Exact Equations Research Articles

Nonlinear petrophysical amplitude variation with offset inversion with spatially variable pore aspect ratio

Petrophysical amplitude variation with offset (AVO) inversion, which combines a rock-physics model and prestack inversion, aims at effectively quantifying reservoir-property parameters from prestack seismic data. However, for a reservoir with complex pore structures, this approach may suffer from difficulties in rock-physics modeling, which hinders solving the relevant inverse problem. We have proposed a petrophysical AVO inversion method that considers the complex pore structures within reservoir rocks. The method integrates a differential effective medium model, the Gassmann equation, and the exact Zoeppritz equation as the forward operator. In particular, the pore complexity is addressed by favoring the spatial variation of pore type. Differing from conventional sequential inversion methods, the pore-type variation is introduced by updating the aspect ratio together with the reservoir parameters during the inversion process. To enhance the stability of the results, the estimation of aspect ratio and reservoir parameters is jointly conditioned to seismic and elastic data under a Bayesian framework, which leads to the objective function with joint data misfit terms. The objective function is optimized with a simulated annealing algorithm that includes the full nonlinearity of the forward operator. The proposed method is demonstrated by testing and applying on a carbonate reservoir. It generally outperforms methods that are either purely conditioned on the seismic data or based on a constant aspect ratio.

Fixed-point quantum Monte Carlo method: A combination of density-matrix quantum Monte Carlo method and stochastic unravellings

We present an alternative approach to the study of equilibrium properties in many-body quantum physics. Our method takes inspiration from density-matrix quantum Monte Carlo in dealing with the sign problem and incorporates additional features. First of all, the dynamics is transferred to the Laplace representation where an exact equation can be derived and solved using an evolution step that, unlike most Monte Carlo methods, is not a priori physically bounded. Moreover, the spawning events are formulated in terms of two-process stochastic unravellings of quantum master equations, a formalism that is particularly useful when working with density matrices. Last, our approach is perturbative in nature, with the free part being integrated exactly and, as such, the convergence is greatly accelerated when the interaction parameter is small. We benchmark our method by applying it to two case studies in condensed-matter physics, perform some ground-state calculations, show its accuracy, and further discuss its efficiency.

P-wave reflectivity parameterization and nonlinear inversion in terms of Young’s modulus and Poisson ratio

Young’s modulus and Poisson ratio are essential parameters for rock brittleness evaluation. At present, prestack inversions of these parameters mostly adopt linear approximation, whereas the nonlinear inversion methods that are based on the exact Zoeppritz equation are seldomly conducted. Nevertheless, in the case of large incident angles and high-impedance contrast, the nonlinear equation has higher accuracy than the linear approximation. It is more in line with the nature of geophysical inversion problems, and it describes the variation of formations’ elastic parameters better than the linear approximation does. Therefore, based on the exact solution of the Zoeppritz equation, we have parameterized and derived a new nonlinear reflection coefficient equation using nonlinear Young’s modulus, Poisson ratio, and density (YPD) as variables. Subsequently, by combining the inverse operator algorithm with the YPD equation, we develop a new nonlinear amplitude-variation-with-offset inversion scheme. This approach can robustly extract Young’s modulus and Poisson ratio from seismic data and identify shale-gas sweet spot locations within the reservoir, as well as predict reservoir fluid properties. We have tested the accuracy and applicability of this approach with synthetic and real data examples.

One-component quantum mechanics and dynamical leakage-free paths

We derive an exact one-component equation of motion for the probability amplitude of a target time-dependent state, and use the equation to reformulate quantum dynamics and control for both closed and open systems. Using the one-component equation, we show that an unexpected time-dependent leakage-free path can be induced and we capture a necessary quantity in determining the effect of decoherence suppression. Our control protocol based on the nonperturbative leakage elimination operator provides a unified perspective connecting some subtle, popular, and important concepts of quantum control, such as dynamical decoupling, quantum Zeno effect, and adiabatic passage. The resultant one-component equation will promise significant advantages in both quantum dynamics and control.

Nonlinear amplitude inversion using a hybrid quantum genetic algorithm and the exact zoeppritz equation

The amplitude versus offset/angle (AVO/AVA) inversion which recovers elastic properties of subsurface media is an essential tool in oil and gas exploration. In general, the exact Zoeppritz equation has a relatively high accuracy in modelling the reflection coefficients. However, amplitude inversion based on it is highly nonlinear, thus, requires nonlinear inversion techniques like the genetic algorithm (GA) which has been widely applied in seismology. The quantum genetic algorithm (QGA) is a variant of the GA that enjoys the advantages of quantum computing, such as qubits and superposition of states. It, however, suffers from limitations in the areas of convergence rate and escaping local minima. To address these shortcomings, in this study, we propose a hybrid quantum genetic algorithm (HQGA) that combines a self-adaptive rotating strategy, and operations of quantum mutation and catastrophe. While the self-adaptive rotating strategy improves the flexibility and efficiency of a quantum rotating gate, the operations of quantum mutation and catastrophe enhance the local and global search abilities, respectively. Using the exact Zoeppritz equation, the HQGA was applied to both synthetic and field seismic data inversion and the results were compared to those of the GA and QGA. A number of the synthetic tests show that the HQGA requires fewer searches to converge to the global solution and the inversion results have generally higher accuracy. The application to field data reveals a good agreement between the inverted parameters and real logs.

Whirl dynamics of an axially functionally graded liquid-filled rotor considering shear deformation and rotary inertia

In this study, whirl characteristics and stability of an axially functionally graded (AFG) liquid-filled rotor are investigated. The rotor is modeled based on the spinning Timoshenko beam theory. The governing equations for flexural vibration are derived via Hamilton’s principle. For pinned–pinned AFG liquid-filled rotor, the analytical solutions are derived for both the exact whirl frequency equation and the stability model. To validate the present formulations, comparative studies by numerical solutions available in the literature are conducted. Some numerical examples are performed to investigate the effects of gradient parameter, mass ratio, cavity ratio, rotary inertia, and shear deformation on the whirl speed, the critical spinning speed, and the stability of the AFG liquid-filled rotor system. The results show that these parameters have noticeable influences on dynamic behavior and stability of the rotor system. In particular, the rotary inertia and shear deformation play an important role in the stability analysis for different length rotors.

A Cell Model of an Ion-Exchange Membrane. Capillary-Osmosis and Reverse-Osmosis Coefficients

The capillary-osmosis and reverse-osmosis coefficients of an ion-exchange membrane have been calculated as the kinetic coefficients of the Onsager matrix within the thermodynamics of nonequilibrium processes and on the basis of the cell model proposed previously by the author for charged porous layers. The membrane is assumed to consist of an ordered set of spherical completely porous charged particles placed into spherical shells filled with a binary electrolyte solution. Boundary value problems have been analytically solved to determine the capillary-osmosis and reverse-osmosis coefficients of such a membrane under the Kuwabara boundary condition imposed on the cell surface. The consideration has been implemented within the framework of small deviations of system parameters from their equilibrium values under the action of external fields. Different particular cases of the obtained exact analytical equations have been studied including the case of a binary symmetric electrolyte and an ideally selective membrane. It has been shown that, for the considered cell model of an ion-exchange membrane, the Onsager reciprocity theorem is violated; i.e., the found kinetic cross coefficients are unequal to each other. The violation is explained by the fact that the reciprocity theorem is valid only for systems implying the linear thermodynamics of irreversible processes, for which generalized flows are equal to zero at nonzero thermodynamic forces.

Exact quantum-mechanical equations for particle beams

The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the well-known paraxial equations in optics (exact Helmholtz equation for light beams) and particle physics and of the previously obtained paraxial equation for a Dirac particle in a uniform magnetic field. Some basic properties of exact wave eigenfunctions of particle beams have been determined.

Nonadiabatic vibronic effects in single-molecule junctions: A theoretical study using the hierarchical equations of motion approach

The interaction between electronic and vibrational degrees of freedom is an important mechanism in nonequilibrium charge transport through molecular nanojunctions. While adiabatic polaron-type coupling has been studied in great detail, new transport phenomena arise for nonadiabatic coupling scenarios corresponding to a breakdown of the Born-Oppenheimer approximation. Employing the numerically exact hierarchical equations of motion approach, we analyze the effect of nonadiabatic electronic-vibrational coupling on electron transport in molecular junctions considering a series of models with increasing complexity. The results reveal a significant influence of nonadiabatic coupling on the transport characteristics and a variety of interesting effects, including negative differential conductance. The underlying mechanisms are analyzed in detail.

EXACT SOLITON SOLUTIONS FOR CONFORMABLE FRACTIONAL SIX WAVE INTERACTION EQUATIONS BY THE ANSATZ METHOD

In this paper, a conformable fractional time derivative of order [Formula: see text] is considered in view of the Lax-pair of nonlinear operators to derive a fractional nonlinear evolution system of partial differential equations, called the Fractional-Six-Wave-Interaction-Equations, which is derived in terms of one temporal plus one and two spatial dimensions. Further, an ansatz consisting of linear combinations of hyperbolic functions with complex coefficients is utilized to obtain an infinite set of exact soliton solutions for this system. Certain numerical examples are introduced to show the effectiveness of the ansatz method in obtaining exact solutions for similar systems of nonlinear evolution equations.

Open-system approach to nonequilibrium quantum thermodynamics at arbitrary coupling

We develop a general theory describing the thermodynamical behavior of open quantum systems coupled to thermal baths beyond perturbation theory. Our approach is based on the exact time-local quantum master equation for the reduced open system states, and on a principle of minimal dissipation. This principle leads to a unique prescription for the decomposition of the master equation into a Hamiltonian part representing coherent time evolution and a dissipator part describing dissipation and decoherence. Employing this decomposition we demonstrate how to define work, heat, and entropy production, formulate the first and second law of thermodynamics, and establish the connection between violations of the second law and quantum non-Markovianity.

Nonperturbative renormalization of quantum thermodynamics from weak to strong couplings

By solving the exact master equation of open quantum systems, we formulate the quantum thermodynamics from weak to strong couplings. The open quantum systems exchange matters, energies and information with their reservoirs through quantum particles tunnelings that are described by the generalized Fano-Anderson Hamiltonians. We find that the exact solution of the reduced density matrix of these systems approaches a Gibbs-type state in the steady-state limit for both the weak and strong system-reservoir coupling strengths. When the couplings become strong, thermodynamic quantities of the system must be renormalized. The renormalization effects are obtained nonperturbatively after exactly traced over all reservoir states through the coherent state path integrals. The renormalized system Hamiltonian is characterized by the renormalized system energy levels and interactions, corresponding to the quantum work done by the system. The renormalized temperature is introduced to characterize the entropy production counting the heat transfer between the system and the reservoir. We further find that only with the renormalized system Hamiltonian and other renormalized thermodynamic quantities, can the exact steady state of the system be expressed as the standard Gibbs state. Consequently, the corresponding exact steady-state particle occupations in the renormalized system energy levels obey the Bose-Einstein and the Fermi-Dirac distributions for bosonic and fermionic systems, respectively. Thus, the conventional statistical mechanics and thermodynamics are thereby rigorously deduced from quantum dynamical evolution.

Approaches to Parameter Estimation from Model Neurons and Biological Neurons

Model optimization in neuroscience has focused on inferring intracellular parameters from time series observations of the membrane voltage and calcium concentrations. These parameters constitute the fingerprints of ion channel subtypes and may identify ion channel mutations from observed changes in electrical activity. A central question in neuroscience is whether computational methods may obtain ion channel parameters with sufficient consistency and accuracy to provide new information on the underlying biology. Finding single-valued solutions in particular, remains an outstanding theoretical challenge. This note reviews recent progress in the field. It first covers well-posed problems and describes the conditions that the model and data need to meet to warrant the recovery of all the original parameters—even in the presence of noise. The main challenge is model error, which reflects our lack of knowledge of exact equations. We report on strategies that have been partially successful at inferring the parameters of rodent and songbird neurons, when model error is sufficiently small for accurate predictions to be made irrespective of stimulation.

Alien suns reversing in exoplanet skies

Earth’s rapid spin, modest tilt, and nearly circular orbit ensure that the sun always appears to move forward, rising in the east and setting in the west. However, for some exoplanets, solar motion can reverse causing alien suns to apparently move backward. Indeed, this dramatic motion marginally occurs for Mercury in our own solar system. For exoplanetary observers, we study the scope of solar motion as a function of eccentricity, spin–orbit ratio, obliquity, and nodal longitude, and we visualize the motion in spatial and spacetime plots. For zero obliquity, reversals occur when a planet’s spin angular speed is between its maximum and minimum orbital angular speeds, and we derive exact nonlinear equations for eccentricity and spin–orbit to bound reversing and non-reversing motion. We generalize the notion of solar day to gracefully handle the most common reversals.

Optimal Resetting Brownian Bridges via Enhanced Fluctuations.

We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time t_{f} is finite and the searcher returns to its starting point at t_{f}. This is simply a Brownian motion with a Poissonian resetting rate r to the origin which is constrained to start and end at the origin at time t_{f}. We unveil a surprising general mechanism that enhances fluctuations of a Brownian bridge, by introducing a small amount of resetting. This is verified for different observables, such as the mean-square displacement, the hitting probability of a fixed target and the expected maximum. This mechanism, valid for a Brownian bridge in arbitrary dimensions, leads to a finite optimal resetting rate that minimizes the time to search a fixed target. The physical reason behind an optimal resetting rate in this case is entirely different from that of resetting Brownian motions without the bridge constraint. We also derive an exact effective Langevin equation that generates numerically the trajectories of a resetting Brownian bridge in all dimensions via a completely rejection-free algorithm.

Enhanced cooperation in multiplayer snowdrift games with random and dynamic groupings.

An analytically tractable generalization of the N-person snowdrift (NSG) game that illustrates how cooperation can be enhanced is proposed and studied. The number of players competing within a NSG varies from one time step to another. Exact equations governing the frequency of cooperation f_{c}(r) as a function of the cost-to-benefit ratio r within an imitation strategy updating scheme are presented. For group sizes g uniformly distributed within the range g∈[1,g_{m}], an analytic formula for the critical value r_{c}(g_{m}), below which the system evolves into a totally cooperative (AllC) state, is derived. In contrast, a fixed group size NSG does not support an AllC state. The result r_{c}(g_{m}) requires the presence of sole-player groups and involves the inverse of the harmonic numbers and, more generally, the inverse first moment of the group size distribution. For r>r_{c}(g_{m}), the equationthat determines the dynamical mixed states f_{c}(r) is given, with exact solutions existing for g_{m}≤5. The exact treatment allows the study of the phase boundary between the AllC state and the mixed states. The analytic results are checked against simulation results and exact agreements are demonstrated. The analytic form of the critical r_{c}(g_{m}) illustrates the necessity of having groups of a sole player in the evolutionary process. This result is supported by simulations with group sizes excluding the sole groups for which no AllC state emerges. A physically transparent picture of the importance of the sole players in inducing an AllC state is further presented based on the last surviving pattern before the AllC state is attained. The exact expression r_{c}(g_{m}) turns out to remain valid for nonuniform group-size distributions. Our analytical tractable generalization, therefore, sheds light on how a competing environment with variable group sizes could enhance cooperation and induce an AllC state.

Emergent pseudo time-irreversibility in the classical many-body system of pair interacting particles

In this paper, the emergence of macroscopic-scale pseudo time-irreversibility is studied in the closed classical many-body system of pair interacting particles. First, exact continuum equations are derived to the Hamiltonian dynamics without the utilisation of statistical mechanics. Next, it is shown that the momentum density field incorporates the thermal degrees of freedom in the considered scaling limit, and therefore initial condition indicated pseudo time-irreversible solutions may exist to the dynamical equations. Numerical evidence for pseudo-irreversible thermal equilibration, heat and momentum transport is provided. Finally, the possible reasons of the numerically obtained non-diffusional relaxation of macroscopic order is discussed.

Theoretical and numerical investigation of internal conical refraction of structured light beams.

We present an ab initio numerical investigation of the internal conical refraction of structured light beams in a biaxial crystal. Starting from the solutions of the Fresnel equation, a theoretical analysis is developed without assuming any analytical approximation, thus obtaining a set of exact equations that can be solved by standard methods of integration for any impinging light beam. As examples of applications, we consider the particular cases of linearly and circularly polarized Gaussian and Bessel beams inside a KTP crystal. Numerical calculations follow the evolution of the refracted beam inside the crystal. It is seen that for realistic boundary conditions, a refraction cone appears in a certain range of distances within the crystal, and its shape is rather sensitive to the initial conditions.

Connector theory for reusing model results to determine materials properties

The success of Density Functional Theory (DFT) is partly due to that of simple approximations, such as the Local Density Approximation (LDA), which uses results of a model, the homogeneous electron gas, to simulate exchange-correlation effects in real materials. We turn this intuitive approximation into a general and in principle exact theory by introducing the concept of a connector: a prescription how to use results of a model system in order to simulate a given quantity in a real system. In this framework, the LDA can be understood as one particular approximation for a connector that is designed to link the exchange-correlation potentials in the real material to that of the model. Formulating the in principle exact connector equations allows us to go beyond the LDA in a systematic way. Moreover, connector theory is not bound to DFT, and it suggests approximations also for other functionals and other observables. We explain why this very general approach is indeed a convenient starting point for approximations. We illustrate our purposes with simple but pertinent examples.

Functional renormalization group for multilinear disordered Langevin dynamics I Formalism and first numerical investigations at equilibrium

This paper aims at using the functional renormalization group formalism to study the equilibrium states of a stochastic process described by a quench–disordered multilinear Langevin equation. Such an equation characterizes the evolution of a time-dependent N-vector q(t) = {q 1(t), ⋯ q N (t)} and is traditionally encountered in the dynamical description of glassy systems at and out of equilibrium through the so-called Glauber model. From the connection between Langevin dynamics and quantum mechanics in imaginary time, we are able to coarse-grain the path integral of the problem in the Fourier modes, and to construct a renormalization group flow for effective Euclidean action. In the large N-limit we are able to solve the flow equations for both matrix and tensor disorder. The numerical solutions of the resulting exact flow equations are then investigated using standard local potential approximation, taking into account the quench disorder. In the case where the interaction is taken to be matricial, for finite N the flow equations are also solved. However, the case of finite N and taking into account the non-equilibrum process will be considered in a companion investigation.