Generalized Langevin Equation Research Articles

Derivation of a generalized Langevin equation from a generic time-dependent Hamiltonian

Abstract The traditional Mori-Zwanzig formalism yields equations of motion, so-called generalized Langevin equations (GLEs), for phase-space observables of interest from the microscopic dynamics of a many-body system governed by a time-independent Hamiltonian using projection techniques. By using time-ordered propagators and time-independent projection operators, we derive the GLE for a scalar observable from a generic time-dependent Hamiltonian. The only restriction in our derivation is that the time-dependent part of the Hamiltonian and the observable of interest depend on spatial phase-space variables only. If the observable obeys Gaussian statistics and the time-dependent part of the Hamiltonian can be expressed as an odd power of the observable, the friction memory kernel in the GLE becomes proportional to the second moment of the complementary force, as is the case for a time-independent Hamiltonian in the Mori-Zwanzig formalism.

Generalized Langevin dynamics for single beads in linear elastic networks

We derive generalized Langevin equations (GLEs) for single beads in linear elastic networks. In particular, the derivations of the GLEs are conducted without employing normal modes, resulting in two distinct representations in terms of resistance and mobility kernels. The fluctuation-dissipation relations are also confirmed for both GLEs. Subsequently, we demonstrate that these two representations are interconnected via Laplace transforms. Furthermore, another GLE is derived by utilizing a projection operator method, and it is shown that the equation obtained through the projection scheme is consistent with the GLE with the resistance kernel. As simple examples, the general theory is applied to the Rouse model and the ring polymer, where the GLEs with the resistance and mobility kernels are explicitly derived for arbitrary positions of the tagged bead in these models. Finally, the GLE with the mobility kernel is also derived for the elastic network with hydrodynamic interactions under the pre-averaging approximation. ©2024 American Physical Society 2024 American Physical Society

On the singular-hybrid type of the Langevin fractional differential equation with a numerical approach

The nonlinear fractional Langevin equation is an important equation used in various areas of physics and mathematics. In this article, we focus on the singular-hybrid case of the generalized Langevin equations and prove the existence of a solution using fixed point theory. We also provide numerical and graphical simulations to demonstrate the convergence and existence of a solution for this singular problem. Finally, we present numerical examples in the form of tables and graphs to examine the accuracy of the method.

Confined fluid dynamics in a viscoelastic, amorphous, and microporous medium: Study of a kerogen by molecular simulations and the generalized Langevin equation.

We combine the use of molecular dynamics simulations and the generalized Langevin equation to study the diffusion of a fluid adsorbed within kerogen, the main organic phase of shales. As a class of microporous and amorphous materials that can exhibit significant adsorption-induced swelling, the dynamics of the kerogen's microstructure is expected to play an important role in the confined fluid dynamics. This role is investigated by conducting all-atom simulations with or without solid dynamics. Whenever the dynamics coupling between the fluid and solid is accounted for, we show that the fluid dynamics displays some qualitative differences compared to bulk fluids, which can be modulated by the amount of adsorbed fluid owing to adsorption-induced swelling. We highlight that working with the memory kernel, the central time correlation function of the generalized Langevin equation, allows the fingerprint of the dynamics of the solid to appear on that of the fluid. Interestingly, we observe that the memory kernels of fluid diffusion in kerogen qualitatively behave as those of tagged particles in supercooled liquids. We emphasize the importance of reproducing the velocity-force correlation function to validate the memory kernel numerically obtained as confinement enhances the numerical instabilities. This route is interesting as it opens the way for modeling the impact of fluid concentration on the diffusion coefficient in such ultra-confining cases.

Statistical Mechanical Theories of Membrane Permeability.

A popular theoretical framework to compute the permeability coefficient of a molecule is provided by the classic Smoluchowski-Kramers treatment of the steady-state diffusive flux across a free-energy barrier. Within this framework, commonly termed "inhomogeneous solubility-diffusion" (ISD), the permeability, P, is expressed in closed form in terms of the potential of mean force and position-dependent diffusivity of the molecule of interest along the membrane normal. In principle, both quantities can be calculated from all-atom MD simulations. Although several methods exist for calculating the position-dependent diffusivity, each of these is at best an estimate. In addition, the ISD model does not account for memory effects along the chosen reaction coordinate. For these reasons, it is important to seek alternative theoretical formulations to determine the permeability coefficient that are able to account for the factors ignored by the ISD approximation. Using Green-Kubo linear response theory, we establish the familiar constitutive relation between the flux density across the membrane and the difference in the concentration of a permeant molecule, j = PΔC. On this basis, we derive a time-correlation function expression for the nonequilibrium flux across a membrane that is reminiscent of the transmission coefficient in the reactive flux formalism treatment of transition rates. An analysis based on the transition path theory framework is exploited to derive alternative expressions for the permeability coefficient. The different strategies are illustrated with stochastic simulations based on the generalized Langevin equation in addition to unbiased molecular dynamics simulations of water permeation of a lipid bilayer.

Coarse-Graining Conformational Dynamics with Multidimensional Generalized Langevin Equation: How, When, and Why.

A data-driven ab initio generalized Langevin equation (AIGLE) approach is developed to learn and simulate high-dimensional, heterogeneous, coarse-grained (CG) conformational dynamics. Constrained by the fluctuation-dissipation theorem, the approach can build CG models in dynamical consistency (DC) with all-atom molecular dynamics. We also propose practical criteria for AIGLE to enforce long-term DC. Case studies of a toy polymer, with 20 CG sites, and the alanine dipeptide, with two dihedral angles, elucidate why one should adopt AIGLE or its Markovian limit for modeling CG conformational dynamics in practice.

Extensive study of droplet dynamics with end-grafted polymer chains in Pickering emulsions: Molecular Dynamics simulation and a generic model derived from the generalized Langevin equation

This paper investigates Pickering emulsions (PEs), formed by suspending oil droplets as the dispersed phase in water as the continuous phase using uncharged particles at their interface. To prevent coalescence, the particles are grafted with polymer chains, as emulsifier, not only to provide steric repulsion forces through the excluded volume between monomers floating in water, compensating for the absence of Coulombic repulsive forces but also control emulsion stabilization/destabilization. We combined molecular dynamics (MD) simulations and a generalized Langevin equation (GLE) model to study the dynamics of hairy-droplets using two dynamic quantities: mean square displacement (MSD) and velocity autocorrelation function (VACF). The number of grafted-polymer-chains, f, was the main parameter of interest, while the other parameters of the system were kept constant. A statistical approach was used to estimate theoretical GLE-based model parameters along with their uncertainties, providing insight into the diffusion behavior of these hairy-droplets and specifically addressing the transition between different observed regimes. we capture three specific values of grafted-polymer-chains as, fb=47 and ft=185 describe the transitions of mushroom-brush conformation of grafted-polymer-chains and viscous-viscoelastic behavior of hairy-droplets, respectively, and fg=257 representing the gel-like state number.

Data-Driven Learning of the Generalized Langevin Equation with State-Dependent Memory.

We present a data-driven method to learn stochastic reduced models of complex systems that retain a state-dependent memory beyond the standard generalized Langevin equation with a homogeneous kernel. The constructed model naturally encodes the heterogeneous energy dissipation by jointly learning a set of state features and the non-Markovian coupling among the features. Numerical results demonstrate the limitation of the standard generalized Langevin equation and the essential role of the broadly overlooked state-dependency nature in predicting molecule kinetics related to conformation relaxation and transition.

A theoretical perspective on segmental relaxation dynamics of model dendrimer

The internal segmental diffusion and the relaxation modes are investigated, and the relationship between the topology of model dendrimers is established, all within the bounds of analytical and simulation approaches. The precise theoretical and simulation approaches are thoroughly described after sufficient data is provided for model theories and their predictions. The analytical framework of the Generalised Langevin Equation is exactly replicated in the results. The primary focus of this review is on the NMR technique for forecasting the occurrence of the prominent processes, such as bond pulsation, segment reorientation, and total dendrimer rotation, utilizing the relaxation spectrum at various time scales. The utilization of two autocorrelation functions, namely time and orientational, is given significant emphasis. These functions illustrate the mobility of branches and their dependence on the generation number. The text also includes discussions of the approximations, assumptions, and limitations of each author’s chosen technique, when relevant.

A theory of phonon-induced friction on molecular adsorbates

In this manuscript, we provide a general theory for how surface phonons couple to molecular adsorbates. Our theory maps the extended dynamics of a surface's atomic vibrational motions to a generalized Langevin equation, and by doing so captures these dynamics in a single quantity: the non-Markovian friction. The different frequency components of this friction are the phonon modes of the surface slab weighted by their coupling to the adsorbate degrees of freedom. Using this formalism, we demonstrate that physisorbed species couple primarily to acoustic phonons while chemisorbed species couple to dispersionless local vibrations. We subsequently derive equations for phonon-adjusted reaction rates using transition state theory and demonstrate that these corrections improve agreement with experimental results for CO desorption rates from Pt(111).

Derivation of the nonequilibrium generalized Langevin equation from a time-dependent many-body Hamiltonian.

It has become standard practice to describe systems that remain far from equilibrium even in their steady state by Langevin equationswith colored noise which is chosen independently from the friction contribution. Since these Langevin equationsare typically not derived from first-principle Hamiltonian dynamics, it is not clear whether they correspond to physically realizable scenarios. By exact Mori projection in phase space we derive the nonequilibrium generalized Langevin equation(GLE) for an arbitrary phase-space dependent observable A from a generic many-body Hamiltonian with a time-dependent external force h(t) acting on the same observable A. This is the same Hamiltonian from which the standard fluctuation-dissipation theorem is derived, which reflects the generality of our approach. The observable A could, for example, be the position of an atom, of a molecule or of a macroscopic object, the distance between two such entities or a more complex phase-space function such as the reaction coordinate of a chemical reaction or of the folding of a protein. The Hamiltonian could, for example, describe a fluid, a solid, a viscoelastic medium, or even a turbulent inhomogeneous environment. The GLE, which is a closed-form equationof motion for the observable A, is obtained in explicit form to all orders in h(t) and without restrictions on the type of many-body Hamiltonian or the observable A. If the dynamics of the observable A corresponds to a Gaussian process, the resultant GLE has a similar form as the equilibrium Mori GLE, and in particular the friction memory kernel is given by the two-point autocorrelation function of the sum of the complementary and the external force h(t). This is a nontrivial and useful result, as many observables that characterize nonequilibrium systems display Gaussian statistics. For non-Gaussian nonequilibrium observables correction terms appear in the GLE and in the relation between the force autocorrelation and the friction memory kernel, which are explicitly given in terms of cubic correlation functions of A. Interpreting the external force h(t) as a stochastic process, we derive nonequilibrium corrections to the fluctuation-dissipation theorem and present methods to extract all GLE parameters from experimental or simulation time-series data, thus making our nonequilibrium GLE a practical tool to study and model general nonequilibrium systems.

Effects of Colored Noise in the Dynamic Motions and Conformational Exploration of Enzymes

The intracellular environment displays complex dynamics influenced by factors such as molecular crowding and the low Reynolds number of the cytoplasm. Enzymes exhibiting active matter properties further heighten this complexity which can lead to memory effects. Molecular simulations often neglect these factors, treating the environment as a “thermal bath” using the Langevin equation (LE) with white noise. One way to consider these factors is by using colored noise instead within the generalized Langevin equation (GLE) framework, which allows for the incorporation of memory effects that have been observed in experimental data. We investigated the structural and dynamic differences in Shikimate kinase (SK) using LE and GLE simulations. Our results suggest that GLE simulations, which reveal significant changes, could be utilized for assessing conformational motions’ impact on catalytic reactions.

Memory effects in colloidal motion under confinement and driving

The transport of individual particles in inhomogeneous environments is complex and exhibits non-Markovian responses. The latter may be quantified by a memory function within the framework of the linear generalised Langevin equation (GLE). Here, we exemplify the implications of steady driving on the memory function of a colloidal model system for Brownian motion in a corrugated potential landscape, specifically, for one-dimensional motion in a sinusoidal potential. To this end, we consider the overdamped limit of the GLE, which is facilitated by separating the memory function into a singular (Markovian) and a regular (non-Markovian) part. Relying on exact solutions for the investigated model, we show that the random force entering the GLE must display a bias far from equilibrium, which corroborates a recent general prediction. Based on data for the mean-square displacement (MSD) obtained from Brownian dynamics simulations, we estimate the memory function for different driving strengths and show that already moderate driving accelerates the decay of the memory function by several orders of magnitude in time. We find that the memory may persist on much longer timescales than expected from the convergence of the MSD to its long-time asymptote. Furthermore, the functional form of the memory function changes from a monotonic decay to a non-monotonic, damped oscillatory behaviour, which can be understood from a competition of confined motion and depinning. Our analysis of the simulation data further reveals a pronounced non-Gaussianity, which questions the Gaussian approximation of the random force entering the GLE.

The challenge of non-Markovian energy balance models in climate.

We first review the way in which Hasselmann's paradigm, introduced in 1976 and recently honored with the Nobel Prize, can, like many key innovations in complexity science, be understood on several different levels. It can be seen as a way to add variability into the pioneering energy balance models (EBMs) of Budyko and Sellers. On a more abstract level, however, it used the original stochastic mathematical model of Brownian motion to provide a conceptual superstructure to link slow climate variability to fast weather fluctuations, in a context broader than EBMs, and led Hasselmann to posit a need for negative feedback in climate modeling. Hasselmann's paradigm has still much to offer us, but naturally, since the 1970s, a number of newer developments have built on his pioneering ideas. One important one has been the development of a rigorous mathematical hierarchy that embeds Hasselmann-type models in the more comprehensive Mori-Zwanzig generalized Langevin equation (GLE) framework. Another has been the interest in stochastic EBMs with a memory that has slower decay and, thus, longer range than the exponential form seen in his EBMs. In this paper, we argue that the Mori-Kubo overdamped GLE, as widely used in statistical mechanics, suggests the form of a relatively simple stochastic EBM with memory for the global temperature anomaly. We also explore how this EBM relates to Lovejoy et al.'s fractional energy balance equation.

ML-GLE: A machine learning enhanced Generalized Langevin equation framework for transient anomalous diffusion in polymer dynamics

In this work we introduce ML-GLE, a machine learning framework to generate long-term single-polymer dynamics by exploiting short-term trajectories from molecular dynamics (MD) simulations of polymer melts. Even with current advances in machine learning for MD, these polymeric materials remain difficult to simulate and characterize due to prohibitive computational costs when long relaxation timescales are involved. Our method relies on a 3D neural auto-regressive model for single polymer lower dimensional collective variables, called normal modes. This enhances the Generalized Langevin Equation (GLE) capabilities in modelling diffusion phenomena. We exploit a particular GLE solution which is known to reproduce the mean square displacement curve relative to transient anomalous diffusion and connect it with the normal modes collective variables. ML-GLE is capable of emulating the single polymer statistical properties in the long-term, predicting the diffusion coefficient. As a consequence, this results in an enormous acceleration in terms of simulation time with respect to the full-size simulation. Moreover, this approach is also scalable with system size.

Standard form of master equations for general non-Markovian jump processes: The Laplace-space embedding framework and asymptotic solution

We present a standard form of master equations (MEs) for general one-dimensional non-Markovian (history-dependent) jump processes, complemented by an asymptotic solution derived from an expanded system-size approach. The ME is obtained by developing a general Markovian embedding using a suitable set of auxiliary field variables. This Markovian embedding uses a Laplace-convolution operation applied to the velocity trajectory. We introduce an asymptotic method tailored for this ME standard, generalizing the system-size expansion for these jump processes. Under specific stability conditions tied to a single noise source, upon coarse graining, the generalized Langevin equation (GLE) emerges as a universal approximate model for point processes in the weak-coupling limit. This methodology offers a unified analytical tool set for general non-Markovian processes, reinforcing the universal applicability of the GLE founded in microdynamics and the principles of statistical physics. Published by the American Physical Society 2024

Scaling limits for the generalized Langevin equation via diffusion approximation theory

Scaling limits for the generalized Langevin equation via diffusion approximation theory

Energy fluctuations of a Brownian particle freely moving in a liquid

We study the statistical properties of the variation of the kinetic energy of a spherical Brownian particle that freely moves in an incompressible fluid at constant temperature. Based on the underdamped version of the generalized Langevin equation that includes the inertia of both the particle and the displaced fluid, we derive an analytical expression for the probability density function of such a kinetic energy variation during an arbitrary time interval, which exactly amounts to the energy exchanged with the fluid in absence of external forces. We also determine all the moments of this probability distribution, which can be fully expressed in terms of a function that is proportional to the velocity autocorrelation function of the particle. The derived expressions are verified by means of numerical simulations of the stochastic motion of a particle in a viscous liquid with hydrodynamic backflow for representative values of the time-scales of the system. Furthermore, we also investigate the effect of viscoelasticity on the statistics of the kinetic energy variation of the particle, which reveals the existence of three distinct regimes of the energy exchange process depending on the values of the viscoelastic parameters of the fluid.

Anomalous and ultraslow diffusion of a particle driven by power-law-correlated and distributed-order noises

We study the generalised Langevin equation (GLE) approach to anomalous diffusion for a harmonic oscillator and a free particle driven by different forms of internal noises, such as power-law-correlated and distributed-order noises that fulfil generalised versions of the fluctuation-dissipation theorem. The mean squared displacement and the normalised displacement correlation function are derived for the different forms of the friction memory kernel. The corresponding overdamped GLEs for these cases are also investigated. It is shown that such models can be used to describe anomalous diffusion in complex media, giving rise to subdiffusion, superdiffusion, ultraslow diffusion, strong anomaly, and other complex diffusive behaviours.

A Gauss-Newton method for iterative optimization of memory kernels for generalized Langevin thermostats in coarse-grained molecular dynamics simulations.

In molecular dynamics simulations, dynamically consistent coarse-grained (CG) models commonly use stochastic thermostats to model friction and fluctuations that are lost in a CG description. While Markovian, i.e., time-local, formulations of such thermostats allow for an accurate representation of diffusivities/long-time dynamics, a correct description of the dynamics on all time scales generally requires non-Markovian, i.e., non-time-local, thermostats. These thermostats typically take the form of a Generalized Langevin Equation (GLE) determined by a memory kernel. In this work, we use a Markovian embedded formulation of a position-independent GLE thermostat acting independently on each CG degree of freedom. Extracting the memory kernel of this CG model from atomistic reference data requires several approximations. Therefore, this task is best understood as an inverse problem. While our recently proposed approximate Newton scheme allows for the iterative optimization of memory kernels (IOMK), Markovian embedding remained potentially error-prone and computationally expensive. In this work, we present an IOMK-Gauss-Newton scheme (IOMK-GN) based on IOMK that allows for the direct parameterization of a Markovian embedded model.