Numerical Path Integration Method Research Articles

Systematic matrix formulation for efficient computational path integration

In this work we introduce a novel methodological treatment of the numerical path integration method, used for computing the response probability density function of stochastic dynamical systems. The method is greatly accelerated by transforming the corresponding Chapman-Kolmogorov equation to a matrix multiplication. With a systematic formulation we split the numerical solution of the Chapman-Kolmogorov equation into three separate parts: we interpolate the probability density function, we approximate the transitional probability density function of the process and evaluate the integral in the Chapman-Kolmogorov equation. We provide a thorough error and efficiency analysis through numerical experiments on a one, two, three and four dimensional problem. By comparing the results obtained through the Path Integration method with analytical solutions and with previous formulations of the path integration method, we demonstrate the superior ability of this formulation to provide accurate results. Potential bottlenecks are identified and a discussion is provided on how to address them.

On the Effect of the Electrical Load on Vibration Energy Harvesting Under Stochastic Resonance

Abstract Vibration energy harvesting (VEH) is a promising alternative for powering wireless electronics in many practical applications. Ambient vibration energy in the surrounding space of a target application often involves an inescapable randomness in the exciting vibrations, which may lead to deterioration of the expected power gains due to insufficient tuning and limited optimal designs. Stochastic resonance (SR) is a concept that has recently been considered for exploiting this randomness toward improving power generation from vibrating systems, based on the coexistence of near-harmonic vibrations with broadband noise excitations in a variety of practical mechanical systems. This paper is concerned with the optimal conditions for SR in vibration energy harvesters, exploring the frequently neglected effect of realistic architectures of the electrical circuit on the system dynamics and the achievable power output. A parametric study is conducted using a numerical path integration (PI) method to compute the response probability density functions (PDFs) of vibration energy harvesters, focusing on the effect of standard electrical components; namely, a load resistor, a rectifier, and a capacitor. It is found that the conditions for SR exhibit a nonlinear dependence on the weak harmonic excitation amplitude. Moreover, the modified nonlinear dissipation properties introduced by the rectifier and the capacitor lead to a tradeoff between the power output and the nonconducting dynamics that is essential in order to determine optimal harvesting designs.

Path integral solution of vibratory energy harvesting systems

A transition Fokker-Planck-Kolmogorov (FPK) equation describes the procedure of the probability density evolution whereby the dynamic response and reliability evaluation of mechanical systems could be carried out. The transition FPK equation of vibratory energy harvesting systems is a four-dimensional nonlinear partial differential equation. Therefore, it is often very challenging to obtain an exact probability density. This paper aims to investigate the stochastic response of vibration energy harvesters (VEHs) under the Gaussian white noise excitation. The numerical path integration method is applied to different types of nonlinear VEHs. The probability density function (PDF) from the transition FPK equation of energy harvesting systems is calculated using the path integration method. The path integration process is introduced by using the Gauss-Legendre integration scheme, and the short-time transition PDF is formulated with the short-time Gaussian approximation. The stationary probability densities of the transition FPK equation for vibratory energy harvesters are determined. The procedure is applied to three different types of nonlinear VEHs under Gaussian white excitations. The approximately numerical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulation (MCS).

Chernoff approximation of subordinate semigroups

This note is devoted to the approximation of evolution semigroups generated by some Markov processes and hence to the approximation of transition probabilities of these processes. The considered semigroups correspond to processes obtained by subordination (i.e. by a time-change) of some original (parent) Markov processes with respect to some subordinators, i.e. Lévy processes with a.s. increasing paths (they play the role of the new time). If the semigroup, corresponding to a parent Markov process, is not known explicitly then neither the subordinate semigroup, nor even the generator of the subordinate semigroup are known explicitly too. In this note, some (Chernoff) approximations are constructed for subordinate semigroups (in the case when subordinators have either known transitional probabilities, or known and bounded Lévy measure) under the condition that the parent semigroups are not known but are already Chernoff-approximated. As it has been shown in the recent literature, this condition is fulfilled for several important classes of Markov processes. This fact allows, in particular, to use the constructed Chernoff approximations of subordinate semigroups, in order to approximate semigroups corresponding to subordination of Feller processes and (Feller type) diffusions in Euclidean spaces, star graphs and Riemannian manifolds. Such approximations can be used for direct calculations and simulation of stochastic processes. The method of Chernoff approximation is based on the Chernoff theorem and can be interpreted also as a construction of Markov chains approximating a given Markov process and as the numerical path integration method of solving the corresponding PDE/SDE.

Rotating shaft's non-linear response statistics under biaxial random excitation, by path integration

The response of rotating shaft to random excitations is of practical concern for various rotor type engine design applications, with high level of potential external forces of stochastic nature.Authors have studied extreme value statistics of random vibrations of a Jeffcott type rotor, modeled as multidimensional dynamic system with non-linear restoring forces, under biaxial white noise excitation. The latter type of dynamic system is of wide use in stability studies of rotating machinery – from automotive to rocket turbo engine design. In particular, the design of liquid-propellant turbo pump rocket engines may be a potential application area of the studied system, due to the biaxial nature of the mechanical excitation, caused by surrounding liquid turbulent pressure field.In this paper, the extreme statistics of the rotor's non-linear oscillations has been studied by applying an enhanced implementation of the numerical path integration method, benchmarked by a known analytical solution. The obtained response probability distributions can serve as input for a wide range of system reliability issues, for example in extreme response study of turbo-pumps for liquid-propellant rocket engines. Predicting extreme transverse random vibrations of shafts in rotating machinery is of importance for applications with high environmental dynamic loads on Jeffcott type rotor supports.The major advantage of using path integration technique rather than direct Monte Carlo simulation is ability to estimate the probability distribution tail with high accuracy. The latter is of critical importance for extreme value statistics and first passage probability calculations.The main contribution of this paper is a reliable and independent confirmation of the path integration technique as a tool for assessing the dynamics of the kind of stochastic mechanical models considered in this paper. By the latter authors mean application of a unique and nontrivial analytical solution, that yields exact dynamic system response distribution, therefore it provides absolutely reliable reference to be compared to. The potential for providing a good qualitative understanding of the behavior of such systems is therefore available.

On numerical density approximations of solutions of SDEs with unbounded coefficients

We study a numerical method to compute probability density functions of solutions of stochastic differential equations. The method is sometimes called the numerical path integration method and has been shown to be fast and accurate in application oriented fields. In this paper we provide a rigorous analysis of the method that covers systems of equations with unbounded coefficients. Working in a natural space for densities, $L^1$, we obtain stability, consistency, and new convergence results for the method, new well-posedness and semigroup generation results for the related Fokker-Planck-Kolmogorov equation, and a new and rigorous connection to the corresponding probability density functions for both the approximate and the exact problems. To prove the results we combine semigroup and PDE arguments in a new way that should be of independent interest.

Long-Lived Coherence Originating from Electronic-Vibrational Couplings in Light-Harvesting Complexes

We theoretically investigate the evolutions of two-dimensional, third-order, nonlinear photon echo rephasing spectra with population time by using an exact numerical path integral method. It is shown that for the same system, the coherence time and relaxation time of excitonic states are short, however, if the couplings of electronic and intra-pigment vibrational modes are considered, the coherence time and relaxation time of this vibronic states are greatly extended. It means that the couplings between electronic and vibrational modes play important roles in keeping long-lived coherence in light-harvesting complexes. Particularly, by using the method we can fix the transition path of the energy transfer in bio-molecular systems.

GPU computing for accelerating the numerical Path Integration approach

The paper discusses a novel approach of accelerating the numerical Path Integration method, used for generating a stationary joint response probability density function of a dynamic system subjected to a random excitation, by the GPU computing. The paper proposes the parallelization of nested loops technique and demonstrates the advantages of GPU computing. Two, three and four dimensional in space problems are investigated as a part of the pilot project and the achieved maximum accelerations are reported. Three degree-of-freedom system (6D) is approached by the Path Integration technique for the first time. The application of the proposed GPU methodology for problems of stochastic dynamics and reliability are discussed.

Investigation on Rattle of Gears under Random Loading

This paper investigates the random rattle motion of a single stage spur gear pair subjected to deterministic and random excitation. The motion is treated separately as free flight between the boundaries and the instant impact at the two boundaries. Numerical path integration method is applied to obtain the probability density of the response during free flight. Monte Carlo simulation is also conducted to verify the efficiency and accuracy of path integration in solving this strongly nonlinear dynamic problem. Comparison between results from path integration and Monte Carlo simulation is made and good agreement is achieved.

Persistent draining crossover in DNA and other semi-flexible polymers: Evidence from hydrodynamic models and extensive measurements on DNA solutions.

Although the scaling theory of polymer solutions has had many successes, this type of argument is deficient when applied to hydrodynamic solution properties. Since the foundation of polymer science, it has been appreciated that measurements of polymer size from diffusivity, sedimentation, and solution viscosity reflect a convolution of effects relating to polymer geometry and the strength of the hydrodynamic interactions within the polymer coil, i.e., "draining." Specifically, when polymers are expanded either by self-excluded volume interactions or inherent chain stiffness, the hydrodynamic interactions within the coil become weaker. This means there is no general relationship between static and hydrodynamic size measurements, e.g., the radius of gyration and the hydrodynamic radius. We study this problem by examining the hydrodynamic properties of duplex DNA in solution over a wide range of molecular masses both by hydrodynamic modeling using a numerical path-integration method and by comparing with extensive experimental observations. We also considered how excluded volume interactions influence the solution properties of DNA and confirm that excluded volume interactions are rather weak in duplex DNA in solution so that the simple worm-like chain model without excluded volume gives a good leading-order description of DNA for molar masses up to 10(7) or 10(8) g/mol or contour lengths between 5 μm and 50 μm. Since draining must also depend on the detailed chain monomer structure, future work aiming to characterize polymers in solution through hydrodynamic measurements will have to more carefully consider the relation between chain molecular structure and hydrodynamic solution properties. In particular, scaling theory is inadequate for quantitative polymer characterization.

Numerical path integration method based on bubble grids for nonlinear dynamical systems

A new numerical path integration method based on bubble grids for nonlinear dynamical systems is presented in this paper. The ordinary differential equations for the first and second order moments are derived on the basis of the Gaussian closure method. Then the probability density values on the bubble nodes in the computational domain can be calculated via the obtained method. The good performance of the resulting method is finally shown in the numerical examples by using some specific nonlinear dynamical systems: Duffing oscillator subjected to harmonic and stochastic excitations, and Duffing–Rayleigh oscillator subjected to harmonic and stochastic excitations.

Influence of the pulse shape and the dot size on the decay and reappearance of Rabi rotations in laser driven quantum dots

We study the dynamics of strongly confined semiconductor quantum dots coupled to acoustic phonons and driven by external laser pulses by a numerical path integral method. The field-dependent damping, caused by the non-Markovian processes of pure dephasing and manifesting itself in the peculiar decay and reappearance phenomenon of Rabi rotations is found to depend notably on the dot size and the shape of the applied laser pulses. In the limit of strong fields rectangular pulses yield a significant weaker damping than Gaussian or other bell-shaped profiles. As a consequence, the undamping of Rabi rotations at high pulse areas is most clearly visible for rectangular pulses.

Excitation energy transfer: Study with non-Markovian dynamics

In this paper, we investigate the non-markovian dynamics of a model to mimic the excitation energy transfer (EET) between chromophores in photosynthesis systems. The numerical path integral method is used. This method includes the non-Markovian effects of the environmental affects, and it does not need the perturbation approximation in solving the dynamics of systems of interest. It implies that the coherence helps the EET between chromophores through lasting the transfer time rather than enhancing the transfer rate of the EET. In particular, the non-markovian environment greatly increases the efficiency of the EET in the photosynthesis systems.

Transport Properties of Wormlike Chains with Applications to Double Helical DNA and Carbon Nanotubes

Many extended threadlike polymeric structures (e.g., certain synthetic polymers, carbon nanotubes, double helical DNA, and certain polypeptides) exhibit a degree of chain-stiffness intermediate between the idealized rod and the random coil. Such structures can be reasonably well described by the wormlike chain model, and we apply our previously developed numerical path-integration method to determine both the intrinsic viscosity [η] and the friction coefficient f of wormlike chains. We have also determined finite-thickness contributions to the volume and the radius of gyration and developed approximants for [η] and f that should be useful in characterization work. Our results are compared to previous computations of [η] and f and the uncertainties of the various computational methods for calculating these transport properties are considered. Our results are also compared to experimental viscometric, sedimentation, and light scattering results for double helical DNA, where we obtain accurate fits by assumi...

Decoherence and relaxation of qubits coupled to low- and medium-frequency Ohmic baths directly and via a harmonic oscillator

Using the numerical path integral method we investigate the decoherence and relaxation of qubits in spin-boson (SB) and spin-intermediate harmonic oscillator (IHO)-bath (SIB) models. The cases that the environment baths with low and medium frequencies are investigated. It is shown that the qubits in SB and SIB models have the same decoherence and relaxation as the baths with low frequencies. However, the qubits in the two models have different decoherence and relaxation as the baths with medium frequencies. The decoherence and relaxation of the qubit in SIB model can be modulated through changing the coupling coefficients of the qubit-IHO and IHO-bath and the oscillation frequency of the IHO.

Decoherence and relaxation of a qubit coupled to an Ohmic bath directly and via an intermediate harmonic oscillator

Using the numerical path integral method we investigate the decoherence and relaxation of qubits coupled to an Ohmic bath directly and via an intermediate harmonic oscillator (IHO). Here, we suppose the oscillation frequencies of the bath modes are higher than the IHO’s. When we choose suitable parameters the qubits in the two models may have almost same decoherence and relaxation times. However, the decoherence and relaxation times of the qubit in the qubit-IHO-bath model can be modulated through changing the coupling coefficients of the qubit-IHO and IHO-bath and the oscillation frequency of the IHO.

Numerical meshfree path integration method for non-linear dynamic systems

We present a new numerical meshfree path integration (MPI) method for non-linear dynamic systems. The obtained MPI method can be performed in the irregular computational domain and the probability density values of the random nodes in the domain can be calculated via the MPI method and the ordinary differential equations for the first and second-order moments on the basis of Gaussian closure method. The piecewise linear interpolation based on adaptive least squares is utilized as a post processor to approximate the probability density values on arbitrary positions. The good performance of the resulting method is finally shown in the numerical examples by using three specific non-linear dynamic systems: Duffing oscillator subjected to both harmonic and stochastic excitations, Duffing–Rayleigh oscillator subjected to both harmonic and stochastic excitations, and CHEN system driven by three different Gaussian white noises.

Nonmonotonic Field Dependence of Damping and Reappearance of Rabi Oscillations in Quantum Dots

The dynamics of strongly confined laser driven semiconductor quantum dots coupled to phonons is studied theoretically by calculating the time evolution of the reduced density matrix using a numerical path integral method. We explore the cases of long pulses, strong dot-phonon and dot-laser coupling, and high temperatures, which, up to now, have been inaccessible. We find that the phonon-induced damping of Rabi rotations is a nonmonotonic function of the laser field that is increasing at low fields and decreasing at high fields. This results in a reappearance of Rabi rotations at high fields. This phenomenon is of a general nature which occurs for all temperatures and carrier-phonon coupling strengths.

High pulse area undamping of Rabi oscillations in quantum dots coupled to phonons

AbstractWe study the dynamics of strongly confined semiconductor quantum dots, coupled to acoustic phonons and driven by external laser pulses, by a numerical path integral method. Our method fully explores the relevant memory of the system thus accounting for all non‐Markovian features of the dynamics. The decay time and the frequency of Rabi oscillations are found to depend notably on the temperature and the applied field. The field dependence is non‐monotonic leading to an undamping of Rabi oscillations at high pulse areas. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical path integration of a non-homogeneous Markov process

The numerical path integration method, based on Gauss–Legendre integration scheme, is applied to a Duffing oscillator subject to both sinusoidal and white noise excitations. The response of the system is a Markov process with one of the drift coefficients being periodic. It is a non-homogeneous Markov process that does not have a stationary probability distribution. When applying the numerical procedure, the values of transition probability density at the Gaussian–Legendre quadrature points need only be calculated for time steps of the first period of the sinusoidal excitation, and they can be saved for use in all subsequent periods. The numerical procedure is capable of capturing the evolution of the probability density from an initial distribution to one that is changing and rotating periodically in the phase space.