Generalized Langevin Equation Approach Research Articles

Cooperative Dynamics of Highly Entangled Linear Polymers within the Entanglement Tube.

We present a quantitative comparison of the dynamic structure factors from unentangled and strongly entangled poly(butylene oxide) (PBO) melts. As expected, the low molecular weight PBO displays Rouse dynamics, however, with very significant subdiffusive center-of-mass diffusion. The spectra from high molecular weight entangled PBO can be very well described by the dynamic structure factor based on the concept of local reptation, including the Rouse dynamics within the tube and allowing for non-Gaussian corrections. Comparing quantitatively the spectra from both polymers leads to the surprising result that their spectra differ only by the contribution of classical Rouse diffusion for the low molecular weight melt. The subdiffusive component is common for both the low and high molecular weight PBO melts, indicating that in both melts the same interchain potential is active, thereby supporting the validity of the Generalized Langevin Equation approach.

A generalized Langevin equation approach for barrier crossing dynamics in conformational transitions of proteins

Barrier crossing dynamics in conformational transitions of proteins are investigated in the framework of the inertial generalized Langevin equation with an exponential memory kernel in a parabolic potential. This approach yields an exact analytical expression for the time dependent Grote–Hynes rate and the transmission coefficient, which typically determines the kinetics of such transitions. The complete transition path time distribution (TPTD) and the mean transition path time (MTPT) are evaluated as a function of the frictional coefficient and barrier curvature. The results of TPTD show an excellent agreement with the experimental results of TPTD for the PrP prion protein and theoretical results in the high friction limit, while they exhibit a considerable deviation from the results of theory at the intermediate and low friction limits. The inertial terms significantly affect the short time dynamics of such transitions. The results of the TPTD and MTPT are discussed at low and high frictional limits with varying curvatures of the potential barrier. The TPTD decreases with an increase in the barrier curvature at the high friction limit and exhibits an exponential decay at long times. The MTPT decreases with an increase in the curvature of the barrier.

The multi-dimensional generalized Langevin equation for conformational motion of proteins.

Using the generalized Langevin equation (GLE) is a promising approach to build coarse-grained (CG) models of molecular systems since the GLE model often leads to more accurate thermodynamic and kinetic predictions than Brownian dynamics or Langevin models by including a more sophisticated friction with memory. The GLE approach has been used for CG coordinates such as the center of mass of a group of atoms with pairwise decomposition and for a single CG coordinate. We present a GLE approach when CG coordinates are multiple generalized coordinates, defined, in general, as nonlinear functions of microscopic atomic coordinates. The CG model for multiple generalized coordinates is described by the multidimensional GLE from the Mori-Zwanzig formalism, which includes an exact memory matrix. We first present a method to compute the memory matrix in a multidimensional GLE using trajectories of a full system. Then, in order to reduce the computational cost of computing the multidimensional friction with memory, we introduce a method that maps the GLE to an extended Markovian system. In addition, we study the effect of using a nonconstant mass matrix in the CG model. In particular, we include mass-dependent terms in the mean force. We used the proposed CG model to describe the conformational motion of a solvated alanine dipeptide system, with two dihedral angles as the CG coordinates. We showed that the CG model can accurately reproduce two important kinetic quantities: the velocity autocorrelation function and the distribution of first passage times.

Segment-scale, force-level theory of mesoscopic dynamic localization and entropic elasticity in entangled chain polymer liquids.

We develop a segment-scale, force-based theory for the breakdown of the unentangled Rouse model and subsequent emergence of isotropic mesoscopic localization and entropic elasticity in chain polymer liquids in the absence of ergodicity-restoring anisotropic reptation or activated hopping motion. The theory is formulated in terms of a conformational N-dynamic-order-parameter generalized Langevin equation approach. It is implemented using a universal field-theoretic Gaussian thread model of polymer structure and closed at the level of the chain dynamic second moment matrix. The physical idea is that the isotropic Rouse model fails due to the dynamical emergence, with increasing chain length, of time-persistent intermolecular contacts determined by the combined influence of local uncrossability, long range polymer connectivity, and a self-consistent treatment of chain motion and the dynamic forces that hinder it. For long chain melts, the mesoscopic localization length (identified as the tube diameter) and emergent entropic elasticity predictions are in near quantitative agreement with experiment. Moreover, the onset chain length scales with the semi-dilute crossover concentration with a realistic numerical prefactor. Distinctive novel predictions are made for various off-diagonal correlation functions that quantify the full spatial structure of the dynamically localized polymer conformation. As the local excluded volume constraint and/or intrachain bonding spring are softened to allow chain crossability, the tube diameter is predicted to swell until it reaches the radius-of-gyration at which point mesoscopic localization vanishes in a discontinuous manner. A dynamic phase diagram for such a delocalization transition is constructed, which is qualitatively consistent with simulations and the classical concept of a critical entanglement degree of polymerization.

Microscopic Theory of the Long-Time Diffusivity and Intermediate-Time Anomalous Transport of a Nanoparticle in Polymer Melts

The transport of a single nanoparticle in unentangled and entangled polymer melts is studied based on a microscopic, force-level, self-consistent generalized Langevin equation approach. Nanoparticle self-motion and length-scale-dependent polymer relaxation enter as two parallel and competing channels of force relaxation. For entangled melts, particle long-time mobility exhibits size-dependent non-Stokes–Einstein behaviors controlled by (i) fast diffusion and length-scale-dependent collective relaxation of the unentangled matrix for particles smaller than the tube diameter and (ii) polymer relaxation via reptation for nanoparticles larger than the tube diameter. The two regimes are connected by a relatively sharp but continuous crossover. Recovery of hydrodynamic behavior requires the nanoparticle to be roughly 10 times larger than the tube diameter. Theoretical predictions for the diffusion constant are in good agreement with recent simulations of unentangled and lightly entangled melts, including the crossover regime where the nanoparticle and tube diameters are roughly equal. The calculations are also in reasonable accord with several experiments in entangled synthetic polymer melts and DNA solutions, although the activated hopping motion not included in the present theory may be important in a narrow intermediate crossover regime for sufficiently entangled polymers. The predictions of the theory for intermediate-time anomalous diffusion are of mixed quality based on comparison to simulations. The significant limitations identified are associated with our minimalist treatment of collective density fluctuation relaxation on intermediate time and length scales, including the possible importance of non-reptation relaxation processes in entangled melts.

Brownian Motion of a Rayleigh Particle Confined in a Channel: A Generalized Langevin Equation Approach

We study confined Brownian motion by investigating the memory function of a $$d$$ -dimensional hypercube ( $$d\ge 2$$ ), which is subject to a harmonic potential and suspended in an ideal gas confined by two parallel walls. For elastic walls and under the infinite-mass limit, we obtain analytic expressions for the force autocorrelation function and the memory function. The transverse-direction memory function possesses a nonnegative tail decaying like $$t^{-(d-1)}$$ , from which anomalous diffusion is expected for $$d=2$$ . For $$d=3$$ , the position-dependent friction coefficient becomes larger than the unconfined case and the increment is inversely proportional to the square of the distance from the wall. We also perform molecular dynamics simulations with thermal walls and/or a finite-mass hypercube. We observe faster decay due to the thermal wall ( $$t^{-3}$$ for $$d=2$$ and $$t^{-5}$$ for $$d=3$$ under the fully thermalizing wall) and convergence behaviors of the finite-mass memory function, which are different from the unconfined case.

Temperature-driven irreversible generalized Langevin equation can capture the nonequilibrium dynamics of two dissipated coupled oscillators

Kawai and Komatsuzaki [J. Chem. Phys. 134, 114523 (2011)] recently derived the nonequilibrium generalized Langevin equation (GLE) for a nonstationary system using the projection operator technique. In the limit when the environment is slowly changing (that is, a quasi-equilibrium bath), it should reduce to the irreversible GLE approach (iGLE) [J. Chem. Phys. 111, 7701 (1999)]. Kawai and Komatsuzaki, however, found that the driven harmonic oscillator, an example of a nonequilibrium system does not obey the iGLE presumably because it did not quite satisfy the limiting conditions of the latter. Notwithstanding the lack of a massive quasi-equilibrium bath (one of the conditions under which the iGLE had been derived earlier), we found that the temperature-driven iGLE (T-iGLE) [J. Chem. Phys. 126, 244506 (2007)] can reproduce the nonequilibrium dynamics of a driven dissipated pair of harmonic oscillators. It requires a choice of the function representing the coupling between the oscillator coordinate and the bath and shows that the T-iGLE representation is consistent with the projection operator formalism if only dominant bath modes are taken into account. Moreover, we also show that the more readily applicable phenomenological iGLE model is recoverable from the Kawai and Komatsuzaki model beyond the adiabatic limit used in the original T-iGLE theory.

Memory effect and fluctuating anomalous dynamics of a tagged monomer

We analyze the anomalous dynamics of a tagged monomer under external navigation. The memory effect causing the anomaly is elucidated, which depends on the magnitude of the force. In particular, the nonlinear and nonequilibrium memory effect under strong force is characterized by the force-dependent self-affine process for the tension transmission along the connectivity. Utilizing such knowledge, a generalized Langevin equation approach is proposed to quantify the fluctuating dynamics of driven anomalous walkers.

Origin of Hyperdiffusion in Generalized Brownian Motion

We study a minimal non-Markovian model of superdiffusion which originates from long-range velocity correlations within the generalized Langevin equation approach. The model allows for a three-dimensional Markovian embedding. The emergence of a transient hyperdiffusion, (Δx2(t))∝t(2+λ), with λ∼1-3 is detected in tilted washboard potentials before it ends up in a ballistic asymptotic regime. We relate this phenomenon to a transient heating of particles T(kin)(t)∝t(λ) from the thermal bath temperature T to some maximal kinetic temperature Tmax. This hyperdiffusive transient regime ceases when the particles arrive at the maximal kinetic temperature.

Fractional generalized Langevin equation approach to single-file diffusion

Fractional generalized Langevin equation with external force is used to model single-file diffusion. It is found that for external force that varies with power law the solution for such a fractional Langevin equation gives the correct short and long time behavior for the mean square displacement of single-file diffusion when appropriate choice of parameters associated with fractional generalized Langevin equation are used. By considering some special cases of the fractional generalized Langevin equation, a new class of closed analytic expressions for the mean square displacement of single-file diffusion can be obtained. The effective Fokker–Planck equation associated with single-file diffusion is briefly considered.

Viscoelastic subdiffusion: From anomalous to normal

We study viscoelastic subdiffusion in bistable and periodic potentials within the generalized Langevin equation approach. Our results justify the (ultra)slow fluctuating rate view of the corresponding bistable non-Markovian dynamics which displays bursting and anticorrelation of the residence times in two potential wells. The transition kinetics is asymptotically stretched exponential when the potential barrier V0 several times exceeds thermal energy k(B)T [V(0) approximately (2-10)k(B)T] and it cannot be described by the non-Markovian rate theory (NMRT). The well-known NMRT result approximates, however, ever better with the increasing barrier height, the most probable logarithm of the residence times. Moreover, the rate description is gradually restored when the barrier height exceeds a fuzzy borderline which depends on the power-law exponent of free subdiffusion alpha . Such a potential-free subdiffusion is ergodic. Surprisingly, in periodic potentials it is not sensitive to the barrier height in the long time asymptotic limit. However, the transient to this asymptotic regime is extremally slow and it does profoundly depend on the barrier height. The time scale of such subdiffusion can exceed the mean residence time in a potential well or in a finite spatial domain by many orders of magnitude. All these features are in sharp contrast with an alternative subdiffusion mechanism involving jumps among traps with the divergent mean residence time in these traps.

Modeling real dynamics in the coarse-grained representation of condensed phase systems

This work presents a systematic multiscale methodology to provide a more faithful representation of real dynamics in coarse-grained molecular simulation models. The theoretical formalism is based on the recently developed multiscale coarse-graining (MS-CG) method [S. Izvekov and G. A. Voth, J. Phys. Chem. B. 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] and relies on the generalized Langevin equation approach and its simpler Langevin equation limit. The friction coefficients are determined in multiscale fashion from the underlying all-atom molecular dynamics simulations using force-velocity and velocity-velocity correlation functions for the coarse-grained sites. The diffusion properties in the resulting CG Brownian dynamics simulations are shown to be quite accurate. The time dependence of the velocity autocorrelation function is also well-reproduced relative to the all-atom model if sufficient resolution of the CG sites is implemented.

The rheology of a dilute suspension of Brownian dipolar spheroids in a simple shear flow under the action of an external force

The effect of rotational Brownian motion on the rheology of a dilute suspension of dipolar spheroids in a simple shear flow under the action of an external force field, is investigated through a generalized Langevin equation approach. The force field is assumed to be either constant or periodic. In the case of constant external fields earlier results in the literature are reproduced, while for the case of periodic forcing certain parametric regimes corresponding to weak Brownian diffusion are identified where the rheological parameters evolve chaotically and settle onto a low dimensional attractor. The response of the system to variations in the strengths of the force field and diffusion is also analyzed through numerical experiments. These results correspond to the region of weak Brownian motion where usual methods render the problem intractable.

3D Generalized langevin equation approach to gas–surface reactive scattering: model H+H→H 2/Si(100)-(2×1)

Classical trajectory calculation has been performed for the H+H→H 2/Si(100)-(2×1) reaction by the 3D Generalized Langevin Equation (GLE) approach. The implementation of the 3D GLE approach to the H+H→H 2/Si(100)-(2×1) reaction is presented. Reaction probabilities are calculated for given surface temperatures and given collision energies. We also calculated vibrational and rotational distributions of product H 2 molecules from the reaction. About 80% of the product hydrogen molecules are in the ground vibrational state and the remaining 20% of the products are in the excited states. The rotational state shows non-Boltzmann distribution which can be seen in the direct collision process. Vibrational and Rotational distributions are strongly related to the impact parameter. The vibrational distribution is correlated with the x-component of the impact parameter and reflects the dimer nature of the silicon (2×1) surface. Details of the dynamics involving vibrational and rotational transitions are discussed.

Generalized Langevin equation approach to higher-order classical response: second-order-response time-resolved Raman experiment in CS2.

A simple, systematic generalized Langevin equation approach for calculating classical nonlinear response functions is formulated and discussed. The two-time Poisson brackets appearing at second and higher order are rendered tractable by a physically motivated approximation. The method is used to calculate the fifth order (second order response) Raman response of liquid CS2. Agreement with simulation is good, but the simplicity of the theoretical expression suggests that the path to obtaining qualitatively new information about liquids with the fifth order experiment is uncertain. Further applications of the basic approach are suggested.

A novel approach to computing the orientation moments of spheroids in simple shear flow at arbitrary Péclet number

A Langevin approach to computing the orientation moments of a dilute suspension of spheroids in a simple shear flow at arbitrary Péclet number is presented. In this method we obtain the equations governing the time evolution of the orientation averages using a generalized Langevin equation approach and develop a computational technique for computing the evolution of the moments from these equations. These results are compared with those available in the literature obtained from other methods and show good agreement. The approach presented here can be easily generalized to a number of similar systems such as forced suspensions of dipolar particles in shear flows and can be applied to other flow problems governed by appropriate Fokker–Planck equations.

Long-time tracer diffusion of nonspherical brownian particles

The long-time tracer-diffusion properties of a nonspherical Brownian particle that interacts with a suspension of spherical particles are studied in terms of an idealized but nontrivial two-dimensional model system. For this system, the predictions of the generalized Langevin equation approach to tracer diffusion can be calculated, and compared with the (extrapolated) results of a computer simulation experiment. In the model, the nonspherical particle is represented by a rigid linear array of N(T) (=2 or 3) spherical particles with nearest-neighbor separation DeltaL. We calculate the long-time rotational and (transverse and longitudinal) translational diffusion coefficients. The theory is found to reproduce qualitatively and quantitatively the main features of the extrapolated results. Finally, we also present theoretical results that derive from still simpler approximate theoretical schemes.

Intermediate-time tracer-diffusion of nonspherical Brownian particles

The time-dependent tracer-diffusion properties of a nonspherical Brownian particle that interacts with a suspension of spherical particles are studied in terms of an idealized but nontrivial model system for which the predictions of the generalized Langevin equation approach to tracer diffusion can be calculated, and compared with the results of a computer simulation experiment. In the model, the nonspherical particle is represented by a linear array of NT (=2 or 3) spherical particles with nearest-neighbor separation ΔL. For this model, we calculate the rotational and the (transversal and longitudinal) translational mean squared displacements, both, directly from the computer simulation, and approximately using the generalized Langevin equation approach. The theory is found to reproduce qualitatively and quantitatively the main features of the results of the simulation experiment for these properties.

Long-time self-diffusion in a model ferrofluid

The long-time self-diffusion coefficients of a concentrated ferrofluid are determined theoretically with the use of the generalized Langevin equation approach. The dependences of the translational and rotational diffusions on the concentration and the dipolar strength are studied. These dependences are expressed through the static structural information of the ferrofluid suspension. It is found that the effect of the dipolar interaction leads to the strong suppression of both long-time coefficients compared to the free diffusion coefficients of the particles in the infinite dilute case.

Dynamics of Gas−Surface Interactions: Reaction of Atomic Oxygen with Chemisorbed Hydrogen on Tungsten

The reaction of gas-phase atomic oxygen with hydrogen atoms chemisorbed on a tungsten surface is studied by means of classical trajectory procedures. Flow of energy between the reaction zone and the bulk solid phase is treated in the generalized Langevin equation approach. Most reactive events occur in small-impact-parameter collisions, in which the incident gas atom undergoes only one impact with the adatom, producing vibrationally excited OH radicals on a subpicosecond scale via the Eley−Rideal mechanism. Short-time calculations show that, in these collisions, the formation of the OH bond and subsequent dissociation of the H−surface bond occur in ∼100 fs. A small fraction of reactive events occurs in a multiple-impact collision, forming a long-lived complex on the surface. As the impact parameter increases, both reaction probability and vibrational excitation decrease, thus producing a population inversion. The dissipation of reaction energy to the heat bath can be adequately described using a 10-atom chain with the chain end bound to the rest of the solid. The probability of OH(g) formation is not sensitive to the variation of surface temperatures between 0 and 300 K, whereas it rapidly rises with the gas temperature up to 1000 K, above which it remains nearly constant. A modified LEPS potential energy surface is used for the reaction zone interaction, whereas the framework of harmonic representation is used in modeling chain atoms.