Approximation Of Path Integral Research Articles

Approximate solution to the photon migration problem in a turbid medium based on the path-integral approach

The functions characterizing the directional and spatial distribution of photons propagating through an infinite scattering medium can be expressed in the form of specific path-integrals. However, these are not solvable analytically and have to be approximated by a more suitable form, while we require the preservation of only a few relaxed constraints imposed on the photon transport. The path integral solutions, i.e. the sought distribution functions, are found in the form of simple closed-form expressions. They are compared with the standard diffusion solution and with the second-order cumulant approximation, which solves an equivalent problem. Photon path length profiles of the distribution functions obtained by the path-integral approximation and the cumulant approximation are almost identical, but the first formulae are much simpler. The numerical examples indicate that the path-integral solution could serve as an improvement of the diffusion equation solution under appropriate conditions.

A new algorithm for computing path integrals and weak approximation of SDEs inspired by large deviations and Malliavin calculus

The paper gives a novel path integral formula inspired by large deviation theory and Malliavin calculus. The proposed finite-dimensional approximation of integrals on path space will be a new higher-order weak approximation of multidimensional stochastic differential equations where the dominant part of the local expansion is governed by Varadhan's geodesic distance and the correction terms are given as Malliavin weights. An optimal truncation of asymptotic expansion is used to reduce computational effort. Kusuoka's estimate is applied to justify the finite-dimensional approximation of path integrals. An efficient simulation method is provided with the algorithm. Numerical results are shown to verify the effectiveness.

Path integral description and direct interaction approximation for elastic plate turbulence

In this work, we apply the Martin–Siggia–Rose path integral formalism to the equations of a thin elastic plate. Using a diagrammatic technique, we obtain the direct interaction approximation (DIA) equations to describe the evolutions of the correlation function and the response function of the fields. Consistent with previous results, we show that DIA equations for elastic plates can be derived from a non-markovian stochastic process and that in the weakly nonlinear limit, the DIA equations lead to the kinetic equation of wave turbulence theory. We expect that this approach will allow a better understanding of the statistical properties of wave turbulence and that DIA equations can open new avenues for understanding the breakdown of weakly nonlinear turbulence for elastic plates.

Imaginary-time open-chain path-integral approach for two-state time correlation functions and applications in charge transfer.

Quantum time correlation functions (TCFs) involving two states are important for describing nonadiabatic dynamical processes such as charge transfer (CT). Based on a previous single-state method, we propose an imaginary-time open-chain path-integral (OCPI) approach for evaluating the two-state symmetrized TCFs. Expressing the forward and backward propagation on different electronic potential energy surfaces as a complex-time path integral, we then transform the path variables to average and difference variables such that the integration over the difference variables up to the second order can be performed analytically. The resulting expression for the symmetrized TCF is equivalent to sampling the open-chain configurations in an effective potential that corresponds to the average surface. Using importance sampling over the extended OCPI space via open path-integral molecular dynamics, we tested the resulting path-integral approximation by calculating the Fermi's golden rule CT rate constant within a widely used spin-boson model. Comparing with the real-time linearized semiclassical method and analytical result, we show that the imaginary-time OCPI provides an accurate two-state symmetrized TCF and rate constant in the typical turnover region. It is shown that the first bead of the open chain corresponds to physical zero-time and that the endpoint bead corresponds to final time t; oscillations of the end-to-end distance perfectly match the nuclear mode frequency. The two-state OCPI scheme is seen to capture the tested model's electronic quantum coherence and nuclear quantum effects accurately.

A quadratic Wiener path integral approximation for stochastic response determination of multi-degree-of-freedom nonlinear systems

A Wiener path integral (WPI) technique is developed for determining the stochastic response of multi-degree-of-freedom (MDOF) nonlinear systems. Specifically, the nonlinear system response joint transition probability density function (PDF) is expressed as a WPI over the space of paths satisfying the initial and final conditions in time. Next, a functional series expansion is considered for the WPI and a quadratic approximation is employed. Further, relying on a variational principle yields a functional optimization problem to be solved for the most probable path, which is used for determining approximately the joint response transition PDF. It is shown that compared to the standard (semiclassical) WPI solution approach, which accounts only for the most probable path, the quadratic approximation developed herein exhibits enhanced accuracy. This is due to the fact that fluctuations around the most probable path are also accounted for by considering a localized state-dependent factor in the calculation of the WPI. Furthermore, the PDF normalization step of the most probable path approach is bypassed, and thus, probabilities of rare events (e.g., failures) can be determined in a direct manner without the need for obtaining the complete joint response PDF first. The herein developed technique can be construed as an extension of earlier efforts in the literature to account for MDOF systems. Several numerical examples are considered for demonstrating the accuracy of the technique. These pertain to various dynamical systems exhibiting diverse nonlinear behaviors. Comparisons with pertinent Monte Carlo simulation data are included as well.

Microcanonical action and the entropy of Hawking radiation

The island formula---an extremization prescription for generalized entropy---is known to result in a unitary Page curve for the entropy of Hawking radiation. This semiclassical entropy formula has been derived for Jackiw-Teitelboim (JT) gravity coupled to conformal matter using the ``replica trick'' to evaluate the Euclidean path integral. Alternatively, for eternal Anti--de Sitter black holes, we derive the extremization of generalized entropy from minimizing the microcanonical action of an entanglement wedge. The on-shell action is minus the entropy and arises in the saddle-point approximation of the (nonreplicated) microcanonical path integral. When the black hole is coupled to a bath, islands emerge from maximizing the entropy at fixed energy, consistent with the island formula. Our method applies to JT gravity as well as other two-dimensional dilaton gravity theories.

Path-integral approximations to quantum dynamics

Imaginary-time path-integral or ‘ring-polymer’ methods have been used to simulate quantum (Boltzmann) statistical properties since the 1980s. This article reviews the more recent extension of such methods to simulate quantum dynamics, summarising the chain of approximations that links practical path-integral methods, such as centroid molecular dynamics (CMD) and ring-polymer molecular dynamics (RPMD), to the exact quantum Kubo time-correlation function. We focus on single-surface Born–Oppenheimer dynamics, using the infrared spectrum of water as an illustrative example, but also survey other recent applications and practical techniques, as well as the limitations of current methods and their scope for future development.Graphic abstract

Computing the CEV option pricing formula using the semiclassical approximation of path integral

The CEV model allows volatility to change with the underlying price, capturing a basic empirical regularity very relevant for option pricing, such as the volatility smile. Nevertheless, the standard CEV solution, using the non-central chi-square approach, still presents high computational times. In this paper, the CEV option pricing formula is computed using the semiclassical approximation of Feynman’s path integral. Our simulations show that the method is quite efficient and accurate compared to the standard CEV solution considering the pricing of European call options.

Physics and Derivatives: Effective-Potential Path-Integral Approximations of Arrow-Debreu Densities

We show how effective-potential path-integrals methods, stemming on a simple and nice idea originally due to Feynman and successfully employed in Physics for a variety of quantum thermodynamics applications, can be used to develop an accurate and easy-to-compute semi-analytical approximation of transition probabilities and Arrow-Debreu densities for arbitrary diffusions. We illustrate the accuracy of the method by presenting results for the Black-Karasinski and the GARCH linear models, for which the proposed approximation provides remarkably accurate results, even in regimes of high volatility, and for multi-year time horizons. The accuracy and the computational efficiency of the proposed approximation makes it a viable alternative to fully numerical schemes for a variety of derivatives pricing applications.

A path-integral approximation for non-linear diffusions

Using the path-integral formalism we develop an accurate and easy-to-compute semi-analytical approximation to transition probabilities and Arrow–Debreu densities for arbitrary diffusions. We illustrate the accuracy of the method by presenting results for the Black–Karasinski model for which the proposed approximation provides remarkably accurate results, even in regimes of high volatility and for multi-year time horizons. The accuracy and the computational efficiency of the proposed approximation makes it a viable alternative to fully numerical schemes for a variety of applications, ranging from maximum-likelihood estimation in econometrics to derivatives pricing.

Maximum entropy searching

This study presents a new perspective for autonomous mobile robots path searching by proposing a biasing direction towards causal entropy maximisation during random tree generation. Maximum entropy-biased rapidly-exploring random tree (ME-RRT) is proposed where the searching direction is computed from random path sampling and path integral approximation, and the direction is incorporated into the existing rapidly-exploring random tree (RRT) planner. Properties of ME-RRT including degenerating conditions and additional time complexity are also discussed. The performance of the proposed approach is studied, and the results are compared with conventional RRT/RRT* and goal-biased approach in 2D/3D scenarios. Simulations show that trees are generated efficiently with fewer iteration numbers, and the success rate within limited iterations has been greatly improved in complex environments.

Path integral approach for Stark broadening of Lyman lines in hydrogen plasma

New results for Lyman lines from hydrogen plasmas are presented using the path integral approach. The influence of plasma components (electrons and ions) on the radiator is analysed separately. The ionic contribution is treated within the path integral approach, while the electronic contribution is estimated by the standard collision operator. The Stark effect, including the ion quadrupole contribution, is considered. The time‐dependent ionic microfield is treated within the path integral approximation using the model microfield method (MMM). The comparison with the quantum statistical approach is performed using a wide range of temperatures (T = 104–107 K) and electron densities (Ne = 1023–1026 m−3). Good agreement is mainly obtained for low density and high temperature.

Harmonic-phase path-integral approximation of thermal quantum correlation functions.

We present an approximation to the thermal symmetric form of the quantum time-correlation function in the standard position path-integral representation. By transforming to a sum-and-difference position representation and then Taylor-expanding the potential energy surface of the system to second order, the resulting expression provides a harmonic weighting function that approximately recovers the contribution of the phase to the time-correlation function. This method is readily implemented in a Monte Carlo sampling scheme and provides exact results for harmonic potentials (for both linear and non-linear operators) and near-quantitative results for anharmonic systems for low temperatures and times that are likely to be relevant to condensed phase experiments. This article focuses on one-dimensional examples to provide insights into convergence and sampling properties, and we also discuss how this approximation method may be extended to many-dimensional systems.

Dynamics of Fluctuactions in the BCS-BEC Crossover in Superfluid Fermi Gases

The BCS-BEC crossover phenomenon in a 3D attractive Fermi gas is revisited by using finite temperature Green functions formalism derived from the quantum path integral technique and Mean Field Approximation (MFA). Collective modes for weak and strong coupling particle-particle interaction g are obtained after recovering the dynamical coefficients in the nonlinear Time Dependent Ginzburg Landau (TDGL) equation, and numerically simulated under those particular states for the filling energy g ~ 0 and order parameter Δ(x,τ) ~ Δ0. Explicit formulas for the first order average fluctuations ⟨η2q,ω⟩ and the scaling law for the thermal energy associated to the non-linear collective modes are also reported.

Action-Amplitude Approach to Controlled Entropic Self-Organization

Motivated by the notion of perceptual error, as a core concept of the perceptual control theory, we propose an action-amplitude model for controlled entropic self-organization (CESO). We present several aspects of this development that illustrate its explanatory power: (i) a physical view of partition functions and path integrals, as well as entropy and phase transitions; (ii) a global view of functional compositions and commutative diagrams; (iii) a local geometric view of the Kähler–Ricci flow and time-evolution of entropic action; and (iv) a computational view using various path-integral approximations.

Quantum mechanics onSO(3)via noncommutative dual variables

We formulate quantum mechanics on SO(3) using a non-commutative dual space representation for the quantum states, inspired by recent work in quantum gravity. The new non-commutative variables have a clear connection to the corresponding classical variables, and our analysis confirms them as the natural phase space variables, both mathematically and physically. In particular, we derive the first order (Hamiltonian) path integral in terms of the non-commutative variables, as a formulation of the transition amplitudes alternative to that based on harmonic analysis. We find that the non-trivial phase space structure gives naturally rise to quantum corrections to the action for which we find a closed expression. We then study both the semi-classical approximation of the first order path integral and the example of a free particle on SO(3). On the basis of these results, we comment on the relevance of similar structures and methods for more complicated theories with group-based configuration spaces, such as Loop Quantum Gravity and Spin Foam models.

Canonical Quantization of Higher‐Order Lagrangians

After reducing a system of higher‐order regular Lagrangian into first‐order singular Lagrangian using constrained auxiliary description, the Hamilton‐Jacobi function is constructed. Besides, the quantization of the system is investigated using the canonical path integral approximation.

A new path integral approximation to photon propagation in turbid media

The path integral approach is applied to describe the propagation of photons in bounded homogeneous turbid media. Introducing a new controlled perturbative expansion centred on the diffusion limit, new analytic solutions for continuous wave reflectance and transmittance are determined, as relevant for non-invasive measurements of tissue optical properties based on continuous wave photon migration instruments. A comparative analysis with the diffusion approximation of radiative transfer theory is made to show that non-diffusive effects are enhanced in turbid regions with increasing absorption coefficients and decreasing reduced scattering coefficients as well as when the distance between interface boundaries increases. The relation between the transport equation (which is the starting point for the path integral approach) and multiple scattering theory is also discussed.

Comparison of ocean-acoustic horizontal coherence predicted by path-integral approximations and parabolic-equation simulation results

A line-integral approximation to the acoustic path integral has been used to generate predictions for the characteristic length scale of horizontal, cross-range coherence in long-range ocean-acoustic propagation. These estimates utilize a single range-independent sound-speed profile and the mean variance, as a function of depth, of fractional sound-speed perturbations due to internal waves. The length scales predicted by the integral approximation have been compared to the values generated by parabolic-equation simulations through multiple realizations of Garrett-Munk internal waves. One of the simulation environments approximates the Slice89 experiment; transmissions from a 250-Hz source were simulated in a deep-water transect to a maximum range of 1000km. The second environment corresponds to one of the propagation paths in the North Pacific Acoustic Laboratory (NPAL) experiment. The source in this experiment was bottom-mounted near Kauai, Hawaii and the relevant receiver consisted of five vertical line arrays oriented transverse to the propagation path with cross-range separations ranging from approximately 500 to 3500m. The receiver was at a range of 3889.8km from the source. The predicted length scales are consistently shorter than the parabolic-equation results by 30%–80%, depending on the range and environment examined.

Estimation of the electronic excitation cross sections of the 1s → 2s, 2p, 3s, 3p, and 3d transitions in the hydrogen atom on the basis of the Gaussian approximation of the Feynman integral

We continue to verify the method for calculating the scattering cross sections that is based on the Gaussian approximation of path integrals upon calculation of the electronic excitation cross sections. For this purpose, we calculate the electronic excitation cross sections of the 1s → 2s, 2p, 3s, 3p, and 3d transitions in the hydrogen atom using more exact Gaussian approximations for the Coulomb interaction and atomic states. The results show that this method allows one to obtain good agreement with data available in the literature. Comparison with our previous data shows that the use of the refined Gaussian expansions improves the calculation results.