Stochastic Langevin Equation Research Articles

РАЗРАБОТКА АГЕНТНОЙ ИМИТАЦИОННОЙ МОДЕЛИ ПРОЦЕССА, ОПИСЫВАЮЩЕГО ГИДРАВЛИЧЕСКУЮ РАЗРАБОТКУ ГРУНТА ПРИ ДНОУГЛУБЛЕНИИ

The article describes the process of hydraulic soil development when performing dredging operations in terms of a probabilistic process. To give the mathematical description of this process, the stochastic differential Langevin equation is used, which makes it possible to determine the velocity of the particle moving along the spatial axis in case of multiple collisions with other soil particles. In order to establish the adequacy of this approach with the experimental data obtained through the utility model there has been developed an agent simulation model. A population of agents containing a module of the system dynamics realizing the differential equation is integrated into the structure of the model.

Predicting size-dependent heating efficiency of magnetic nanoparticles from experiment and stochastic Néel-Brown Langevin simulation

Abstract Magnetic nanoparticles (MNP) have been investigated for generating therapeutic heat when subjected to an alternating magnetic field (AMF) and applied for tumor-confined cancer therapy, so-called magnetic fluid hyperthermia (MFH). For application of MFH, a key requirement is the reduction of MNP dosing by maximizing the heat generation within medically safe limits of the applied AMF. Therefore, reliable and accurate predictions of particle heating are required for the advancement of therapy planning. In this study, we compare size-dependent particle heating data from calorimetric measurements to stochastic Neel-Brown Langevin equation Monte Carlo simulations, finding good agreement between them for various AMF amplitudes and frequencies. Within medical safety constraints of the AMF, our simulations predict maximum particle heating for magnetite particle core sizes above 25 nm with effective anisotropy constants K = 4000 J/m3 at frequencies of ∼ 100 kHz and field amplitudes ∼ 10 mT/μ0. These simulations could help to predict the optimal combination of medically safe AMF parameters and MNP intrinsic properties, such as core size and effective anisotropy, to maximize heat generation and reduce MNP dosing in the application of MFH.

Analogue stochastic gravity in strongly-interacting Bose–Einstein condensates

Collective modes propagating in a moving superfluid are known to satisfy wave equations in a curved space–time, with a metric determined by the underlying superflow. We use the Keldysh technique in a curved space–time to develop a quantum geometric theory of fluctuations in superfluid hydrodynamics. This theory relies on a “quantized” generalization of the two-fluid description of Landau and Khalatnikov, where the superfluid component is viewed as a quasi-classical field coupled to a normal component — the collective modes/phonons representing a quantum bath. This relates the problem in the hydrodynamic limit to the “quantum friction” problem of Caldeira–Leggett type. By integrating out the phonons, we derive stochastic Langevin equations describing a coupling between the superfluid component and phonons. These equations have the form of Euler equations with additional source terms expressed through a fluctuating stress–energy tensor of phonons. Conceptually, this result is similar to stochastic Einstein equations that arise in the theory of stochastic gravity. We formulate the fluctuation–dissipation theorem in this geometric language and discuss possible physical consequences of this theory.

Dynamical evolution of spectator systems produced in ultrarelativistic heavy-ion collisions

In peripheral heavy-ion collisions at ultra-relativistic energies, usually only parts of the colliding nuclei effectively interact with each other. In the overlapping zone, a fireball or quark-gluon plasma is produced. The excitation energy of the heavy remnant can range from a few tens to several hundreds of MeV, depending on the impact parameter. The decay of these excited spectators is investigated in this work for the first time within a dynamical approach based on the multi-dimensional stochastic Langevin equation. The potential of this exploratory work to understand the connection between electromagnetic fields generated by the heavy spectators and measured pion distributions is discussed.

On fast and slow Earth’s magnetospheric dynamics during geomagnetic storms: a stochastic Langevin approach

The Earth’s magnetosphere responds to the external changes of interplanetary magnetic field and solar wind conditions showing a multiscale dynamics, manifesting in the occurrence of fluctuations over a very wide range of timescales. Here, using an approach based on a Langevin/Fokker-Planck description we investigate the nature of the fast (short-) and slow (long-timescale) fluctuations of SYM-H index during geomagnetic storms. The results point towards a different origin of the fast (τ < 200 min) and slow (τ > 200 min) fluctuations, which are characterized by state functions of different nature. In detail, the state function associated with the slow dynamics shows the evidence of the occurrence of first-order-like topological phase transition during the different phases of a geomagnetic storm, while the fast dynamics seems to be characterized by a quasi-invariant quadratic state function. A modeling in terms of stochastic Langevin equation is discussed and the relevance of our results in the framework of Space Weather studies is outlined.

Analytic solution to space-fractional Fokker–Planck equations for tempered-stable Lévy distributions with spatially linear, time-dependent drift

We derive analytic solutions for the full time dependence of space-fractional Fokker–Planck equations corresponding to stochastic Langevin equations with additive tempered-stable Lévy noise terms. The drift terms are generalised to be spatially linear, but may contain arbitrary time dependence such that no steady-state solution is available, even for the deterministic system.

Exponential ergodicity for stochastic Langevin equation with partial dissipative drift

By using the Krylov estimate, we prove the exponential ergodicity of the invariant measure for stochastic Langevin equation with singular coefficients, where the classical Lyapunov condition cannot be verified.

General framework for fluctuating dynamic density functional theory

We introduce a versatile bottom-up derivation of a formal theoretical framework to describe (passive) soft-matter systems out of equilibrium subject to fluctuations. We provide a unique connection between the constituent-particle dynamics of real systems and the time evolution equation of their measurable (coarse-grained) quantities, such as local density and velocity. The starting point is the full Hamiltonian description of a system of colloidal particles immersed in a fluid of identical bath particles. Then, we average out the bath via Zwanzig’s projection-operator techniques and obtain the stochastic Langevin equations governing the colloidal-particle dynamics. Introducing the appropriate definition of the local number and momentum density fields yields a generalisation of the Dean–Kawasaki (DK) model, which resembles the stochastic Navier–Stokes description of a fluid. Nevertheless, the DK equation still contains all the microscopic information and, for that reason, does not represent the dynamical law of observable quantities. We address this controversial feature of the DK description by carrying out a nonequilibrium ensemble average. Adopting a natural decomposition into local-equilibrium and nonequilibrium contribution, where the former is related to a generalised version of the canonical distribution, we finally obtain the fluctuating-hydrodynamic equation governing the time-evolution of the mesoscopic density and momentum fields. Along the way, we outline the connection between the ad hoc energy functional introduced in previous DK derivations and the free-energy functional from classical density-functional theory. The resultant equation has the structure of a dynamical density-functional theory (DDFT) with an additional fluctuating force coming from the random interactions with the bath. We show that our fluctuating DDFT formalism corresponds to a particular version of the fluctuating Navier–Stokes equations, originally derived by Landau and Lifshitz. Our framework thus provides the formal apparatus for ab initio derivations of fluctuating DDFT equations capable of describing the dynamics of soft-matter systems in and out of equilibrium.

Dressed diffusion and friction coefficients in inhomogeneous multicomponent self-gravitating systems

General self-consistent expressions for the coefficients of diffusion and dynamical friction in a stable, bound, multicomponent self-gravitating and inhomogeneous system are derived. They account for the detailed dynamics of the colliding particles and their self-consistent dressing by collective gravitational interactions. The associated Fokker-Planck equation is shown to be fully consistent with the corresponding inhomogeneous Balescu-Lenard equation and, in the weak self-gravitating limit, to the inhomogeneous Landau equation. Hence it provides an alternative derivation to both and demonstrates their equivalence. The corresponding stochastic Langevin equations are presented: they can be a practical alternative to numerically solving the inhomogeneous Fokker-Planck and Balescu-Lenard equations. The present formalism allows for a self-consistent description of the secular evolution of different populations covering a spectrum of masses, with a proper accounting of the induced secular mass segregation, which should be of interest to various astrophysical contexts, from galactic centers to protostellar discs.

High Order Conformal Symplectic and Ergodic Schemes for the Stochastic Langevin Equation via Generating Functions

In this paper, we consider the stochastic Langevin equation with additive noises, which possesses both conformal symplectic geometric structure and ergodicity. We propose a methodology of constructing high weak order conformal symplectic schemes by converting the equation into an equivalent autonomous stochastic Hamiltonian system and modifying the associated generating function. To illustrate this approach, we construct a specific second order numerical scheme, and prove that its symplectic form dissipates exponentially. Moreover, for the linear case, the proposed scheme is also shown to inherit the ergodicity of the original system, and the temporal average of the numerical solution is a proper approximation of the ergodic limit over long time. Numerical experiments are given to verify these theoretical results.

Harmonic Simulation Study of Simultaneous Nanoparticle Size and Viscosity Differentiation

The signal harmonics of magnetic nanoparticles in solution under an ac magnetic field are highly sensitive to their physical properties and the surrounding environment. Here, we use Monte Carlo simulations of coupled stochastic Langevin equations to demonstrate a possible technique that can simultaneously characterize different particle sizes and viscous environments. When combined with a constant bias field, the relative amplitudes of the harmonics change significantly, and are shown to be uniquely sensitive to changes in particle size and viscosity. Spectroscopic experiments verify the size-dependent harmonic behavior. With this technique, particles with sizes differing by only ~1-2 nm in diameter can theoretically be identified, while separately quantifying changes in viscosity. This concurrent characterization could be implemented in magnetic particle imaging to create a colorized image.

Non-Markovian Magnetization Dynamics for Uniaxial Nanomagnets

A stochastic approach for the description of the time evolution of the magnetization of nanomagnets is proposed, that interpolates between the Landau-Lifshitz-Gilbert and the Landau-Lifshitz-Bloch approximations, by varying the strength of the noise. Its finite autocorrelation time, i.e. when it may be described as colored, rather than white, is, also, taken into account and the consequences, on the scale of the response of the magnetization are investigated. It is shown that the hierarchy for the moments of the magnetization can be closed, by introducing a suitable truncation scheme, whose validity is tested by direct numerical solution of the moment equations and compared to the averages obtained from a numerical solution of the corresponding colored stochastic Langevin equation. This comparison is performed on magnetic systems subject to both an external uniform magnetic field and an internal one-site uniaxial anisotropy.

WE‐H‐207A‐09: Theoretical Limits to Molecular Biomarker Detection Using Magnetic Nanoparticles

Purpose:Estimate the limits of molecular biomarker detection using magnetic nanoparticle methods like in vivo ELISA.Methods:Magnetic nanoparticles in an alternating magnetic field produce a magnetization that can be detected at exceedingly low levels because the signal at the harmonic frequencies is uniquely produced by the nanoparticles. Because the magnetization can also be used to characterize the nanoparticle rotational freedom, the bound state can be found. If the nanoparticles are coated with molecules that bind the desired biomarker, the rotational freedom reflects the biomarker concentration. The irreducible noise limit is the thermal noise or Johnson noise of the tissue and the contrast that can be measured must be larger than that limit. The contrast produced is a function of the applied field and depends strongly on nanoparticle volume. We have estimated the contrast using a Langevin function of a single composite variable to approximate the full stochastic Langevin equation for nanoparticle dynamics.Results:The thermal noise for a bandwidth reasonable for spectroscopy suggests mid zeptomolar (10–21) to low attomolar (10–18) concentrations can be measured in a volume that is 10cm in scale. The suggested sensitivity is far below the physiologically concentrations of almost all critical biomarkers including cytokines (picomolar), hormones (nanomolar) and heat shock proteins.Conclusion:The sensitivity of in vivo ELISA concentration measurements should be sufficient to measure physiological concentrations of critical biomarkers like cytokines in vivo. Further the sensitivity should be sufficient to measure concentrations of other biomarkers that are six to eight orders of magnitude lower in concentration than immune signaling molecules like cytokines.NIH ‐ 1U54CA151662‐01Department of Radiology

Allowance for the tunnel effect in the entrance channel of fusion–fission reactions

A two-stage model is developed in order to describe fusion–fission reactions. The process in the course of which colliding ions approach each other is simulated at the first stage, the deformations and relative orientations of the ions being taken into account. The first stage of the calculation is completed as soon as colliding nuclei touch each other. A continuous nuclear system (monosystem) is formed at this instant. The emerging distributions of the angular momenta of this system and of its potential and internal energies at the point of touching are used as input data that are necessary for triggering the second stage of the calculation. The evolution of collective coordinates that describe the shape of the monosystem is calculated at the second stage. The description of this evolution is terminated either at the instant of its fission or upon the release of a major part of its excess energy via particle and photon emission. In the latter case, the probability for the fission of the monosystem or a further decrease in its excitation energy becomes extremely small. The ion-collision process and the evolution of the monosystem formed after primary nuclei come into contact are simulated on the basis of stochastic Langevin equations. The quantities appearing in them (which include the potential energy and inertial and friction parameters) are determined with allowance for the shell structure of nuclei. The tunneling of colliding nuclei through the Coulomb barrier is taken into account, and the effect of this phenomenon on model predictions is studied.

Multipole electrodynamic ion trap geometries for microparticle confinement under standard ambient temperature and pressure conditions

Trapping of microparticles and aerosols is of great interest for physics and chemistry. We report microparticle trapping in case of multipole linear Paul trap geometries, operating under standard ambient temperature and pressure conditions. An 8- and 12-electrode linear trap geometries have been designed and tested with an aim to achieve trapping for larger number of particles and to study microparticle dynamical stability in electrodynamic fields. We report emergence of planar and volume ordered structures of microparticles, depending on the a.c. trapping frequency and particle specific charge ratio. The electric potential within the trap is mapped using the electrolytic tank method. Particle dynamics is simulated using a stochastic Langevin equation. We emphasize extended regions of stable trapping with respect to quadrupole traps, as well as good agreement between experiment and numerical simulations.

On the connection between dissipative particle dynamics and the Itô-Stratonovich dilemma.

Dissipative Particle Dynamics (DPD) is a popular simulation model for investigating hydrodynamic behavior of systems with non-negligible equilibrium thermal fluctuations. DPD employs soft core repulsive interactions between the system particles, thus allowing them to overlap. This supposedly permits relatively large integration time steps, which is an important feature for simulations on large temporal scales. In practice, however, an increase in the integration time step leads to increasingly larger systematic errors in the sampling statistics. Here, we demonstrate that the prime origin of these systematic errors is the multiplicative nature of the thermal noise term in Langevin's equation, i.e., the fact that it depends on the instantaneous coordinates of the particles. This lead to an ambiguity in the interpretation of the stochastic differential Langevin equation, known as the Itô-Stratonovich dilemma. Based on insights from previous studies of the dilemma, we propose a novel algorithm for DPD simulations exhibiting almost an order of magnitude improvement in accuracy, and nearly twice the efficiency of commonly used DPD Langevin thermostats.

Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes.

We investigate the ensemble and time averaged mean squared displacements for particle diffusion in a simple model for disordered media by assuming that the local diffusivity is both fluctuating in time and has a deterministic average growth or decay in time. In this study we compare computer simulations of the stochastic Langevin equation for this random diffusion process with analytical results. We explore the regimes of normal Brownian motion as well as anomalous diffusion in the sub- and superdiffusive regimes. We also consider effects of the inertial term on the particle motion. The investigation of the resulting diffusion is performed for unconfined and confined motion.

Combined Néel and Brown rotational Langevin dynamics in magnetic particle imaging, sensing, and therapy.

Magnetic nanoparticles have been studied intensely because of their possible uses in biomedical applications. Biosensing using the rotational freedom of particles has been used to detect biomarkers for cancer, hyperthermia therapy has been used to treat tumors, and magnetic particle imaging is a promising new imaging modality that can spatially resolve the concentration of nanoparticles. There are two mechanisms by which the magnetization of a nanoparticle can rotate, a fact that poses a challenge for applications that rely on precisely one mechanism. The challenge is exacerbated by the high sensitivity of the dominant mechanism to applied fields. Here, we demonstrate stochastic Langevin equation simulations for the combined rotation in magnetic nanoparticles exposed to oscillating applied fields typical to these applications to both highlight the existing relevant theory and quantify which mechanism should occur in various parameter ranges.

Closing the hierarchy for non-Markovian magnetization dynamics

We propose a stochastic approach for the description of the time evolution of the magnetization of nanomagnets, that interpolates between the Landau–Lifshitz–Gilbert and the Landau–Lifshitz–Bloch approximations, by varying the strength of the noise. In addition, we take into account the autocorrelation time of the noise and explore the consequences, when it is finite, on the scale of the response of the magnetization, i.e. when it may be described as colored, rather than white, noise and non-Markovian features become relevant. We close the hierarchy for the moments of the magnetization, by introducing a suitable truncation scheme, whose validity is tested by direct numerical solution of the moment equations and compared to the average deduced from a numerical solution of the corresponding stochastic Langevin equation. In this way we establish a general framework that allows both coarse-graining simulations and faster calculations beyond the truncation approximation used here.

Quantifying the non-ergodicity of scaled Brownian motion

We examine the non-ergodic properties of scaled Brownian motion (SBM), a non-stationary stochastic process with a time dependent diffusivity of the form . We compute the ergodicity breaking parameter EB in the entire range of scaling exponents α, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths T and short lag times Δ the EB parameter as function of the scaling exponent α has no divergence at α = 1/2 and present the asymptotes for EB in different limits. We generalize the analytical and simulations results for the time averaged and ergodic properties of SBM in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion.