Stochastic Langevin Equation Research Articles

Stochastic Langevin Model for Flow and Transport in Porous Media

We present a new model for fluid flow and solute transport in porous media, which employs smoothed particle hydrodynamics to solve a Langevin equation for flow and dispersion in porous media. This allows for effective separation of the advective and diffusive mixing mechanisms, which is absent in the classical dispersion theory that lumps both types of mixing into dispersion coefficient. The classical dispersion theory overestimates both mixing-induced effective reaction rates and the effective fractal dimension of the mixing fronts associated with miscible fluid Rayleigh-Taylor instabilities. We demonstrate that the stochastic (Langevin equation) model overcomes these deficiencies.

Improved configuration space sampling: Langevin dynamics with alternative mobility

We present a new and efficient method for determining optimal configurations of a large number (N) of interacting particles. We use a coarse-grained stochastic Langevin equation in the overdamped limit to describe the dynamics of this system and replace the standard mobility by an effective space dependent inverse Hessian correlation matrix. Due to the analogy of the drift term in the Langevin equation and the update scheme in Newton's method, we expect accelerated dynamics or improved convergence in the convex part of the potential energy surface Phi. The stochastic noise term, however, is not only essential for proper thermodynamic sampling but also allows the system to access transition states in the concave parts of Phi. We employ a Broyden-Fletcher-Goldfarb-Shannon method for updating the local mobility matrix. Quantitative analysis for one and two dimensional systems shows that the new method is indeed more efficient than standard methods with constant effective friction. Due to the construction, our effective mobility adapts high values/low friction in configurations which are less optimal and low values/high friction in configurations that are more optimal.

Anomalous diffusion, structural relaxation and shear thinning in glassy hard spherefluids

The stochastic nonlinear Langevin equation theory of the single-particle dynamics ofglassy hard sphere fluids and suspensions has been applied to address severalissues raised by recent experimental and simulation studies. The theoreticallypredicted degree of non-Fickian behaviour at intermediate times, and the slowstructural relaxation component of the small wavevector incoherent dynamic structurefactor, are compared quantitatively with recent measurements and reveal goodagreement. A roughly power law growth, with a common exponent, of the classic andalternative non-Gaussian parameter amplitudes with mean alpha relaxation timeis predicted, and qualitative similarities with a multi-point susceptibility thatquantifies dynamic heterogeneity effects are noted. The nonlinear rheology version ofthe theory has been quantitatively applied to explain recent measurements ofshear-induced acceleration of the single-particle relaxation time on the local cage scale.The resulting no adjustable parameter calculations are in remarkable agreementwith the experimental observations for the absolute magnitude, volume fractiondependence and fractional power law scaling aspects of the shear thinning phenomenon.

Center-of-mass tomographic approach to quantum dynamics

In the tomography representation we propose a new approach, which describes the dynamics of quantum particles by the Kolmogorov equations for non-negative propagators. To solve the Kolmogorov equations we use a diffusive Markovian random processes described by the related nonlinear stochastic Langevin equations. As a result the dynamics of quantum particles is described by the proposed numerical scheme combining both Langevin dynamics and Monte Carlo methods. We test the developed approach by applying it to the wave packet dynamics in harmonic potentials and to particle tunneling through a barrier.

Large-amplitude jumps and non-Gaussian dynamics in highly concentrated hard sphere fluids

Our microscopic stochastic nonlinear Langevin equation theory of activated dynamics has been employed to study the real-space van Hove function of dense hard sphere fluids and suspensions. At very short times, the van Hove function is a narrow Gaussian. At sufficiently high volume fractions, such that the entropic barrier to relaxation is greater than the thermal energy, its functional form evolves with time to include a rapidly decaying component at small displacements and a long-range exponential tail. The "jump" or decay length scale associated with the tail increases with time (or particle root-mean-square displacement) at fixed volume fraction, and with volume fraction at the mean alpha relaxation time. The jump length at the alpha relaxation time is predicted to be proportional to a measure of the decoupling of self-diffusion and structural relaxation. At long times corresponding to mean displacements of order a particle diameter, the volume fraction dependence of the decay length disappears. A good superposition of the exponential tail feature based on the jump length as a scaling variable is predicted at high volume fractions. Overall, the theoretical results are in good accord with recent simulations and experiments. The basic aspects of the theory are also compared with a classic jump model and a dynamically facilitated continuous time random-walk model. Decoupling of the time scales of different parts of the relaxation process predicted by the theory is qualitatively similar to facilitated dynamics models based on the concept of persistence and exchange times if the elementary event is assumed to be associated with transport on a length scale significantly smaller than the particle size.

Theory of gelation, vitrification, and activated barrier hopping in mixtures of hard and sticky spheres

Naive mode coupling theory (NMCT) and the nonlinear stochastic Langevin equation theory of activated dynamics have been generalized to mixtures of spherical particles. Two types of ideal nonergodicity transitions are predicted corresponding to localization of both, or only one, species. The NMCT transition signals a dynamical crossover to activated barrier hopping dynamics. For binary mixtures of equal diameter hard and attractive spheres, a mixture composition sensitive "glass-melting" type of phenomenon is predicted at high total packing fractions and weak attractions. As the total packing fraction decreases, a transition to partial localization occurs corresponding to the coexistence of a tightly localized sticky species in a gel-like state with a fluid of hard spheres. Complex behavior of the localization lengths and shear moduli exist because of the competition between excluded volume caging forces and attraction-induced physical bond formation between sticky particles. Beyond the NMCT transition, a two-dimensional nonequilibrium free energy surface emerges, which quantifies cooperative activated motions. The barrier locations and heights are sensitive to the relative amplitude of the cooperative displacements of the different species.

Universal shape law of stochastic supercritical bifurcations: Theory and experiments

A universal analytical expression for the supercritical bifurcation shape of transverse one-dimensional (1D) systems in the presence of additive noise is given. The stochastic Langevin equation of such systems is solved by using a Fokker-Planck equation, leading to the expression for the most probable amplitude of the critical mode. From this universal expression, the shape of the bifurcation, its location, and its evolution with the noise level are completely defined. Experimental results obtained for a 1D transverse Kerr-type slice subjected to optical feedback are in excellent agreement.

Asymptotic velocity of one dimensional diffusions with periodic drift

We consider the asymptotic behavior of the solution of one dimensional stochastic differential equations and Langevin equations in periodic backgrounds with zero average. We prove that in several such models, there is generically a non-vanishing asymptotic velocity, despite of the fact that the average of the background is zero.

Kinetics of phase separation in the critical spherical model and local scale invariance

The scaling forms of the space- and time-dependent two-time correlation and responsefunctions are calculated for the kinetic spherical model with a conserved orderparameter and quenched to its critical point from a completely disordered initialstate. The stochastic Langevin equation can be split into a noise part and into adeterministic part which has local scale transformations with a dynamical exponentz = 4 as a dynamical symmetry. An exact reduction formula allows us to express any physicalaverage in terms of averages calculable from the deterministic part alone. The exactspherical model results are shown to agree with these predictions of local scale invariance.The results also include kinetic growth with mass conservation as described by theMullins–Herring equation.

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.

Spatial correlation functions of one-dimensional Bose gases at equilibrium

The dependence of the three lowest order spatial correlation functions of a harmonically confined Bose gas on temperature and interaction strength is presented at equilibrium. Our analysis is based on a stochastic Langevin equation for the order parameter of a weakly-interacting gas. Comparison of the predicted first order correlation functions to those of appropriate mean field theories demonstrates the potentially crucial role of density fluctuations on the equilibrium coherence length. Furthermore,the change in both coherence length and shape of the correlation function, from gaussian to exponential, with increasing temperature is quantified. Moreover, the presented results for higher order correlation functions are shown to be in agreeement with existing predictions. Appropriate consideration of density-density correlations is shown to facilitate a precise determination of quasi-condensate density profiles, providing an alternative approach to the bimodal density fits typically used experimentally.

A new approach to IR bimaterial detectors theory

In this paper we propose a new theoretical approach to the analysis of bimaterial infrared thermal detector (BMD) performance. In order to determine the basic parameters of a BMD—sensitivity, noise equivalent power, detectivity—we considered the BMD as a mechanical oscillating system and introduced an equivalent “thermomechanical” excitation force. This force is proportional with temperature changes and causes mechanical oscillations of the bimaterial cantilever. After identifying all of the relevant noise mechanisms (temperature fluctuations, Brownian motion), we solved the appropriate Langevin stochastic equation and obtained the mean square deflection of the bimaterial cantilever oscillator. This enabled us to determine all of the important BMD parameters. These parameters depend on the relevant thermal, mechanical and geometrical properties of the constituent parts of the detector and the chosen materials, as well as on the gas type and pressure inside the housing. Our analysis is focused on the study of pressure influence to the BMD performance. We showed that detectivity can approach the ideal value with pressure decrease if other bimaterial microcantilever parameters are properly chosen. Finally, we applied our theory on a BMD fabricated at ORNL.

Activated hopping and dynamical fluctuation effects in hard sphere suspensions and fluids

Single particle Brownian dynamics simulation methods are employed to establish the full trajectory level predictions of our nonlinear stochastic Langevin equation theory of activated hopping dynamics in glassy hard sphere suspensions and fluids. The consequences of thermal noise driven mobility fluctuations associated with the barrier hopping process are determined for various ensemble-averaged properties and their distributions. The predicted mean square displacements show classic signatures of transient trapping and anomalous diffusion on intermediate time and length scales. A crossover to a stronger volume fraction dependence of the apparent nondiffusive exponent occurs when the entropic barrier is of order the thermal energy. The volume fraction dependences of various mean relaxation times and rates can be fitted by empirical critical power laws with parameters consistent with ideal mode-coupling theory. However, the results of our divergence-free theory are largely a consequence of activated dynamics. The experimentally measurable alpha relaxation time is found to be very similar to the theoretically defined mean reaction time for escape from the barrier-dominated regime. Various measures of decoupling have been studied. For fluid states with small or nonexistent barriers, relaxation times obey a simple log-normal distribution, while for high volume fractions the relaxation time distributions become Poissonian. The product of the self-diffusion constant and mean alpha relaxation time increases roughly as a logarithmic function of the alpha relaxation time. The cage scale incoherent dynamic structure factor exhibits nonexponential decay with a modest degree of stretching. A nearly universal collapse of the different volume fraction results occurs if time is scaled by the mean alpha relaxation time. Hence, time-volume fraction superposition holds quite well, despite the presence of stretching and volume fraction dependent decoupling associated with the stochastic barrier hopping process. The relevance of other origins of dynamic heterogeneity (e.g., mesoscopic domains), and comparison of our results with experiments, simulations, and alternative theories, is discussed.

Effect of strongly anharmonic longitudinal and transverse vibrations with wave vectork=2∕3(111)on the structural stability ofβ−Zrunder pressure

The pressure and temperature dependence of the vibrational frequency of two interacting strongly anharmonic longitudinal and transverse modes with wave vector $\mathbf{k}=2∕3(111)$ in $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{Zr}$ is studied by solving a set of stochastic differential Langevin equations with a thermostat of the white noise type. The appropriate effective potential is calculated within the electron density functional theory, taking into account the contributions to the free energy from the electronic entropy depending on the atomic displacements. An analysis of the changes in the spectral density of vibrations with the pressure and temperature allows us to determine the stability region of the bcc phase of zirconium at pressures up to $35\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. A good agreement is obtained with the experimental data available. From the calculation performed it follows that the structural instability of the Zr bcc lattice with respect to the displacements characteristic of vibrations with wave vector $\mathbf{k}=2∕3(111)$ is of significant importance not only for the $\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\omega}$ transition, but also for the $\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\alpha}$ transformation observed at pressures less than $5\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$.

Reliability of the Langevin perturbative solution in stochastic inflation

A method to estimate the reliability of a perturbative expansion of the stochastic inflationary Langevin equation is presented and discussed. The method is applied to various inflationary scenarios, as large field, small field, and running mass models. It is demonstrated that the perturbative approach is more reliable than could be na\"{\i}vely suspected and, in general, only breaks down at the very end of inflation.

A LES-Langevin model for turbulence

We propose a new model of turbulence for use in large-eddy simulations (LES). The turbulent force, represented here by the turbulent Lamb vector, is divided in two contributions. The contribution including only subfilter fields is deterministically modeled through a classical eddy-viscosity. The other contribution including both filtered and subfilter scales is dynamically computed as solution of a generalized (stochastic) Langevin equation. This equation is derived using Rapid Distortion Theory (RDT) applied to the subfilter scales. The general friction operator therefore includes both advection and stretching by the resolved scale. The stochastic noise is derived as the sum of a contribution from the energy cascade and a contribution from the pressure. The LES model is thus made of an equation for the resolved scale, including the turbulent force, and a generalized Langevin equation integrated on a twice-finer grid. The model is validated by comparison to DNS and is tested against classical LES models for isotropic homogeneous turbulence, based on eddy viscosity. We show that even in this situation, where no walls are present, our inclusion of backscatter through the Langevin equation results in a better description of the flow.

Derivation of a microscopic theory of barriers and activated hopping transport in glassy liquids and suspensions

A recently proposed microscopic activated barrier hopping theory [K. S. Schweizer and E. J. Saltzman, J. Chem. Phys. 119, 1181 (2003)] of slow single-particle dynamics in glassy liquids, suspensions, and gels is derived using nonequilibrium statistical mechanics. Fundamental elements underlying the stochastic nonlinear Langevin equation description include an inhomogeneous liquid or locally solid-state perspective, dynamic density-functional theory (DDFT), a local equilibrium closure, and a coarse-grained free-energy functional. A dynamic Gaussian approximation is not adopted which is the key for avoiding a kinetic ideal glass transition. The relevant excess free energy is of a nonequilibrium origin and is related to dynamic force correlations in the fluid. The simplicity of the approach allows external perturbations to be rather easily incorporated. Dynamic heterogeneity enters naturally via mobility fluctuations associated with the stochastic barrier-hopping process. The derivation both identifies the limitations of the theory and suggests new avenues for its systematic improvement. Comparisons with ideal mode-coupling theory, alternative DDFT approaches and a field theoretic path-integral formulation are presented.

Decoherence during inflation: The generation of classical inhomogeneities

We show how the quantum to classical transition of the cosmological fluctuations produced during inflation can be described by means of the influence functional and the master equation. We split the inflaton field into the system-field (long-wavelength modes), and the environment, represented by its own short-wavelength modes. We compute the decoherence times for the system-field modes and compare them with the other time scales of the model. We present the renormalized stochastic Langevin equation for an homogeneous system-field and then we analyze the influence of the environment on the power spectrum for some modes in the system.

Brownian dynamics simulation and experimental study of colloidal particle deposition in a microchannel flow

This paper reports an analysis of the irreversible deposition of colloidal particles from the pressure-driven flow in a microchannel within the framework of DLVO theory. A theoretical model is presented on the basis of the stochastic Langevin equation, incorporating the random Brownian motion of colloidal particles. Brownian dynamics simulation is used to compute the particle deposition in terms of the surface coverage. To validate the theoretical model, experiments are carried out using the parallel-plate flow cell technique, enabling direct videomicroscopic observation of the deposition kinetics of polystyrene latex particles in NaCl electrolytes. The theoretical predictions are compared with the experimental results, and good agreement is found.

An Alternative Numerical Approach to Solve the Langevin Equation Applied to Air Pollution Dispersion

In this work an alternative numerical approach to solve the three-dimensional stochastic Langevin equation applied to air pollution is proposed and tested. The method leads to a first-order differential equation whose solution is obtained through the method of successive approximations or Picard’s iteration method. Langevin models for Gaussian and non-Gaussian turbulence are obtained, considering Gaussian and bi-Gaussian probability density functions (PDF) of the turbulent velocity, respectively. The proposed approach is evaluated through the comparison with experimental data and results from three different models. A statistical analysis shows a good agreement between the comparisons.