In this paper, we consider the iterative algorithm for a boundary value problem of n -order fractional differential equation with mixed integral and multipoint boundary conditions. Using an iterative technique, we derive an existence result of the uniqueness of the positive solution, then construct the iterative scheme to approximate the positive solution of the equation, and further establish some numerical results on the estimation of the convergence rate and the approximation error.

The phenomena, molecular path in a liquid or a gas, fluctuating price stoke, fission and fusion, quantum field theory, relativistic wave motion, etc., are modeled through the nonlinear time fractional clannish random Walker’s parabolic equation, nonlinear time fractional SharmaTassoOlver equation, and the nonlinear space-time fractional KleinGordon equation. The fractional derivative is described in the sense of conformable derivative. From there, the G ′ / G, 1 / G -expansion method is found to be (...) ensuing, effective, and capable to provide functional solutions to nonlinear models concerning physical and engineering problems. In this study, an extension of the G ′ / G, 1 / G -expansion method has been introduced. This enhancement establishes broad-ranging and adequate fresh solutions. In addition, some existing solutions attainable in the literature also confirm the validity of the suggested extension. We believe that the extension might be added to the literature as a reliable and efficient technique to examine a wide variety of nonlinear fractional systems with parameters including solitary and periodic wave solutions to nonlinear FDEs. (shrink)

In this article, we make analysis of the implicit fractional differential equations involving integral boundary conditions associated with Stieltjes integral and its corresponding coupled system. We use some sufficient conditions to achieve the existence and uniqueness results for the given problems by applying the Banach contraction principle, Schaefer’s fixed point theorem, and Leray–Schauder result of the cone type. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability by using (...) the classical technique of functional analysis. At the end, the results are verified with the help of examples. (shrink)

This paper is to investigate the existence and uniqueness of solutions for an integral boundary value problem of new fractional differential equations with a sign-changed parameter in Banach spaces. The main used approach is a recent fixed point theorem of increasing Ψ − h, r -concave operators defined on ordered sets. In addition, we can present a monotone iterative scheme to approximate the unique solution. In the end, two simple examples are given to illustrate our main results.

This article concerns with the existence and uniqueness theory of solutions for sequential fractional differential system involving Caputo fractional derivatives of order 1

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This paper involves extended b − metric versions of a fractional differential equation, a system of fractional differential equations and two-dimensional linear Fredholm integral equations. By various given hypotheses, exciting results are established in the setting of an extended b − metric space. Thereafter, by making consequent use of the fixed point technique, short and simple proofs are obtained for solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional (...) linear Fredholm integral equation. (shrink)

In this paper, we investigate the numerical solution of the coupled fractional massive Thirring equation with the aid of He’s fractional complex transform. This study plays a significant aspect in the field of quantum physics, weakly nonlinear thrilling waves, and nonlinear optics. The main advantage of FCT is that it converts the fractional differential equation into its traditional parts and is also capable to handle the fractional order, whereas the homotopy perturbation method is employed (...) to tackle the nonlinear terms in the coupled fractional massive Thirring equation. An example is illustrated to present the efficiency and validity of the two-scale theory. The solutions are obtained in the form of series with simple and easy computations which confirm that the present approach is good in agreement and is easy to implement for such type of complex systems in science and engineering. (shrink)

To solve fractional delay differential equation systems, the Laguerre Wavelets Method is presented and coupled with the steps method in this article. Caputo fractional derivative is used in the proposed technique. The results show that the current procedure is accurate and reliable. Different nonlinear systems have been solved, and the results have been compared to the exact solution and different methods. Furthermore, it is clear from the figures that the LWM error converges quickly when compared to other (...) approaches. When compared with the exact solution to other approaches, it is clear that LWM is more accurate and gets closer to the exact solution faster. Moreover, on the basis of the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations. (shrink)

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. The fractional derivative is defined in Caputo manner. The proposed method is simple, effective, and straightforward as compared to other numerical techniques. To check the validity and accuracy of the proposed method, some illustrative examples are solved by using the present scenario. The obtained results have confirmed the greater accuracy than the modified Laguerre wavelet (...) method, the Chebyshev wavelet method, and the modified wavelet-based algorithm. Moreover, based on the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations. (shrink)

In this article, we solve nonlinear systems of third order KdV Equations and the systems of coupled Burgers equations in one and two dimensions with the help of two different methods. The suggested techniques in addition with Laplace transform and Atangana–Baleanu fractional derivative operator are implemented to solve four systems. The obtained results by implementing the proposed methods are compared with exact solution. The convergence of the method is successfully presented and mathematically proved. The results we get are compared (...) with exact solution through graphs and tables which confirms the effectiveness of the suggested techniques. In addition, the results obtained by employing the proposed approaches at different fractional orders are compared, confirming that as the value goes from fractional order to integer order, the result gets closer to the exact solution. Moreover, suggested techniques are interesting, easy, and highly accurate which confirm that these methods are suitable methods for solving any partial differential equations or systems of partial differential equations as well. (shrink)

Quantum mechanics posits that the wave function of a one-particle system evolves with time according to the Schrödinger equation, and furthermore has a square modulus that serves as a probability density function for the position of the particle. It is natural to wonder if this stochastic characterization of the particle's position can be framed as a univariate continuous Markov process, sometimes also called a classical diffusion process, whose temporal evolution is governed by the classically transparent equations of Langevin (...) and Fokker-Planck. It is shown here that this cannot generally be done in a consistent way, despite recent suggestions to the contrary. (shrink)

To construct fractional rogue waves, this paper first introduces a conformable fractional partial derivative. Based on the conformable fractional partial derivative and its properties, a fractional Schrödinger equation with Lax integrability is then derived and first- and second-order fractional rogue wave solutions of which are finally obtained. The obtained fractional rogue wave solutions possess translational coordination, providing, to some extent, the degree of freedom to adjust the position of the rogue waves on the (...) coordinate plane. It is shown that the obtained first- and second-order fractional rogue wave solutions are steeper than those of the corresponding NLS equation with integer-order derivatives. Besides, the time the second-order fractional rogue wave solution undergoes from the beginning to the end is also short. As for asymmetric fractional rogue waves with different backgrounds and amplitudes, this paper puts forward a way to construct them by modifying the obtained first- and second-order fractional rogue wave solutions. (shrink)

The demand of many scientific areas for the usage of fractional partial differential equations to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential (...) equations. The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation. Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference method and standard finite difference technique, which are popular in the literature for solving engineering problems. (shrink)

This study presents a modified simplest equation method to investigate some real and exact solutions of conformable time fractional Benjamin-Bona-Mahony equation and Chan-Hilliard equation. We use traveling wave transformation to obtain the results in the form of series solution. Some calculations are performed through Mathematica software to analyze the accuracy of this approach. Graphical representations are reported for more significant results at different fractional-order which demonstrates that this approach is very simple, adequate, and legitimate.

A new class of polynomials investigates the numerical solution of the fractional pantograph delay ordinary differential equations. These polynomials are equipped with an auxiliary unknown parameter a, which is obtained using the collocation and least-squares methods. In this study, the numerical solution of the fractional pantograph delay differential equation is displayed in the truncated series form. The upper bound of the solution as well as the error analysis and the rate of convergence theorem are also investigated in (...) this study. In five examples, the numerical results of the present method have been compared with other methods. For the first time, a -polynomials are used in this study to numerically solve delay equations, and accurate approximations have been displayed. (shrink)

Drinking kills a significant proportion of individuals every year, particularly in low-income communities. An impulsive differential equation system is used to explore the effectiveness of activated charcoal in detoxifying the body after methanol poisoning. Our impression of activated charcoal is shaped by the fractional dynamics of the problem, which leads to speedy and low-cost first aid. The adsorption capacity of activated charcoal is investigated using impulsive differential equations. The ABC fractional operator’s findings paint a more realistic image (...) of first aid in public and primary health centers, which can assist to reduce methanol poisoning deaths. Numerical simulations are provided using generalized Adams–Bashforth–Moulton method. (shrink)

A nonlocal equation of motion with damping is derived by means of a Mori-Zwanzig renormalization process. The treatment is analogous to that of Mori in deriving the Langevin equation. For the case of electrodynamics, a local approximation yields the Lorentz equation; a relativistic generalization gives the Lorentz-Dirac equation. No self-acceleration or self-mass difficulties occur in the classical treatment, although runaway solutions are not eliminated. The nonrelativistic quantum case does not exhibit runaways, however, provided one remains (...) within a weak damping approximation. The correspondence limit shows that a classical limit may be taken, again within the same approximation. (shrink)

In this paper, DSEK model with fractional derivatives of the Atangana-Baleanu Caputo is proposed. This paper gives a brief overview of the ABC fractional derivative and its attributes. Fixed point theory has been used to establish the uniqueness and existence of solutions for the fractional DSEK model. According to this theory, we will define two operators based on Lipschitzian and prove that they are contraction mapping and relatively compact. Ulam-Hyers stability theorem is implemented to prove the (...) class='Hi'>fractional DSEK model’s stability in Banach space. Also, fractional Euler’s numerical method is derived for initial value problems with ABC fractional derivative and implemented on fractional DSEK model. The symmetric properties contribute to determining the appropriate method for finding the correct solution to fractional differential equations. The numerical solutions generated using fractional Euler’s method have been plotted for different values of α where α ∈ 0,1 and different step sizes h. Result discussion will be given, describing the changes that occur due to the step size h. (shrink)

In this paper, we deal with an alternative analytical analysis of fractional-order partial differential equations with proportional delay, achieved by applying Yang decomposition method, where the fractional derivative is taken in Caputo sense. The suggested series results are discovered to quickly converge to an exact solution. The computation of three test problems of fractional-order with proportional delay partial differential equations was presented to confirm the validity and efficiency of suggested method. The system appears to be a very (...) dependable, effective, and powerful method for solving a variety of physical problems that arise in engineering and science. (shrink)

This paper presents the design, simulation, and experimental verification of the fractional-order multiscroll Lü chaotic system. We base them on op-amp-based approximations of fractional-order integrators and saturated series of nonlinear functions. The integrators are first-order active realizations tuned to reduce the inaccuracy of the frequency response. By an exponential curve fitting, we got a convenient design equation for realizing fractional-order integrators of orders from 0.1 to 0.95. The results include simulations in SPICE of the mathematical description (...) and the electronic implementation and experimental measurements that confirm them. Monte Carlo and sensitivity tests revealed a robust realization. Contrary to its passive counterparts, the suggested realizations significantly reduce design and implementation efforts by favoring resistors and capacitors with commercial values and reducing hardware requirements. (shrink)

This study employs a newly developed methodology called the variational homotopy perturbation transformation method to study fractional acoustic wave equations. The motivation for this study is to extend the variational homotopy perturbation technique to the variational homotopy perturbation transformation technique in the sense of the Yang–Caputo–Fabrizio operator. The suggested method demonstrated a straightforward and accurate technique for investigating fractional-order partial differential equations. The technique’s validity is demonstrated through the use of several illustrative instances. The obtained answers were found (...) to be extremely near to the precise solutions. Additionally, the proposed strategy achieves the best degree of accuracy. Indeed, the current technique can be seen as one of the analytic strategies for solving nonlinear fractional partial differential equations compared to other analytical techniques. (shrink)

Fractional calculus is nowadays an efficient tool in modelling many interesting nonlinear phenomena. This study investigates, in a novel way, the Ulam–Hyers and Ulam–Hyers–Rassias stability of differential equations with general conformable derivative. In our analysis, we employ some version of Banach fixed-point theory. In this way, we generalize several earlier interesting results. Two examples are given at the end to illustrate our results.

Equation in the published paper is incorrect. The divergence of a vector field is obviously a scalar field, even when this operator is acting on a D-dimensional space.

In this research, the He-Laplace algorithm is extended to generalized third order, time-fractional, Korteweg-de Vries models. In this algorithm, the Laplace transform is hybrid with homotopy perturbation and extended to highly nonlinear fractional KdVs, including potential and Burgers KdV models. Time-fractional derivatives are taken in Caputo sense throughout the manuscript. Convergence and error estimation are confirmed theoretically as well as numerically for the current model. Numerical convergence and error analysis is also performed by computing residual errors in (...) the entire fractional domain. Graphical illustrations show the effect of fractional parameter on the solution as 2D and 3D plots. Analysis reveals that the He-Laplace algorithm is an efficient approach for time-fractional models and can be used for other families of equations. (shrink)

Why might it be beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse (...) primes for the equation that immediately followed it. Students with relatively high math ability showed relational priming both with and without high perceptual similarity. Students with relatively low math ability also showed priming, but only when the structurally inverse equation pairs were supported by high perceptual similarity between numbers. Several additional experiments established boundary conditions on relational priming with fractions. These findings are interpreted in terms of componential processing of fractions in a relational multiplication context that takes advantage of their inherent connections to a multiplicative schema for whole numbers. (shrink)

In this paper, we investigate non-homogeneous wave equations in fractional space-time domains of space dimension _D_, \(0 and time dimension \(\beta\), \(0. We write the wave equations in terms of potential functions and non-zero source terms. For scalar source terms, the potential functions are also scalar functions, and for vector source terms, the potential functions are vector functions. We derived an expression for the wave to propagate from the source point to the observation point. The study shows that the (...) time for a wave to propagate from the source point to the observation point in a fractional space-time domain could be different from that in an integer order space-time domain. (shrink)

On the basis of the previous publications, a new fractional-order prey-predator model is set up. Firstly, we discuss the existence, uniqueness, and nonnegativity for the involved fractional-order prey-predator model. Secondly, by analyzing the characteristic equation of the considered fractional-order Lotka–Volterra model and regarding the delay as bifurcation variable, we set up a new sufficient criterion to guarantee the stability behavior and the appearance of Hopf bifurcation for the addressed fractional-order Lotka–Volterra system. Thirdly, we perform the (...) computer simulations with Matlab software to substantiate the rationalisation of the analysis conclusions. The obtained results play an important role in maintaining the balance of population in natural world. (shrink)

Memristor is the fourth basic electronic element discovered in addition to resistor, capacitor, and inductor. It is a nonlinear gadget with memory features which can be used for realizing chaotic, memory, neural network, and other similar circuits and systems. In this paper, a novel memristor-based fractional-order chaotic system is presented, and this chaotic system is taken as an example to analyze its dynamic characteristics. First, we used Adomian algorithm to solve the proposed fractional-order chaotic system and yield a (...) chaotic phase diagram. Then, we examined the Lyapunov exponent spectrum, bifurcation, SE complexity, and basin of attraction of this system. We used the resulting Lyapunov exponent to describe the state of the basin of attraction of this fractional-order chaotic system. As the local minimum point of Lyapunov exponential function is the stable point in phase space, when this stable point in phase space comes into the lowest region of the basin of attraction, the solution of the chaotic system is yielded. In the analysis, we yielded the solution of the system equation with the same method used to solve the local minimum of Lyapunov exponential function. Our system analysis also revealed the multistability of this system. (shrink)

Using the position as an independent variable, and time as the dependent variable, we derive the function $${\mathcal{P}}^{(\pm )}=\pm \sqrt{2m({\mathcal{H}}-{\mathcal{V}}(q))}$$, which generates the space evolution under the potential $${\mathcal{V}}(q)$$ and Hamiltonian $${\mathcal{H}}$$. No parametrization is used. Canonically conjugated variables are the time and minus the Hamiltonian ( $$-{\mathcal{H}}$$ ). While the classical dynamics do not change, the corresponding Quantum operator $${{{\hat{\mathcal P}}}}^{(\pm )}$$ naturally leads to a 1/2-fractional time evolution, consistent with a recent proposed space–time symmetric formalism of the (...) Quantum Mechanics. Using Dirac’s procedure, separation of variables is possible, and while the two-coupled position-independent Dirac equations depend on the 1/2-fractional derivative, the two-coupled time-independent Dirac equations lead to positive and negative shifts in the potential, proportional to the force. Both equations couple the (±) solutions of $${{{\hat{\mathcal P}}}}^{(\pm )}$$ and the kinetic energy $${\mathcal{K}}_{0}$$ (separation constant) is the coupling strength. Thus, we obtain a pair of coupled states for systems with finite forces, not necessarily stationary states. The potential shifts for the harmonic oscillator (HO) are $$\pm {\hbar {\omega}} /2$$, and the corresponding pair of states are coupled for $${\mathcal{K}}_{0}\ne 0$$. No time evolution is present for $${\mathcal{K}}_{0}=0$$, and the ground state with energy $${\hbar {\omega}} /2$$ is stable. For $${\mathcal{K}}_{0}>0$$, the ground state becomes coupled to the state with energy $$-{\hbar {\omega}} /2$$, and this coupling allows to describe higher excited states in the HO. Energy quantization of the HO leads to the quantization of $${\mathcal{K}}_{0}=k{\hbar {\omega}}$$ ( $$k=1,2,\ldots$$ ). For the one-dimensional Hydrogen atom, the potential shifts become imaginary and position-dependent. Decoupled case $${\mathcal{K}}_{0}=0$$ leads to plane-waves-like solutions at the threshold. Above the threshold ( $${\mathcal{K}}_{0}>0$$ ), we obtain a plane-wave-like solution, and for the bounded states ( $${\mathcal{K}}_{0}<0$$ ), the wave-function becomes similar to the exact solutions but squeezed closer to the nucleus. (shrink)

Relativistic geometrical action for a quantum particle in the superspace is analyzed from theoretical group point of view. To this end an alternative technique of quantization outlined by the authors in a previous work and that is based in the correct interpretation of the square root Hamiltonian, is used. The obtained spectrum of physical states and the Fock construction consist of Squeezed States which correspond to the representations with the lowest weights $\lambda=\frac{1}{4}$ and $\lambda=\frac{3}{4}$ with four possible (non-trivial) fractional (...) representations for the group decomposition of the spin structure. From the theory of semigroups the analytical representation of the radical operator in the superspace is constructed, the conserved currents are computed and a new relativistic wave equation is proposed and explicitly solved for the time-dependent case. The relation with the Relativistic Schrödinger equation and the Time-dependent Harmonic Oscillator is analyzed and discussed. (shrink)

The umbrella-term ‘executive functions’ includes various domain-general, goal-directed cognitive abilities responsible for behavioral self-regulation. The influential unity and diversity model of EF posits the existence of three correlated yet separable executive domains: inhibition, shifting and updating. These domains may be influenced by factors such as socioeconomic status and culture, possibly due to the way EF tasks are devised and to biased choice of stimuli, focusing on first-world testees. Here, we propose a FREE test battery that includes two open-access tasks for (...) each of the three abovementioned executive domains to allow latent variables to be obtained. The tasks were selected from those that have been shown to be representative of each domain, that are not copyrighted and do not require special hardware/software to be administered. These tasks were adapted for use in populations with varying SES/schooling levels by simplifying tasks/instructions and using easily recognized stimuli such as pictures. Items are answered verbally and tasks are self-paced to minimize interference from individual differences in psychomotor and perceptual speed, to better isolate executive from other cognitive abilities. We tested these tasks on 146 early adolescents of both sexes and varying SES, because this is the age group in which the executive domains of interest become distinguishable and in order to confirm that SES effects were minimized. Performance was determined by Rate Correct Scores, which consider speed-accuracy trade-offs. Scores were sensitive to the expected improvement in performance with age and rarely/inconsistently affected by sex and SES, as expected, with no floor or ceiling effects, or skewed distribution, thus suggesting their adequacy for diverse populations in these respects. Using structural equation modeling, evidence based on internal structure was obtained by replicating the three correlated-factor solution proposed by the authors of the model. We conclude that the FREE test battery, which is open access and described in detail, holds promise as a tool for research that can be adapted for a wide range of populations, as well as altered and/or complemented in coming studies. (shrink)

This work builds on the Volterra series formalism presented in Dreisigmeyer and Young to model nonconservative systems. Here we treat Lagrangians and actions as ‘time dependent’ Volterra series. We present a new family of kernels to be used in these Volterra series that allow us to derive a single retarded equation of motion using a variational principle.

The fluctuation-dissipation theorem is a central theorem in nonequilibrium statistical mechanics by which the evolution of velocity fluctuations of the Brownian particle under a fluctuating environment is intimately related to its dissipative behavior. This can be illuminated in particular by an example of Brownian motion in an ohmic environment where the dissipative effect can be accounted for by the first-order time derivative of the position. Here we explore the dynamics of the Brownian particle coupled to a supraohmic environment by considering (...) the motion of a charged particle interacting with the electromagnetic fluctuations at finite temperature. We also derive particle’s equation of motion, the Langevin equation, by minimizing the corresponding stochastic effective action, which is obtained with the method of Feynman-Vernon influence functional. The fluctuation-dissipation theorem is established from first principles. The backreaction on the charge is known in terms of electromagnetic self-force given by a third-order time derivative of the position, leading to the supraohmic dynamics. This self-force can be argued to be insignificant throughout the evolution when the charge barely moves. The stochastic force arising from the supraohmic environment is found to have both positive and negative correlations, and it drives the charge into a fluctuating motion. Although positive force correlations give rise to the growth of the velocity dispersion initially, its growth slows down when correlation turns negative, and finally halts, thus leading to the saturation of the velocity dispersion. The saturation mechanism in a supraohmic environment is found to be distinctly different from that in an ohmic environment. The comparison is discussed. (shrink)

Free and weakly interacting particles are described by a second-quantized nonlinear Schrödinger equation, or relativistic versions of it. They describe Gaussian random walks with collisions. By contrast, the fields of strongly interacting particles are governed by effective actions, whose extremum yields fractional field equations. Their particle orbits perform universal Lévy walks with heavy tails, in which rare events are much more frequent than in Gaussian random walks. Such rare events are observed in exceptionally strong windgusts, monster or rogue (...) waves, earthquakes, and financial crashes. While earthquakes may destroy entire cities, the latter have the potential of devastating entire economies. (shrink)

This article constitutes the new fixed point results of dynamic process D through FIC-integral contractions of the Ciric kind and investigates the said contraction to iterate a fixed point of set-valued mappings in the module of metric space. To do so, we use the dynamic process instead of the conventional Picard sequence. The main results are examined by tangible nontrivial examples which display the motivation for such investigation. The work is completed by giving an application to Liouville‐Caputo fractional differential (...) equations. (shrink)

Analytical models describing the motion of colloidal particles in given force fields are presented. In addition to local approaches, leading to well known master equations such as the Langevin and the Fokker–Planck equations, a global description based on path integration is reviewed. A new result is presented, showing that under very broad conditions, during its evolution a dissipative system tends to minimize its energy dissipation in such a way to keep constant the Hamiltonian time rate, equal to the difference (...) between the flux-based and the force-based Rayleigh dissipation functions. In fact, the Fokker–Planck equation can be interpreted as the Hamilton–Jacobi equation resulting from such minumum principle. At steady state, the Hamiltonian time rate is maximized, leading to a minimum resistance principle. In the unsteady case, we consider the relaxation to equilibrium of harmonic oscillators and the motion of a Brownian particle in shear flow, obtaining results that coincide with the solution of the Fokker–Planck and the Langevin equations. (shrink)

Can Brownian motion arise from a deterministic system of particles? This paper addresses this question by analysing the derivation of Brownian motion as the limit of a deterministic hard-spheres gas with Lanford’s theorem. In particular, we examine the role of the Boltzmann-Grad limit in the loss of memory of the deterministic system and compare this derivation and the derivation of Brownian motion with the Langevin equation.

We present a formal derivation of the Mino–Sasaki–Tanaka–Quinn–Wald (MSTQW) equation describing the self-force on a (semi-) classical relativistic point mass moving under the influence of quantized linear metric perturbations on a curved background space–time. The curvature of the space–time implies that the dynamics of the particle and the field is history-dependent and as such requires a non-equilibrium formalism to ensure the consistent evolution of both particle and field, viz., the worldline influence functional and the closed- time-path (CTP) coarse-grained effective (...) action. In the spirit of effective field theory, we regularize the formally divergent self-force by smearing the local part of the retarded Green’s function and employing a quasi-local expansion. We derive the MSTQW–Langevin equations describing the perturbations of the particle’s worldline about its semi-classical trajectory resulting from interactions with the quantum fluctuations of the linear metric perturbations. Finally, we demonstrate that the quantum fluctuations of the field could, in principle, leave imprints on gravitational waveforms expected to be observed by gravitational interferometers, thereby encoding information about tree-level perturbative quantum gravity. (shrink)

The Wigner distribution and its equation of motion in the scalar potential case are arrived at in an unusual way. This in turn suggests (a) a departure from the standard Wigner distribution treatment for a charged particle in a magnetic field and (b) a new approach to quantization of nonconservative systems. Suggestion (a) is found to be, like the standard treatment, in agreement with Schrödinger's equation but, unlike it, also satisfies local classical-type conservation laws and employs a distribution (...) which is gauge-invariant rather than merely gauge-covariant. Suggestion (b) gives a clear result only in the case of resistance proportional to velocity, when it agrees with the Schrödinger-Langevin equation; for other dissipative systems a fresh assumption is required, and a proposal in that direction is put forward. (shrink)

We review studies of an evolution operator ℒ for a discrete Langevin equation with a strongly hyperbolic classical dynamics and a Gaussian noise. The leading eigenvalue of ℒ yields a physically measurable property of the dynamical system, the escape rate from the repeller. The spectrum of the evolution operator ℒ in the weak noise limit can be computed in several ways. A method using a local matrix representation of the operator allows to push the corrections to the escape (...) rate up to order eight in the noise expansion parameter. These corrections then appear to form a divergent series. Actually, via a cumulant expansion, they relate to analogous divergent series for other quantities, the traces of the evolution operators ℒn. Using an integral representation of the evolution operator ℒ, we then investigate the high order corrections to the latter traces. Their asymptotic behavior is found to be controlled by sub-dominant saddle points previously neglected in the perturbative expansion, and to be ultimately described by a kind of trace formula. (shrink)

We present a framework for analyzing black hole backreaction from the point of view of quantum open systems using influence functional formalism. We focus on the model of a black hole described by a radially perturbed quasi-static metric and Hawking radiation by a conformally coupled massless quantum scalar field. It is shown that the closed-time-path (CTP) effective action yields a non-local dissipation term as well as a stochastic noise term in the equation of motion, the Einstein–Langevin equation. (...) Once the thermal Green's function in a Schwarzschild background becomes available to the required accuracy, the strategy described here can be applied to obtain concrete results on backreaction. We also present an alternative derivation of the CTP effective action in terms of the Bogolyubov coefficients, thus making a connection with the interpretation of the noise term as measuring the difference in particle production in alternative histories. (shrink)