Path Integral Techniques Research Articles

Path integral contour deformations for noisy observables

Monte Carlo studies of many quantum systems face exponentially severe signal-to-noise problems. We show that noise arising from complex phase fluctuations of observables can be reduced without introducing bias using path integral contour deformation techniques. A numerical study of contour deformations for correlation functions in Abelian gauge theory and complex scalar field theory demonstrates that variance can be reduced by orders of magnitude without modifying Monte Carlo sampling.

Distribution of the time between maximum and minimum of random walks.

We consider a one-dimensional Brownian motion of fixed duration T. Using a path-integral technique, we compute exactly the probability distribution of the difference τ=t_{min}-t_{max} between the time t_{min} of the global minimum and the time t_{max} of the global maximum. We extend this result to a Brownian bridge, i.e., a periodic Brownian motion of period T. In both cases, we compute analytically the first few moments of τ, as well as the covariance of t_{max} and t_{min}, showing that these times are anticorrelated. We demonstrate that the distribution of τ for Brownian motion is valid for discrete-time random walks with n steps and with a finite jump variance, in the limit n→∞. In the case of Lévy flights, which have a divergent jump variance, we numerically verify that the distribution of τ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "τ=n" is exactly 1/(2n) for any finite n, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of (1+1)-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size L. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.200201].

Functional Series Expansions and Quadratic Approximations for Enhancing the Accuracy of the Wiener Path Integral Technique

A novel Wiener path integral (WPI) technique is developed for determining the response of stochastically excited nonlinear oscillators. This is done by employing functional series expansions in conjunction with quadratic approximations. The technique can be construed as an extension and enhancement in terms of accuracy of the standard (semiclassical) WPI solution approach where only the most probable path connecting initial and final states is considered for determining the joint response probability density function (PDF). In contrast, the technique developed herein accounts also for fluctuations around the most probable path, thus yielding an increased accuracy degree. An additional significant advantage of the proposed enhancement as compared to the most probable path approach relates to the fact that low-probability events (e.g., failure probabilities) can be estimated directly in a computationally efficient manner by determining only a few points of the joint response PDF. Specifically, the normalization step in the standard approach, which requires the evaluation of the joint response PDF over its entire effective domain, is circumvented. The performance of the technique is assessed in several numerical examples pertaining to various oscillators exhibiting diverse nonlinear behaviors, where analytical results are set vis-à-vis pertinent Monte Carlo simulation data.

Inexpensive modeling of quantum dynamics using path integral generalized Langevin equation thermostats

The properties of molecules and materials containing light nuclei are affected by their quantum mechanical nature. Accurate modeling of these quantum nuclear effects requires computationally demanding path integral techniques. Considerable success has been achieved in reducing the cost of such simulations by using generalized Langevin dynamics to induce frequency-dependent fluctuations. Path integral generalized Langevin equation methods, however, have this far been limited to the study of static, thermodynamic properties due to the large perturbation to the system's dynamics induced by the aggressive thermostatting. Here, we introduce a post-processing scheme, based on analytical estimates of the dynamical perturbation induced by the generalized Langevin dynamics, which makes it possible to recover meaningful time correlation properties from a thermostatted trajectory. We show that this approach yields spectroscopic observables for model and realistic systems that have an accuracy comparable to much more demanding approximate quantum dynamics techniques based on full path integral simulations.

An exact closed-form solution for linear multi-degree-of-freedom systems under Gaussian white noise via the Wiener path integral technique

The exact joint response transition probability density function (PDF) of linear multi-degree-of-freedom oscillators under Gaussian white noise is derived in closed-form based on the Wiener path integral (WPI) technique. Specifically, in the majority of practical implementations of the WPI technique, only the first couple of terms are retained in the functional expansion of the path integral related stochastic action. The remaining terms are typically omitted since their evaluation exhibits considerable analytical and computational challenges. Obviously, this approximation affects, unavoidably, the accuracy degree of the technique. However, it is shown herein that, for the special case of linear systems, higher than second order variations in the path integral functional expansion vanish, and thus, retaining only the first term (most probable path approximation) yields the exact joint response transition Gaussian PDF. Both single- and multi-degree-of-freedom linear systems are considered as illustrative examples for demonstrating the exact nature of the derived solutions. In this regard, the herein derived analytical results are also compared with readily available in the literature closed-form exact solutions obtained by alternative stochastic dynamics techniques. In addition to the mathematical merit of the derived exact solution, the closed-form joint response transition PDF can also serve as a benchmark for assessing the performance of alternative numerical solution methodologies.

Thermodynamics of a generalized graphene-motivated (2+1) D Gross–Neveu model beyond the mean field within the Beth–Uhlenbeck approach

Abstract We investigate the thermodynamics at finite density of a generalized $(2 + 1)$D Gross–Neveu model of $N$ fermion species with various types of four-fermion interactions. The motivation for considering such a generalized schematic model arises from taking the Fierz transformation of an effective Coulomb current–current interaction and certain symmetry-breaking interaction terms, as considered for graphene-type models in Ref. [29]. We then apply path-integral bosonization techniques, based on the large-$N$ limit, to derive the thermodynamic potential. This includes the leading-order mean-field (saddle point) contribution as well as the next-order contribution of Gaussian fluctuations of exciton fields. The main focus of the paper is then the investigation of the thermodynamic properties of the resulting fermion–exciton plasma. In particular, we derive an extended Beth–Uhlenbeck form of the thermodynamic potential, and discuss the Levinson theorem and the decomposition of the phase of the exciton correlation into resonant and scattering parts.

Dirac oscillator in a uniform electric field: Path integral treatment

In this paper, the propagator of a two-dimensional Dirac oscillator in the presence of a uniform electric field is derived by using the path integral technique. The fact that the globally named approach is used in this work redirects, beforehand, our search for the propagator of the Dirac equation to that of the propagator of its quadratic form. The internal motions relative to the spin are represented by two fermionic oscillators, which are described by Grassmannian variables, according to Schwinger’s fermionic model. Once the integration over the anticommuting variables (Grassmannian variables) is accomplished, the problem becomes the one of finding a non-relativistic propagator with only bosonic variables. The energy spectrum of the electron and the corresponding eigenspinors are also obtained in this work.

Coupled oscillators in non-commutative phase space: Path integral approach

We present the path integral techniques in a non-commutative phase space and illustrate the calculation in the case of an exact problem of the coupled oscillator in two dimensions. The non-commutativity, with respect to Poisson (classical) and Heisenberg (quantum) brackets, in this phase space, is governed by two small constant parameters. They characterize the geometric deformation of this space. The path integral is formulated in a mixed representation due to the non-commutativity of the coordinates on one hand and those of the momentum on the other hand. Using a canonical linear transformation in this non-commutative phase space, we retrieve the commutative phase space properties by which the study becomes more suitable. The case of the non-commutative coupled harmonic oscillator in two dimensions is treated and the thermodynamics properties of the assembly of such oscillators are derived too.

Simulating schlieren and shadowgraph images from LES data

Geometrical optics ray-tracing is used to derive schlieren and shadowgraph images from large-eddy simulation (LES) data of a jet in supersonic crossflow and to compare with experimental data. Including the components of the optical system that forms the image in the simulation is found to be important. The technique produces images that replicate flow physics more faithfully than straight-line path integration and other techniques, and more efficiently than physical-optics techniques. Applications of these simulated images are demonstrated in supersonic flows. Time-correlated pairs of shadowgraph images taken from the LES using this technique are used in conjunction with an image-correlation velocimetry technique to compare the estimated convection velocity field in the LES to that of experiments of the same flow. Agreement between the two is good with a maximum variance of 5% by some metrics. This technique can aid in the validation of LES results, allowing quantitative comparison between experiment and simulation, and to extract information unattainable by experiment alone. Comparisons of simulated and experimental jet penetration into the supersonic freestream are also made.Graphic abstract

Nuclear Motion in the Intramolecular Dihydrogen-Bound Regime of an Aminoborane Complex.

The 1,3-diaza-2,4-diborobutane (NBNB) molecule serves as the smallest model complex of an intramolecular "dihydrogen bond," which involves a nominally hydrogen-bonding interaction between amine and borane hydrogen atoms. In the present study, the role of this dihydrogen bond in influencing the inherent molecular dynamics of NBNB is investigated computationally with ab initio molecular dynamics and path integral molecular dynamics techniques, as well as vibrational spectra analysis and static quantum chemistry computations. These simulations indicate that the dihydrogen-bonding interaction impacts both the high- and low-frequency motions of the molecule, with the dominant motions involving low-frequency backbone isomerization and terminal amine rotation. Geometric isotope effects were found to be modest. The analysis also addresses the paradoxical fostering of amine rotation via a relatively strong dihydrogen bond interaction. Electrostatic and geometric factors most directly explain this effect, and although some orbital evidence was found for a small covalent component of this interaction, the dynamics and electronic structure suggest that electrostatic contributions are the controlling factors for molecular motion in NBNB.

Zero- and high-pressure mechanisms in the complex forming reactions of OH with methanol and formaldehyde at low temperatures.

A recent Ring Polymer Molecular Dynamics study of the reactions of OH with methanol and formaldehyde, at zero pressure and below 100 K, has shown the formation of long lived complexes, with long lifetimes, longer than 100 ns for the lower temperatures studied, 20-100 K (del Mazo-Sevillano et al., 2019). These long lifetimes support the existence of multi collision events with the He buffer-gas atoms under experimental conditions, as suggested by several transition state theory studies of these reactions. In this work we study these secondary collisions, as a dynamical approach to study pressure effects on these reactions. For this purpose, the potential energy surfaces of He with H2CO, OH, H2O and HCO are calculated at highly accurate ab initio level. The stability of some of the complexes is studied using Path Integral Molecular dynamics techniques, determining that OH-H2CO complexes can be formed up to 100 K or higher temperatures, while the weaker He-H2CO complexes dissociate at approximately 50 K. The predicted IR intensity spectra shows new features which could help the identification of the OH-H2CO complex. Finally, the He-H2CO + OH and OH-H2CO + He collisions are studied using quassi-classical trajectories, finding that the cross section to produce HCO + H2O products increases with decreasing collision energy, and that it is ten times higher in the He-H2CO + OH case.

Work fluctuation relations for a dragged Brownian particle in active bath

We study the work distribution of a Brownian particle diffusing in an environment of active particles and being trapped in a harmonic potential, the center of which is subjected to a time-dependent protocol. Employing phase space path integral technique we find an expression of work distribution for any generic model of active noise. Here we consider two active noise models — Gaussian correlated and Poisson white, each of which can represent some physical systems. For both the cases, it is found that transient fluctuation relation of work is not applicable though at steady state it holds by defining a renormalized temperature τr in place of bath temperature. Interestingly, τr is the same for both the models and can be expressed in terms of diffusivities of active and thermal noises. For correlated Gaussian bath, an alternative approach is presented. Analogous to the formalism given by Hatano and Sasa (2001), we obtain a work like quantity from nonequilibrium potential with the inclusion of a new stationary parameter Ω. With proper choice of Ω, a steady-state fluctuation relation, namely Jarzynski equality is satisfied.

On the importance of anharmonicities and nuclear quantum effects in modelling the structural properties and thermal expansion of MOF-5

In this article, we investigate the influence of anharmonicities and nuclear quantum effects (NQEs) in modelling the structural properties and thermal expansion of the empty MOF-5 metal-organic framework. To introduce NQEs in classical molecular dynamics simulations, two different methodologies are considered, comparing the approximate, but computationally cheap, method of generalised Langevin equation thermostatting to the more advanced, computationally demanding path integral molecular dynamics technique. For both methodologies, similar results were obtained for all the properties under investigation. The structural properties of MOF-5, probed by means of radial distribution functions (RDFs), show some distinct differences with respect to a classical description. Besides a broadening of the RDF peaks under the influence of quantum fluctuations, a different temperature dependence is also observed due to a dominant zero-point energy (ZPE) contribution. For the thermal expansion of MOF-5, by contrast, NQEs appear to be only of secondary importance with respect to an adequate modelling of the anharmonicities of the potential energy surface (PES), as demonstrated by the use of two differently parametrised force fields. Despite the small effect in the temperature dependence of the volume of MOF-5, NQEs do however significantly affect the absolute volume of MOF-5, in which the ZPE resulting from the intertwining of NQEs and anharmonicities plays a crucial role. A sufficiently accurate description of the PES is therefore prerequisite when modelling NQEs.

Stochastic dynamics of a nonlinear time-varying spur gear model using an adaptive time-stepping path integration method

This study is purposed to enhance conventional spur gear models by moving from a deterministic to a stochastic representation, based on adding a Gaussian white noise in the loading terms while including the effect of both backlash and time-varying mesh stiffness. Meanwhile, an efficient numerical path integration (PI) technique is used to obtain the probability distribution functions (PDFs) of the response. A novel adaptive time-stepping scheme is proposed for the PI method that momentarily increases its accuracy in order to detect the edge of transition when dealing with the non-smoothness due to backlash. The results demonstrate good agreement with Monte Carlo simulation (MCS). It is observed that if the initial distribution covers a region containing the basins of attraction of coexisting solutions, the stationary PDF will be formed around both of their deterministic solutions. The resulting response PDF contains subtle information about the system, which is essential in the design of its parts and its reliability assessment. The generality of the approach is such that the new path integration method can be implemented for the stochastic analysis of other non-smooth random dynamical systems.

I-PI 2.0: A universal force engine for advanced molecular simulations

Progress in the atomic-scale modeling of matter over the past decade has been tremendous. This progress has been brought about by improvements in methods for evaluating interatomic forces that work by either solving the electronic structure problem explicitly, or by computing accurate approximations of the solution and by the development of techniques that use the Born–Oppenheimer (BO) forces to move the atoms on the BO potential energy surface. As a consequence of these developments it is now possible to identify stable or metastable states, to sample configurations consistent with the appropriate thermodynamic ensemble, and to estimate the kinetics of reactions and phase transitions. All too often, however, progress is slowed down by the bottleneck associated with implementing new optimization algorithms and/or sampling techniques into the many existing electronic-structure and empirical-potential codes. To address this problem, we are thus releasing a new version of the i-PI software. This piece of software is an easily extensible framework for implementing advanced atomistic simulation techniques using interatomic potentials and forces calculated by an external driver code. While the original version of the code (Ceriotti et al., 2014) was developed with a focus on path integral molecular dynamics techniques, this second release of i-PI not only includes several new advanced path integral methods, but also offers other classes of algorithms. In other words, i-PI is moving towards becoming a universal force engine that is both modular and tightly coupled to the driver codes that evaluate the potential energy surface and its derivatives. Program summaryProgram Title: i-PIProgram Files doi:http://dx.doi.org/10.17632/x792grbm9g.1Licensing provisions: GPLv3, MITProgramming language: PythonExternal routines/libraries: NumPyNature of problem: Lowering the implementation barrier to bring state-of-the-art sampling and atomistic modeling techniques to ab initio and empirical potentials programs.Solution method: Advanced sampling methods, including path-integral molecular dynamics techniques, are implemented in a Python interface. Any electronic structure code can be patched to receive the atomic coordinates from the Python interface, and to return the forces and energy that are used to integrate the equations of motion, optimize atomic geometries, etc.Restrictions: This code does not compute interatomic potentials, although the distribution includes sample driver codes that can be used to test different techniques using a few simple model force fields.

Internal quantum dynamics of a nanoparticle in a thermal electromagnetic field: A minimal model

We argue that macroscopic electrodynamics is unsuited to describe the process of radiative thermalization between a levitated nanoparticle in high vacuum and the thermal electromagnetic field. Based on physical arguments, we propose a model to describe such systems beyond the quasi-equilibrium approximation. We use path integral techniques to analytically solve the model and exactly calculate the time evolution of the quantum degrees of freedom of the system. Free parameters of the microscopic quantum model are determined by matching analytical results to well-known macroscopic response functions. The time evolution of the internal energy of a levitated nanoparticle in a thermal electromagnetic field, as described by our model, qualitatively differs from macroscopic electrodynamics, a prediction that can be experimentally tested.

An efficient pair natural orbital based configuration interaction scheme for the calculation of open-shell ionization potentials

A spin adapted configuration interaction scheme is proposed for the evaluation of ionization potentials in α high spin open shell reference functions. There are three different ways to remove an electron from such a reference, including the removal of an alpha or a beta electron from doubly occupied or an alpha electron from singly occupied molecular orbitals. Ionization operators are constructed for each of these cases, and the resulting second quantized expressions are implemented using an automated code generator environment. To achieve greater computational efficiency, the virtual space is reduced using an averaged pair natural orbital machinery developed earlier and applied with great success in the calculation of X-ray absorption spectra [D. Manganas et al., J. Chem. Phys. A 122, 1215 (2018)]. Various approximate integral evaluation schemes including the resolution of identity and seminumerical techniques are also invoked to further enhance the computational efficiency. Although the resulting method is not particularly accurate in terms of predicting absolute energy values, with a simple shift in the ionization potentials, it is still possible to use it for the qualitative characterization of the basic features of X-ray photoionization spectra. While satellite intensities cannot be computed with the current method, the inclusion of vibrational effects using a path integral technique allows for the computation of vibrational transitions corresponding to main peaks.

Short DNA persistence length in a mesoscopic helical model

The flexibility of short DNA chains is investigated via computation of the average correlation function between dimers which defines the persistence length. Path integration techniques have been applied to confine the phase space available to base pair fluctuations and derive the partition function. The apparent persistence lengths of a set of short chains have been computed as a function of the twist conformation both in the over-twisted and the untwisted regimes, whereby the equilibrium twist is selected by free energy minimization. The obtained values are significantly lower than those generally attributed to kilo-base long DNA. This points to an intrinsic helix flexibility at short length scales, arising from large fluctuational effects and local bending, in line with recent experimental indications. The interplay between helical untwisting and persistence length has been discussed for a heterogeneous fragment by weighing the effects of the sequence specificities through the non-linear stacking potential.

Path Integral Approach to Quantum Thermodynamics.

Work belongs to the most basic notions in thermodynamics but it is not well understood in quantum systems, especially in open quantum systems. By introducing a novel concept of the work functional along an individual Feynman path, we invent a new approach to study thermodynamics in the quantum regime. Using the work functional, we derive a path integral expression for the work statistics. By performing the ℏ expansion, we analytically prove the quantum-classical correspondence of the work statistics. In addition, we obtain the quantum correction to the classical fluctuating work. We can also apply this approach to an open quantum system in the strong coupling regime described by the quantum Brownian motion model. This approach provides an effective way to calculate the work in open quantum systems by utilizing various path integral techniques. As an example, we calculate the work statistics for a dragged harmonic oscillator in both isolated and open quantum systems.

Wiener Path Integral based response determination of nonlinear systems subject to non-white, non-Gaussian, and non-stationary stochastic excitation

The recently developed Wiener Path Integral (WPI) technique for determining the joint response probability density function of nonlinear systems subject to Gaussian white noise excitation is generalized herein to account for non-white, non-Gaussian, and non-stationary excitation processes. Specifically, modeling the excitation process as the output of a filter equation with Gaussian white noise as its input, it is possible to define an augmented response vector process to be considered in the WPI solution technique. A significant advantage relates to the fact that the technique is still applicable even for arbitrary excitation power spectrum forms. In such cases, it is shown that the use of a filter approximation facilitates the implementation of the WPI technique in a straightforward manner, without compromising its accuracy necessarily. Further, in addition to dynamical systems subject to stochastic excitation, the technique can also account for a special class of engineering mechanics problems where the media properties are modeled as stochastic fields. Several numerical examples pertaining to both single- and multi-degree-of-freedom systems are considered, including a marine structural system exposed to flow-induced non-white excitation, as well as a bending beam with a non-Gaussian and non-homogeneous Young's modulus. Comparisons with Monte Carlo simulation data demonstrate the accuracy of the technique.