturm–Liouville problems

In the paper, we determine the eigenvalues and eigenfunctions of the generalized Sturm–Liouville problems, whose potentials may be nonlinear functions of eigen-parameter, by developing new iterative algorithms based on the fictitious time integration method (FTIM) and half-interval method (HIM). We derive two eigen-parameter dependent linear shape functions, from which we can transform the generalized Sturm–Liouville problem to an initial value problem for a new variable, and automatically preserve the prescribed eigen-parameter dependent Sturm–Liouville boundary conditions. A nonlinear equation in terms of the relative norm of two consecutive right-end values of the new variable is derived for iteratively determining the eigenvalue by using the FTIM. The resultant sequence of the iterated eigenvalues are monotonically convergent to the desired eigenvalue, and meanwhile the unknown initial values of the eigenfunction can be determined for computing the eigenfunction by integrating the Sturm–Liouville equation. We propose a high precision point target method by using the HIM to determine the eigenvalue and eigenfunction of the generalized Sturm–Liouville problem, which is transformed to a definite initial value problem with a simpler Dirichlet or Neumann boundary condition on the right-end as a point target equation. Several new theoretical results are proved such that more simple generalized Sturm–Liouville problem can be derived with a single point target on the right-end. Depending on different transformation techniques and the derived target equations, seven types FTIM and four types HIM are developed. Numerical examples confirm that the present FTIM and HIM are effective and accurate.

Memory effect & Carroll symmetry, 50 years later

Particles at rest before the arrival of a burst of gravitational wave move, after the wave has passed, with constant velocity along diverging geodesics. As recognized by Souriau 50 years ago and then forgotten, their motion is particularly simple in Baldwin–Jeffery–Rosen (BJR) coordinates (which are however defined only in coordinate patches): they are determined when the first integrals associated with the 5-parameter isometry group (recently identified as Lévy-Leblond’s “Carroll” group with broken rotations) are used. A global description can be given instead in terms of Brinkmann coordinates, however it requires to solve a Sturm–Liouville equation, whereas the relation between BJR and Brinkmann requires to solve yet another Sturm–Liouville equation. The theory is illustrated by geodesic motion in a linearly polarized (approximate) “sandwich” wave proposed by Gibbons and Hawking for gravitational collapse, and by circularly polarized approximate sandwich waves with Gaussian envelope.

Wigner Equations for Phonons Transport and Quantum Heat Flux

Starting from the quantum Liouville equation for the density operator and applying the Weyl quantization, Wigner equations for the acoustic, optical and Z phonons are deduced. The equations are valid for any solid, including 2D crystals like graphene. With the use of Moyal’s calculus and its properties, the pseudo-differential operators are expanded up to the second order in ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document}. An energy transport model is obtained by using the moment method with closure relations based on a quantum version of the Maximum Entropy Principle by employing a relaxation time approximation for the production terms of energy and energy flux. An explicit form of the thermal conductivity with quantum correction up to ħ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar ^2$$\\end{document} order is obtained under a long-time scaling for the most relevant phonon branches.

Deep ensemble geophysics-informed neural networks for the prediction of celestial pole offsets

SUMMARY Celestial Pole Offsets (CPO), denoted by dX and dY, describe the differences in the observed position of the pole in the celestial frame with respect to a certain precession-nutation model. Precession and nutation components are part of the transformation matrix between terrestrial and celestial systems. Therefore, various applications in geodetic science such as high-precision spacecraft navigation require information regrading precession and nutation. For this purpose, CPO can be added to the precession-nutation model to precisely describe the motion of the celestial pole. However, as Very Long Baseline Interferometry (VLBI)—currently the only technique providing CPO—requires long data processing times resulting in several weeks of latency, predictions of CPO become necessary. Here we present a new methodology named Deep Ensemble Geophysics-Informed Neural Networks (DEGINNs) to provide accurate CPO predictions. The methodology has three main elements: (1) deep ensemble learning to provide the prediction uncertainty; (2) broad-band Liouville equation as a geophysical constraint connecting the rotational dynamics of CPO to the atmospheric and oceanic Effective Angular Momentum (EAM) functions and (3) coupled oscillatory recurrent neural networks to model the sequential characteristics of CPO time-series, also capable of handling irregularly sampled time-series. To test the methodology, we use the newest version of the final CPO time-series of International Earth Rotation and Reference Systems Service (IERS), namely IERS 20 C04. We focus on a forecasting horizon of 90 days, the practical forecasting horizon needed in space-geodetic applications. Furthermore, for validation purposes we generate an independent global VLBI solution for CPO since 1984 up to the end of 2022 and analyse the series. We draw the following conclusions. First, the prediction performance of DEGINNs demonstrates up to 25 and 33 percent improvement, respectively, for dX and dY, with respect to the rapid data provided by IERS. Secondly, predictions made with the help of EAM are more accurate compared to those without EAM, thus providing a clue to the role of atmosphere and ocean on the excitation of CPO. Finally, free core nutation period shows temporal variations with a dominant periodicity of around one year, partially excited by EAM.

Heisenberg Spin Exchange Between Nitroxide Probes Diffusing in a Percolation Network

AbstractHeisenberg spin exchange between nitroxide (Tempone) spin probes has been measured as a function of concentration in the aqueous phase of the hydrated ion exchange membrane Nafion 117. The observed fast-motional electron paramagnetic resonance spectra were analyzed in terms of the stochastic Liouville equation lineshape calculation of Freed and coworkers and the “new paradigm” for interpreting spin exchange effects proposed by Salikhov. Differences between the effective spin exchange measured from the spectrum by these methods are presented and compared, and indicate that dipolar interactions make a significant contribution to spin exchange in this system. In acidic Nafion membranes, the spin probes are deactivated over time, allowing simultaneous measurement of the decay kinetics and spin exchange as a function of paramagnetic probe concentration. Both these processes deviate from the behavior that would be expected from classical chemical kinetics in isotropic media. The results are discussed in terms of currently available models for diffusion and reaction in a percolation network.

Some geometric inequalities related to Liouville equation

Some geometric inequalities related to Liouville equation

Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions

Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions

Constructing the Asymptotics of Solutions to Differential Sturm–Liouville Equations in Classes of Oscillating Coefficients

Constructing the Asymptotics of Solutions to Differential Sturm–Liouville Equations in Classes of Oscillating Coefficients

Simple blow-up solutions of singular Liouville equations

In a recent series of important works Wei-Zhang [Adv. Math. 380 (2021), Paper No. 107606, 45; Proc. Lond. Math. Soc. (3) 124 (2022), pp. 106–131; Laplacian vanishing theorem for quantized singular Liouville equation, Preprint, arXiv:2202.10825, 2022] proved several vanishing theorems for non-simple blow-up solutions of singular Liouville equations. It is well known that a non-simple blow-up situation happens when the spherical Harnack inequality is violated near a quantized singular source. In this article, we further strengthen the conclusions of Wei-Zhang by proving that if the spherical Harnack inequality does hold, there exist blow-up solutions with non-vanishing coefficient functions.

Nonuniqueness for the Nonlocal Liouville Equation in \(\mathbb{R}\) and Applications

Nonuniqueness for the Nonlocal Liouville Equation in \(\mathbb{R}\) and Applications