Nonequilibrium Langevin Dynamics Research Articles

Dynamic density functional theory with inertia and background flow.

We present dynamic density functional theory (DDFT) incorporating general inhomogeneous, incompressible, time-dependent background flows and inertia, describing externally driven passive colloidal systems out of equilibrium. We start by considering the underlying nonequilibrium Langevin dynamics, including the effect of the local velocity of the surrounding liquid bath, to obtain the nonlinear, nonlocal partial differential equations governing the evolution of the (coarse-grained) density and velocity fields describing the dynamics of colloids. In addition, we show both with heuristic arguments, and by numerical solution, that our equations and solutions agree with existing DDFTs in the overdamped (high friction) limit. We provide numerical solutions that model the flow of hard spheres, in both unbounded and confined domains, and compare with previously derived DDFTs with and without the background flow.

Out-of-equilibrium dynamics of repulsive ranked diffusions: The expanding crystal.

We study the nonequilibrium Langevin dynamics of N particles in one dimension with Coulomb repulsive linear interactions. This is a dynamical version of the so-called jellium model (without confinement) also known as ranked diffusion. Using a mapping to the Lieb-Liniger model of quantum bosons, we obtain an exact formula for the joint distribution of the positions of the N particles at time t, all starting from the origin. A saddle-point analysis shows that the system converges at long time to a linearly expanding crystal. Properly rescaled, this dynamical state resembles the equilibrium crystal in a time-dependent effective quadratic potential. This analogy allows us to study the fluctuations around the perfect crystal, which, to leading order, are Gaussian. There are however deviations from this Gaussian behavior, which embody long-range correlations of purely dynamical origin, characterized by the higher-order cumulants of, e.g., the gaps between the particles, which we calculate exactly. We complement these results using a recent approach by one of us in terms of a noisy Burgers equation. In the large-N limit, the mean density of the gas can be obtained at any time from the solution of a deterministic viscous Burgers equation. This approach provides a quantitative description of the dense regime at shorter times. Our predictions are in good agreement with numerical simulations for finite and large N.

Education

This issue of SIAM Review presents two papers in the Education section. The first paper, “Time Correlation Functions of Equilibrium and Nonequilibrium Langevin Dynamics: Derivations and Numerics Using Random Numbers,” is coauthored by Xiaocheng Shang and Martin Krog̈er. Langevin dynamics is a term describing a mathematical model of the dynamics of molecular systems which uses stochastic differential equations. This model extends the molecular dynamics to allow for describing effects due to friction caused by solvent or air molecules or other phenomena that perturb the system. In this paper, the authors present the model equation for the behavior of a harmonic oscillator in two or three dimensions in the presence of friction, additive noise, and an external field with both rotational and deformational components. They study the time correlation function of the dynamics analytically and numerically. The paper provides a good illustration of the tools of stochastic calculus on a nontrivial application and helps evaluate the numerical solvers for stochastic differential equations (SDEs). The authors argue that the fair assessment of the efficiency and accuracy of numerical solvers for SDEs requires access to analytical reference solutions. However, deriving relevant analytical formulae is possible only for very simple cases. Here a two-dimensional nontrivial benchmark problem is analyzed and an exact solution is derived. The analysis in the paper includes also a discussion on inertial effects. The model equation formulated in section 2 is interesting because it arises in multiple applications. Furthermore, the absence of the restoring force or the external field in the model leads to popular special cases. Among those are the random walk, diffusion models, the motion of atoms in the presence of gravitational or centrifugal potential, nanomagnets subjected to magnetic fields, Brownian oscillators, rotational relaxation of molecules trapped in a three-dimensional crystal, and many others. Relevant literature describing those models is provided. The authors describe a direct approach to the derivation of the time correlation functions for general Langevin dynamics, as well as an approach that uses Fourier transforms and the Wiener--Khinchin theorem, and a complementary approach via the associated Fokker--Planck equation. The last portion of the paper contains the description of two numerical methods and numerical experience. The second paper presents “An Introduction to Quantum Computing, without the Physics,” written by Giacomo Nannicini. This paper introduces discrete mathematicians and computer scientists to quantum computing. While quantum computing uses certain quantum phenomena to perform computation, many scientists may learn about its principles without acquiring deeper knowledge of quantum physics. The author starts with a set of assumptions that are satisfied by a quantum computing device. From those assumptions further properties are derived in a rigorous way. The necessary concepts are formally introduced and all statements are supplied with mathematical proofs. The author adopts a model of computation, which is known as the quantum circuit model. Quoting from the paper, it works as follows: - The quantum computer has a state that is contained in a quantum register and is initialized in a predefined way. - The state evolves by applying operations specified in advance in the form of an algorithm. - At the end of the computation, some information on the state of the quantum register is obtained by means of a special operation, called a measurement. At the beginning of the paper, some properties of the tensor product are reviewed, which are used to express states and operations in a quantum device. For example, qubits are the quantum counterparts of the bits in our computers and the state of a $q$-qubit quantum register is identified with a unit vector in the space $(\mathbb{C}^2)^{\otimes q}$, where $\mathbb{C}^2$ is the complex Euclidean space. The exposition proceeds with the introduction and analysis of the types of states and their properties, the quantum phenomena of superposition and enlargement, measurement gates, and the input-output model for quantum computations. The second portion of the paper discusses several quantum algorithms including a numerical implementation of Grover's algorithm. It may be of interest that, although quantum computing is able to solve some problems much faster than the classical computer, it is still not known to solve NP-complete problems efficiently. The paper concludes with a section containing notes for further reading. According to the author's experience, the paper provides material for a graduate-level module in quantum computing for mathematicians and computer scientists. It is recommended to split the material into five lectures/meetings of 90--120 minutes each. The last meeting would aim at providing a hands-on numerical experience based on section 6 of the paper. The audience should be familiar with linear algebra and with basic concepts in computing such as Turing machines and algorithmic complexity.

Time Correlation Functions of Equilibrium and Nonequilibrium Langevin Dynamics: Derivations and Numerics Using Random Numbers

We study the time correlation functions of coupled linear Langevin dynamics with and without inertial effects, both analytically and numerically. The model equation represents the physical behavior of a harmonic oscillator in two or three dimensions in the presence of friction, additive noise, and an external field with both rotational and deformational components. This simple model plays pivotal roles in understanding more complicated processes. The analytical solution presented serves as a test of numerical integration schemes, and its derivation is presented in a fashion that allows it to be repeated directly in a classroom. While the results in the absence of fields (equilibrium) or confinement (free particle) are omnipresent in the literature, we write down, apparently for the first time, the full nonequilibrium results that may correspond, e.g., to a Hookean dumbbell embedded in a macroscopically homogeneous shear or mixed flow field. We demonstrate how the inertial results reduce to their noninertial counterparts in the nontrivial limit of vanishing mass. While the results are derived using basic integrations over Dirac delta distributions, we also provide alternative approaches involving (i) Fourier transforms, which seem advantageous only if the measured quantities also reside in Fourier space, and (ii) a Fokker--Planck equation and the moments of the probability distribution. The results, verified by numerical experiments, provide additional means of measuring the performance of numerical methods for such systems. It should be emphasized that this article provides specific details regarding the derivations of the time correlation functions as well as the implementations of various numerical methods, so that it can serve as a standalone piece for lessons in the framework of Itô stochastic differential equations and calculus.

Direct estimation of the parameters of a delayed, intermittent activation feedback model of postural sway during quiet standing

Human postural sway during quiet standing has been characterized as a proportional-integral-derivative controller with intermittent activation. In the model, patterns of sway result from both instantaneous, passive, mechanical resistance and delayed, intermittent resistance signaled by the central nervous system. A Kalman-Filter framework was designed to directly estimate from experimental data the parameters of the model’s stochastic delay differential equations with discrete dynamic switching conditions. Simulations showed that all parameters could be estimated over a variety of possible data-generating configurations with varying degrees of bias and variance depending on their empirical identification. Applications to experimental data reveal distributions of each parameter that correspond well to previous findings, suggesting that many useful, physiological measures may be extracted from sway data. Individuals varied in degree and type of deviation from theoretical expectations, ranging from harmonic oscillation to non-equilibrium Langevin dynamics.

Strong convergence of integrators for nonequilibrium Langevin dynamics

ABSTRACTSeveral numerical schemes are proposed for the solution of Nonequilibrium Langevin Dynamics (NELD), and the strong rate of convergence for each scheme is analyzed. The schemes considered here employ specialised periodic boundary conditions that deform with the flow, namely Lees-Edwards and Kraynik-Reinelt boundary conditions and their generalisations. We show that care must be taken when implementing standard stochastic integration schemes with these boundary conditions in order to avoid a breakdown in the strong order of convergence.

Convergence rates for nonequilibrium Langevin dynamics

We study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap.

Nonequilibrium Langevin dynamics: A demonstration study of shear flow fluctuations in a simple fluid.

The present paper is based on a recent success of the second-order stochastic fluctuation theory in describing time autocorrelations of equilibrium and nonequilibrium physical systems. In particular, it was shown to yield values of the related deterministic parameters of the Langevin equation for a Couette flow in a microscopic molecular dynamics model of a simple fluid. In this paper we find all the remaining constants of the stochastic dynamics, which then is simulated numerically and compared directly with the original physical system. By using these data, we study in detail the accuracy and precision of a second-order Langevin model for nonequilibrium physical systems theoretically and computationally. We find an intriguing relation between an applied external force and cumulants of the resulting flow fluctuations. This is characterized by a linear dependence of an athermal cumulant ratio, an apposite quantity introduced here. In addition, we discuss how the order of a given Langevin dynamics can be raised systematically by introducing colored noise.

Fine tuning classical and quantum molecular dynamics using a generalized Langevin equation.

Generalized Langevin Equation (GLE) thermostats have been used very effectively as a tool to manipulate and optimize the sampling of thermodynamic ensembles and the associated static properties. Here we show that a similar, exquisite level of control can be achieved for the dynamical properties computed from thermostatted trajectories. We develop quantitative measures of the disturbance induced by the GLE to the Hamiltonian dynamics of a harmonic oscillator, and show that these analytical results accurately predict the behavior of strongly anharmonic systems. We also show that it is possible to correct, to a significant extent, the effects of the GLE term onto the corresponding microcanonical dynamics, which puts on more solid grounds the use of non-equilibrium Langevin dynamics to approximate quantum nuclear effects and could help improve the prediction of dynamical quantities from techniques that use a Langevin term to stabilize dynamics. Finally we address the use of thermostats in the context of approximate path-integral-based models of quantum nuclear dynamics. We demonstrate that a custom-tailored GLE can alleviate some of the artifacts associated with these techniques, improving the quality of results for the modeling of vibrational dynamics of molecules, liquids, and solids.

Current fluctuations in nanopores: The effects of electrostatic and hydrodynamic interactions

Using nonequilibrium Langevin dynamics simulations of an electrolyte with explicit solvent particles, we investigate the effect of hydrodynamic interactions on the power spectrum of ionic nanopore currents. At low frequency, we find a power-law dependence of the power spectral density, with an exponent depending on the ion density. Surprisingly, however, the exponent is not affected by the presence of the neutral solvent particles. We conclude that hydrodynamic interactions do not affect the shape of the power spectrum in the frequency range studied.

Langevin Dynamics with Space-Time Periodic Nonequilibrium Forcing

We present results on the ballistic and diffusive behavior of the Langevin dynamics in a periodic potential that is driven away from equilibrium by a space-time periodic driving force, extending some of the results obtained by Collet and Martinez. In the hyperbolic scaling, a nontrivial average velocity can be observed even if the external forcing vanishes in average. More surprisingly, an average velocity in the direction opposite to the forcing may develop at the linear response level -- a phenomenon called negative mobility. The diffusive limit of the non-equilibrium Langevin dynamics is also studied using the general methodology of central limit theorems for additive functionals of Markov processes. To apply this methodology, which is based on the study of appropriate Poisson equations, we extend recent results on pointwise estimates of the resolvent of the generator associated with the Langevin dynamics. Our theoretical results are illustrated by numerical simulations of a two-dimensional system.

Derivation of Langevin dynamics in a nonzero background flow field

We propose a derivation of a nonequilibrium Langevin dynamics for a large particle immersed in a background flow field. A single large particle is placed in an ideal gas heat bath composed of point particles that are distributed consistently with the background flow field and that interact with the large particle through elastic collisions. In the limit of small bath atom mass, the large particle dynamics converges in law to a stochastic dynamics. This derivation follows the ideas of [D. Durr, S. Goldstein, and J. L. Lebowitz, 1981 and 1983; P. Calderoni, D. Durr, and S. Kusuoka, 1989] and provides extensions to handle the nonzero background flow. The derived nonequilibrium Langevin dynamics is similar to the dynamics in [M. McPhie, et al., 2001]. Some numerical experiments illustrate the use of the obtained dynamic to simulate homogeneous liquid materials under flow.

Nonequilibrium Shear Viscosity Computations with Langevin Dynamics

We study the mathematical properties of a nonequilibrium Langevin dynamics which can be used to estimate the shear viscosity of a system. More precisely, we prove a linear response result which allows us to relate averages over the nonequilibrium stationary state of the system to equilibrium canonical expectations. We then write a local conservation law for the average longitudinal velocity of the fluid and show how, under some closure approximation, the viscosity can be extracted from this profile. We finally characterize the asymptotic behavior of the velocity profile, in the limit where either the transverse or the longitudinal friction goes to infinity. Some numerical illustrations of the theoretical results are also presented.

Fluctuation relations in simple examples of non-equilibrium steady states

We discuss fluctuation relations in simple cases of non-equilibrium Langevin dynamics. Inparticular, we show that, close to non-equilibrium steady states with non-vanishingprobability currents, some of these relations reduce to a modified version of thefluctuation–dissipation theorem. The latter may be interpreted as the equilibrium-likerelation in the reference frame moving with the mean local velocity determined by theprobability current.