Langevin Stochastic Differential Equation Research Articles

Nonasymptotic convergence analysis for the unadjusted Langevin algorithm

In this paper, we study a method to sample from a target distribution $\pi$ over $\mathbb{R}^d$ having a positive density with respect to the Lebesgue measure, known up to a normalisation factor. This method is based on the Euler discretization of the overdamped Langevin stochastic differential equation associated with $\pi$. For both constant and decreasing step sizes in the Euler discretization, we obtain non-asymptotic bounds for the convergence to the target distribution $\pi$ in total variation distance. A particular attention is paid to the dependency on the dimension $d$, to demonstrate the applicability of this method in the high dimensional setting. These bounds improve and extend the results of (Dalalyan 2014).

Multiscale temporal integrators for fluctuating hydrodynamics

Following on our previous work [S. Delong, B. E. Griffith, E. Vanden-Eijnden, and A. Donev, Phys. Rev. E 87, 033302 (2013)], we develop temporal integrators for solving Langevin stochastic differential equations that arise in fluctuating hydrodynamics. Our simple predictor-corrector schemes add fluctuations to standard second-order deterministic solvers in a way that maintains second-order weak accuracy for linearized fluctuating hydrodynamics. We construct a general class of schemes and recommend two specific schemes: an explicit midpoint method and an implicit trapezoidal method. We also construct predictor-corrector methods for integrating the overdamped limit of systems of equations with a fast and slow variable in the limit of infinite separation of the fast and slow time scales. We propose using random finite differences to approximate some of the stochastic drift terms that arise because of the kinetic multiplicative noise in the limiting dynamics. We illustrate our integrators on two applications involving the development of giant nonequilibrium concentration fluctuations in diffusively mixing fluids. We first study the development of giant fluctuations in recent experiments performed in microgravity using an overdamped integrator. We then include the effects of gravity and find that we also need to include the effects of fluid inertia, which affects the dynamics of the concentration fluctuations greatly at small wave numbers.

Weak backward error analysis for overdamped Langevin processes

We consider numerical approximations of overdamped Langevin stochastic differential equations by implicit methods. We show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. This implies that every measure of the numerical scheme is close to a modified invariant measure obtained by asymptotic expansion. Moreover, we prove that, up to negligible terms, the dynamic associated with the implicit scheme considered is exponentially mixing.

Analytical Computation of Mean Time to Lose Lock for Langevin Delay-Locked Loops

This paper presents a novel method for the analytical mean time to lose lock (MTLL) computation of coherent second-order Langevin delay-locked loops (DLLs). Analytical MTLL computation is a key task for DLLs, since the computational complexity of numerical MTLL simulations is far too high in many operating ranges of the second-order Langevin DLLs. To obtain the crucial MTLL values analytically without simulations, we rewrite the Langevin stochastic differential equation (SDE) as a vector-valued Ornstein-Uhlenbeck (OU) SDE. It includes a Gaussian noise term, which yields as a solution of the vector-valued OU SDE a time-variant Gaussian distribution. Thus, the complementary error function yields the loss of lock probability and thereby the MTLL. If we replace the complementary error functions by suitable exponential approximations, we obtain a simple MTLL expression with an exponential function as dominant term. The simple exponential MTLL expression yields the optimum loop parameters corresponding to the maximum MTLL. Simulation results confirm that the optimum loop parameters corresponding to our analytical MTLL computation method and to the simplified exponential approximation coincide. Besides the crucial analytical MTLL results, the OU random processes yield additionally the likewise crucial analytical jitter results.

Modelling the IDV Emissions of the BL Lac Objects with a Langevin Type Stochastic Differential Equation

In this paper, we introduce a simplified model for explaining the observations of the optical intraday variability (IDV) of the BL Lac Objects. We assume that the source of the IDV are the stochastic oscillations of an accretion disk around a supermassive black hole. The Stochastic Fluctuations on the vertical direction of the accretion disk are described by using a Langevin type equation with a damping term and a random, white noise type force. Furthermore, the preliminary numerical simulation results are presented, which are based on the numerical analysis of the Langevin stochastic differential equation.

A second-order fluid queue with subordinator input and Markov-modulated linear output

We consider an infinite capacity second-order fluid queue with subordinator input and Markovmodulated linear release rate. The fluid queue level is described by a generalized Langevin stochastic differential equation (SDE). Applying infinitesimal generator, we obtain the stationary distribution that satisfies an integro-differential equation. We derive the solution of the SDE and study the transient level’s convergence in distribution. When the coefficients of the SDE are constants, we deduce the system transient property.

Relativistic diffusion

We discuss relativistic diffusion in proper time in the approach of Schay (Ph.D. thesis, Princeton University, Princeton, NJ, 1961) and Dudley [Ark. Mat. 6, 241 (1965)]. We derive (Langevin) stochastic differential equations in various coordinates. We show that in some coordinates the stochastic differential equations become linear. We obtain momentum probability distribution in an explicit form. We discuss a relativistic particle diffusing in an external electromagnetic field. We solve the Langevin equations in the case of parallel electric and magnetic fields. We derive a kinetic equation for the evolution of the probability distribution. We discuss drag terms leading to an equilibrium distribution. The relativistic analog of the Ornstein-Uhlenbeck process is not unique. We show that if the drag comes from a diffusion approximation to the master equation then its form is strongly restricted. The drag leading to the Tsallis equilibrium distribution satisfies this restriction whereas the one of the Jüttner distribution does not. We show that any function of the relativistic energy can be the equilibrium distribution for a particle in a static electric field. A preliminary study of the time evolution with friction is presented. It is shown that the problem is equivalent to quantum mechanics of a particle moving on a hyperboloid with a potential determined by the drag. A relation to diffusions appearing in heavy ion collisions is briefly discussed.

CEST and MEST: Tools for the simulation of radio frequency electric discharges in waveguides

In this paper we present two software tools for the simulation of electron multiplication processes in radio frequency (RF) waveguides. The electric discharges are caused by the multiplication of a small initial number of electrons. These are accelerated by the RF field and produce new electrons either by collisions with the walls of the waveguide (ripping new electrons from them), or by ionization of the neutral atoms of a gas inside the device. MEST allows simulating the Multipactor effect, a discharge produced in vacuum and generated by the collision of the electrons with the walls. CEST simulates the discharge when in addition a neutral gas is present in the waveguide, at pressures lower than ground levels (often denominated Corona discharge). The main characteristic of both tools is that they implement individual-based, microscopic models, where every electron is individually represented and tracked. In the case of MEST, the simulation is discrete-event, as the trajectory of each electron can be computed analytically. In CEST we use a hybrid simulation approach. The trajectory of each electron is governed by the Langevin stochastic differential equations that take into account a deterministic RF electric force and the random interaction with the neutral atom background. In addition, wall and ionizing collisions are modelled as discrete events. The tools allow performing batches of simulations with different wall coating materials and gases, and have produced results in good agreement with experimental and theoretical data. The different output forms generated at run-time have proven to be very useful in order to analyze the different discharge processes. The tools are valuable for the selection of the most promising coating materials for the construction of the waveguide, as well as for the identification of safe operating parameters.

Statistical mechanical theory for steady state systems. VI. Variational principles

Several variational principles that have been proposed for nonequilibrium systems are analyzed. These include the principle of minimum rate of entropy production due to Prigogine [Introduction to Thermodynamics of Irreversible Processes (Interscience, New York, 1967)], the principle of maximum rate of entropy production, which is common on the internet and in the natural sciences, two principles of minimum dissipation due to Onsager [Phys. Rev. 37, 405 (1931)] and to Onsager and Machlup [Phys. Rev. 91, 1505 (1953)], and the principle of maximum second entropy due to Attard [J. Chem.. Phys. 122, 154101 (2005); Phys. Chem. Chem. Phys. 8, 3585 (2006)]. The approaches of Onsager and Attard are argued to be the only viable theories. These two are related, although their physical interpretation and mathematical approximations differ. A numerical comparison with computer simulation results indicates that Attard's expression is the only accurate theory. The implications for the Langevin and other stochastic differential equations are discussed.

LONG-TERM BEHAVIOUR OF LARGE MECHANICAL SYSTEMS WITH RANDOM INITIAL DATA

We study the long-time behaviour of large systems of ordinary differential equations with random data. Our main focus is a Hamiltonian system which describes a distinguished particle attached to a large collection of heat bath particles by springs. In the limit where the size of the heat bath tends to infinity, the trajectory of the distinguished particle can be weakly approximated, on finite time intervals, by a Langevin stochastic differential equation. We examine the long-term behaviour of these trajectories, both analytically and numerically. We find ergodic behaviour manifest in both the long-time empirical measures and in the resulting auto-correlation functions.

Planck—Wheeler Quantum Foam as White Noise: Metric Diffusion and Congruence Focussing for Fluctuating Spacetime Geometry

It is expected that quantum effects endow spacetime with stochastic properties near the Planck scale as exemplified by random fluctuations of the metric, usually referred to as spacetime foam or geometrodynamics. In this paper, a methodology is presented for incorporating Planck scale stochastic effects and corrections into general relativity within the ADM formalism, by coupling the Riemann 3-metric to white noise. The ADM—Cauchy evolution of a Riemann 3-metric h ij (t) induced on spacelike hypersurface C(t) can be interpreted within pure general relativity as a smooth geodesic flow in superspace, whose points consist of equivalence classes of 3-metrics. Coupling white noise to h ij gives Langevin stochastic differential equations for the Cauchy evolution of h ij, which is now a Brownian motion or diffusion in superspace. A fluctuation h′ij away from h ij is considered to be related to h ij by elements of the diffeomorphism group diff(C). Hydrodynamical Fokker—Planck continuity equations are formulated describing the stochastic Cauchy evolution of h ij as a probability flow. The Cauchy invariant or equilibrium solution gives a stationary probability distribution of fluctuations peaked around the deterministic metric. By selecting a physically viable ansatz for the scale dependent diffusion coefficient, one reproduces the Wheeler uncertainty relation for the metric fluctuations of quantum geometrodynamics. Treating h ij as a random variable, a non-linear Raychaudhuri—Langevin equation is derived describing the “geometro-hydrodynamics” of a congruence of fluid or dust matter propagating on the stochastic spacetime. For an initially converging congruence θ′>0 at s′ the singularity θ=−∞ at future proper time s=3/|θ′|$, which is expected in general relativity, is now smeared out near the Planck scale. Proper time s can be extended indefinitely (s→∞) so that intrinsic metric fluctuations can restore geodesic completeness although the geodesics remain trapped for all time: although a singularity can be removed the collapsing matter still creates a black hole. A Fokker—Planck formulation also gives zero probability that θ→−∞ for s→∞. Essentially, the short distance stochastic corrections to the deterministic equations of general relativity can remove pathologies such as singularities, conjugate points and geodesic incompleteness.

Toward consistent formulation of Reynolds stress and scalar flux closures

Langevin stochastic differential equations provide a consistent basis for Reynolds stress, scalar transport and p.d.f. models. However, the stochastic equations must be capable of representing existing closures, like the General Linear Model, or the Rotta and Monin return to isotropy formulations. A consistent approach to derive both Reynolds stress and scalar flux transport equations, starting from a stochastic differential equation for velocity fluctuations, is presented here. A set of algebraic relations for the dispersion tensor is derived for homogeneous shear flow and for the log-layer.