Stochastic Equations Of Motion Research Articles

Probabilistic stability of uncertain composite plates and stochastic irregularity in their buckling mode shapes: A semi-analytical non-intrusive approach

In this study, the mechanical properties of the composite plate were considered Gaussian random fields and their effects on the buckling load and corresponding mode shapes were studied by developing a semi-analytical non-intrusive approach. The random fields were decomposed by the Karhunen-Loève method. The strains were defined based on the assumptions of the first-order and higher-order shear-deformation theories. Stochastic equations of motion were extracted using Euler–Lagrange equations. The probabilistic response space was obtained by employing the non-intrusive polynomial chaos method. Finally, the effect of spatially varying stochastic properties on the critical load of the plate and the irregularity of buckling mode shapes and their sequences were studied for the first time. Our findings showed that different shear deformation plate theories could significantly influence the reliability of thicker plates under compressive loading. It is suggested that a linear relationship exists between the mechanical properties’ variation coefficient and critical loads’ variation coefficient. Also, in modeling the plate properties as random fields, a significant stochastic irregularity is obtained in buckling mode shapes, which is crucial in practical applications.

Stochastic free vibration of composite plates with temperature-dependent properties under spatially varying stochastic high thermal gradient

The uncertainty of natural frequencies of composite plates due to the uncertainty of high-temperature distribution on the plate is investigated. Mechanical properties are assumed to be uncertain linear functions of temperature with assuming a stochastic linear dependency function (SLDF). The temperature gradient is assumed to be a Gaussian stochastic field with an exponential kernel. Assuming first-order shear deformation theory, the displacement field is defined. The stochastic assumed modes (SAM) method is proposed and discretized stochastic equations of motion are derived. The probability space is studied using an intrusive polynomial chaos approach. Stochastic simulations are verified using the stochastic finite element method (SFEM) based on the Monte Carlo approach. Results show the proposed semi-analytical method is an efficient alternative to time-consuming SFEM with resulting accurate statistical moments. Uncertainty in natural frequencies is quantified for symmetric and anti-symmetric composite plates with different side to thickness ratios. The applicability of linear dependency of composite material properties to the temperature is evaluated by uncertainty propagation in SLDF. Uncertainties in SLDF and temperature distribution have significant effects on predicted natural frequencies. Based on the results, the application of linear dependency of material properties to the temperature can lead to significant errors in predicted natural frequencies.

Nonlinear response of a dilute ferrofluid to an alternating magnetic field

The response of dilute monodisperse ferrofluid to alternating magnetic field is studied theoretically. Brownian rotational relaxation of magnetic nanoparticles is taken into account. Susceptibilities of magnetization harmonics are calculated in a wide range of field amplitudes and frequencies via three common approaches: the Fokker-Planck equation for the one-particle orientation distribution function, the Martsenyuk-Raikher-Shliomis effective field method and the Langevin dynamics simulation which is based on direct numerical integration of particles’ stochastic equations of motion. A comparison of these approaches revealed that all of them are able to describe accurately the behavior of the fundamental magnetization harmonic in a broad range of field parameters. But in the case of high-order harmonics, shortcomings of Langevin dynamics and the effective field method manifest themselves. Due to strong random fluctuations of the system magnetization, Langevin dynamics fails to produce reliable results in the weak-field limit, i.e. when the characteristic magnetic energy scale is lower than the system thermal energy. The Martsenyuk-Raikher-Shliomis equation gives a large systematic error for high-order harmonics if the field period is comparable with (or smaller than) the Brownian relaxation time.

Stochastic equation of motion approach to fermionic dissipative dynamics. I. Formalism.

In this work, we establish formally exact stochastic equation of motion (SEOM) theory to describe the dissipative dynamics of fermionic open systems. The construction of the SEOM is based on a stochastic decoupling of the dissipative interaction between the system and fermionic environment, and the influence of environmental fluctuations on the reduced system dynamics is characterized by stochastic Grassmann fields. Meanwhile, numerical realization of the time-dependent Grassmann fields has remained a long-standing challenge. To solve this problem, we propose a minimal auxiliary space (MAS) mapping scheme with which the stochastic Grassmann fields are represented by conventional c-number fields along with a set of pseudo-levels. This eventually leads to a numerically feasible MAS-SEOM method. The important properties of the MAS-SEOM are analyzed by making connection to the well-established time-dependent perturbation theory and the hierarchical equations of motion theory. The MAS-SEOM method provides a potentially promising approach for the accurate and efficient simulation of fermionic open systems at ultra-low temperatures.

Stochastic equation of motion approach to fermionic dissipative dynamics. II. Numerical implementation

This paper provides a detailed account of the numerical implementation of the stochastic equation of motion (SEOM) method for the dissipative dynamics of fermionic open quantum systems. To enable direct stochastic calculations, a minimal auxiliary space (MAS) mapping scheme is adopted, with which the time-dependent Grassmann fields are represented by c-number noises and a set of pseudo-operators. We elaborate on the construction of the system operators and pseudo-operators involved in the MAS-SEOM, along with the analytic expression for the particle current. The MAS-SEOM is applied to study the relaxation and voltage-driven dynamics of quantum impurity systems described by the single-level Anderson impurity model, and the numerical results are benchmarked against those of the highly accurate hierarchical equations of motion method. The advantages and limitations of the present MAS-SEOM approach are discussed extensively.

Modeling uncertainties of vehicle-track coupled dynamic systems

In this paper, the nonparametric models of the vehicle and track subsystems are established separately by the nonparametric probabilistic method, in which modeling uncertainties are represented by random matrices and the dispersion level of each random matrix is controlled by a corresponding dispersion parameter. The stochastic equations of motion of the vehicle and track subsystems are obtained respectively and coupled by the wheel-rail dynamic interaction model, in which nonlinear wheel-rail contact geometry relation and wheel-rail detachment can be considered. Random responses are obtained with the prediction-based iterative method in the framework of Monte Carlo simulation (MCS). The sensitivities of the random responses to the uncertainties of coefficient matrices of vehicle and track subsystems are analyzed and the confidence intervals obtained by the nonparametric and parametric models are compared. The results show that the lateral acceleration of the car body on a straight track is most sensitive to the vehicle’s mass matrix, while the wheel derailment coefficient is most sensitive to the vehicle’s stiffness matrix at curve negotiation. The confidence intervals obtained by the two probabilistic models with the same dispersion level are close to each other, verifying the validity of the nonparametric model of vehicle-track coupled systems.

Vector Resonant Relaxation of Stars around a Massive Black Hole

Abstract In the vicinity of a massive black hole, stars move on precessing Keplerian orbits. The mutual stochastic gravitational torques between the stellar orbits drive a rapid reorientation of their orbital planes, through a process called vector resonant relaxation. We derive, from first principles, the correlation of the potential fluctuations in such a system, and the statistical properties of random walks undergone by the stellar orbital orientations. We compare this new analytical approach with numerical simulations. We also provide a simple scheme to generate the random walk of a test star’s orbital orientation using a stochastic equation of motion. We finally present quantitative estimations of this process for a nuclear stellar cluster such as that of the Milky Way.

Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: existence, uniqueness, exponential stability and invariant measures

In this work, we consider the two-dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows. We investigate the well-posedness of such models in two-dimensional bounded and unbounded (Poincaré domains) domains, both in deterministic and stochastic settings. The existence and uniqueness of weak solution in the deterministic case is proved via a local monotonicity property of the linear and nonlinear operators and a localized version of the Minty-Browder technique. Some results on the exponential stability of stationary solutions are also established. The global solvability results for the stochastic counterpart are obtained by a stochastic generalization of the Minty-Browder technique. The exponential stability results in the mean square as well as in the pathwise (almost sure) sense are also discussed. Using the exponential stability results, we finally prove the existence of a unique invariant measure, which is ergodic and strongly mixing.

Stochastic Representation of Non-Markovian Fermionic Quantum Dissipation.

Quantum Brownian motion plays a fundamental role in many areas of modern physics. In the path-integral formulation, environmental fluctuations can be characterized by auxiliary stochastic fields. Intriguingly, for fermionic environments the stochastic fields must be Grassmann valued so as to memorize the order of the random forces exerted on the system. Such nonclassical fields cannot be represented by conventional means. We propose a strategy to map the Grassmann-number fields to conventional c-number noises and a set of quantized pseudolevels. The resulting stochastic equation of motion (SEOM) enables direct stochastic simulation of the fermionic dissipative dynamics. The SEOM gives exact physical observables of noninteracting systems, and yields accurate approximate results for interacting systems. The practicality and accuracy of the proposed strategy and the SEOM are exemplified by numerical studies conducted on a single-impurity Anderson model.

Catching drifting pebbles

Turbulence plays a key role in the transport of pebble-sized particles. It also affects the ability of pebbles to be accreted by protoplanets because it stirs pebbles out of the disk midplane. In addition, turbulence suppresses pebble accretion once the relative velocities become too high for the settling mechanism to be viable. Following Paper I, we aim to quantify these effects by calculating the pebble accretion efficiency ε using three-body simulations. To model the effect of turbulence on the pebbles, we derive a stochastic equation of motion (SEOM) applicable to stratified disk configurations. In the strong coupling limit (ignoring particle inertia) the limiting form of this equation agrees with previous works. We conduct a parameter study and calculate ε in 3D, varying pebble and gas turbulence properties and accounting for the planet inclination. We find that strong turbulence suppresses pebble accretion through turbulent diffusion, agreeing closely with previous works. Another reduction of ε occurs when the turbulent rms motions are high and the settling mechanism fails. In terms of efficiency, the outer disk regions are more affected by turbulence than the inner regions. At the location of the H2O iceline, planets around low-mass stars achieve much higher efficiencies. Including the results from Paper I, we present a framework to obtain ε under general circumstances.

Critical test of bead–spring model to resolve the scaling laws of polymer melts: a molecular dynamics study

To examine the intrinsic nature of the bead–spring Kremer–Grest (KG) model, long-time molecular dynamics simulations are performed. Certain scaling laws for representative polymer properties are compared with theoretical predictions. The results for static properties satisfy the expected static Gaussian nature, irrespective of the chain length. In contrast, the results for the dynamic properties of short chains show a clear discrepancy from theoretical predictions that assume ideal chain motion. This is clear evidence that the Gaussian nature of the dynamics of short chains is not necessarily established for the actual KG model, despite it being designed to have Gaussian characteristics by virtue of its stochastic equations of motion. This intrinsic nature of the KG model should be considered carefully when using this model for applications that involve relatively short chains.

The Quadrature Master Equations

In this paper, we derive the non-Markovian stochastic equation of motion (SEM) and master equations (MEs) for the open quantum system by using the non-Markovian stochastic Schrodinger equations (SSEs) for the quadrature unraveling in linear and nonlinear cases. The SSEs for quadrature unraveling arise as a special case of a quantum system. Also we derive the Markovian SEM and ME by using linear and nonlinear Ito SSEs for the measurement probabilities. In linear non-Markovian case, we calculate the convolutionless linear quadrature non-Markovian SEM and ME. We take advantage from example and show that corresponding theory.

Wigner Representation for Investigation of Entangled Photon Evolution in Four-Wave Mixing Process within a Fiber Ring Resonator

The aim of this study is to analyze the quantum correlation of entangled photons for four-wavemixing process within Kerr nonlinear susceptibility χ(3) of an optical fiber ring resonator. The main system Hamiltonian composes of two types of coupling photon modes, including pumping and parametric-down conversion. Wigner representation is applied to the Heisenberg equation of motion derived from the system Hamiltonian in order to obtain the corresponding stochastic equations of motion in a c-number. Quantum trajectories obtained from the stochastic equation of motion of photon states are then derived. Finally, the entanglement inseparability criteria for a pair of entangled photonfrom a numerical approach is also satisfied.

Effect of thermal noise on vesicles and capsules in shear flow

We add thermal noise consistently to reduced models of undeformable vesicles and capsules in shear flow and derive analytically the corresponding stochastic equations of motion. We calculate the steady-state probability distribution function and construct the corresponding phase diagrams for the different dynamical regimes. For fluid vesicles, we predict that at small shear rates thermal fluctuations induce a tumbling motion for any viscosity contrast. For elastic capsules, due to thermal mixing, an intermittent regime appears in regions where deterministic models predict only pure tank treading or tumbling.

Investigations on the numerical solution of the Fokker‐Planck equation with discontinuous Galerkin Methods

AbstractNonlinear dynamic systems under stochastic excitation possess Markov characteristics. Thus, their stochastic equation of motion can be transformed into the Fokker‐Planck equation which describes the evolution of the probability density. A discontinuous Galerkin (DG) method is applied to solve the Fokker‐Planck equation. This method provides numerical stability as well as accuracy and is able to treat discontinuities of the solution. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear effects in the dynamics governed by non-Markovian stochastic Langevin-like equations

The influence of nonlinear effects in stochastic equations of motion with both additive and multiplicative noises is studied. Non-Markovian stochastic dynamics are compared with their corresponding Markovian (local approximations). Non-Markovian effects are implemented through Ornstein-Uhlenbeck and exponential damped harmonic dissipative kernels.

Dynamical density functional theory for molecular and colloidal fluids: A microscopic approach to fluid mechanics

In recent years, a number of dynamical density functional theories (DDFTs) have been developed for describing the dynamics of the one-body density of both colloidal and atomic fluids. In the colloidal case, the particles are assumed to have stochastic equations of motion and theories exist for both the case when the particle motion is overdamped and also in the regime where inertial effects are relevant. In this paper, we extend the theory and explore the connections between the microscopic DDFT and the equations of motion from continuum fluid mechanics. In particular, starting from the Kramers equation, which governs the dynamics of the phase space probability distribution function for the system, we show that one may obtain an approximate DDFT that is a generalization of the Euler equation. This DDFT is capable of describing the dynamics of the fluid density profile down to the scale of the individual particles. As with previous DDFTs, the dynamical equations require as input the Helmholtz free energy functional from equilibrium density functional theory (DFT). For an equilibrium system, the theory predicts the same fluid one-body density profile as one would obtain from DFT. Making further approximations, we show that the theory may be used to obtain the mode coupling theory that is widely used for describing the transition from a liquid to a glassy state.

Relativistic diffusions: A unifying approach

A new, wide class of relativistic stochastic processes is introduced. All relativistic processes considered so far in the literature are members of this class. For each process, the stochastic equations of motion are obtained in an arbitrary Lorentz frame. The associated Kolmogorov equation is also derived for the first time, in both (3+1)- and manifestly covariant forms.

A UNIFYING APPROACH TO RELATIVISTIC DIFFUSIONS AND H-THEOREMS

A new, wide class of relativistic stochastic processes is introduced. All relativistic processes considered so far in the literature (the Relativistic Ornstein–Uhlenbeck Process as well as the Franchi–Le Jan and the Dunkel–Hänggi processes) are members of this class. The stochastic equations of motion and the associated forward Kolmogorov equations are obtained for each process in the class. The corresponding manifestly covariant transport equation is also obtained. In particular, the manifestly covariant equations for the Franchi–Le Jan and the Dunkel–Hänggi processes are derived here for the first time. Finally, the manifestly covariant approach is used to prove a new H-theorem for all processes in the class.

Symmetry-based determination of space-time functions in nonequilibrium growth processes

We study the space-time correlation and response functions in nonequilibrium growth processes described by linear stochastic Langevin equations. Exploiting exclusively the existence of space- and time-dependent symmetries of the noiseless part of these equations, we derive expressions for the universal scaling functions of two-time quantities which are found to agree with the exact expressions obtained from the stochastic equations of motion. The usefulness of the space-time functions is illustrated through the investigation of two atomistic growth models, the Family model and the restricted Family model, which are shown to belong to a unique universality class in 1+1 and 2+1 space dimensions. This corrects earlier studies which claimed that in 2+1 dimensions the two models belong to different universality classes.