High Friction Limit Research Articles

Intermediate inflation in a generalized non-minimal derivative coupling model

In this work, we consider intermediate inflation in the context of the generalized non-minimal derivative coupling (GNMDC) model. In the GNMDC model, inflation is driven by a canonical scalar field that is coupled not only to gravity but also to the derivative of the scalar field. The model introduces new dynamics and features during the inflationary epoch. We find inflationary solutions with a power law scalar field for the power law coupling function. Additionally, we determine the inflaton potential that generates intermediate expansion of the scale factor. We also discuss the background equations in the high friction limit and derive constraints on the parameters of our model. Furthermore, we investigate the cosmological perturbations in the slow roll approximation within the GNMDC model, and we calculate the scalar and tensor spectral index and the tensor-to-scalar ratio during intermediate inflation. We compare the results of this model with observational data that can be used to test the model using cosmic microwave background radiation data. Overall, we establish conditions for the inflaton potential that ensure the continuation of accelerated expansion during slow roll inflation. We numerically analyze the power spectrum and spectral index for scalar and tensor modes in intermediate inflation in the high friction limit, and we use Planck 2018 data to obtain constraints on the parameters of the model. We demonstrate that intermediate inflation in the GNMDC model is successful in evaluation and explanation of background and perturbation quantities using observational data.

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.

Superior tribological and sealing performance of micro-crystalline diamond coated silicon carbide seals under dry friction and high load

Silicon carbide seals suffer severe wear under the conditions of dry friction and high load, posing a great challenge in achieving a stable performance and a long lifetime. In this work, micro-crystalline diamond coatings were deposited on silicon carbide sealing rings to prevent the failure caused by wear. The deposited micro-crystalline diamond coatings exhibit high-quality crystallinity and strong adhesive strength with the silicon carbide substrate. In sealing tests, the micro-crystalline diamond coated silicon carbide sealing pair demonstrates superior tribological and sealing properties under dry friction and high load of 1800 N, featuring a low coefficient of friction of 0.16, a low wear rate of 4.00 × 10−7 mm3·N−1·m−1 and a high friction limit of 5.24 MPa·m·s−1. Characterizations of morphology, chemical composition and microstructure confirm that the smoothening effect on the contact face and the formation of graphite during the sealing test synergistically contribute to the outstanding tribological and sealing performance.

On the high friction limit for the complete Euler system

On the high friction limit for the complete Euler system

Cahn–Hillard and Keller–Segel systems as high-friction limits of Euler–Korteweg and Euler–Poisson equations

Cahn–Hillard and Keller–Segel systems as high-friction limits of Euler–Korteweg and Euler–Poisson equations

Stability and asymptotic analysis for instationary gas transport via relative energy estimates

We consider the transport of gas in long pipes and pipeline networks for which the dynamics are dominated by friction at the pipe walls. The governing equations can be formulated as an abstract dissipative Hamiltonian system which allows us to derive perturbation bounds via relative energy estimates using a problem adapted nonlinear analysis. As particular consequences of these results, we are able to prove stability estimates with respect to initial conditions and model parameters and we conduct a quantitative asymptotic analysis in the high friction limit. Our results are established first for the flow in a single pipe and we then extend our analysis to pipe networks in the spirit of energy-based port-Hamiltonian modelling.

Diffusion in a two-dimensional energy landscape in the presence of dynamical correlations and validity of random walk model.

Diffusion in a multidimensional energy surface with minima and barriers is a problem of importance in statistical mechanics and it also has wide applications, such as protein folding. To understand it in such a system, we carry out theory and simulations of a tagged particle moving on a two-dimensional periodic potential energy surface, both in the presence and absence of noise. Langevin dynamics simulations at multiple temperatures are carried out to obtain the diffusion coefficient of a solute particle. Friction is varied from zero to large values. Diffusive motion emerges in the limit of a long time, even in the absence of noise. Noise destroys the correlations and increases diffusion at small friction. Diffusion thus exhibits a nonmonotonic friction dependence at the intermediate value of the damping, ultimately converging to our theoretically predicted value. The latter is obtained using the well-established relationship between diffusion and random walk. An excellent agreement is obtained between theory and simulations in the high-friction limit but not so in the intermediate regime. We explain the deviation in the low- to intermediate-friction regime using the modified random walk theory. The rate of escape from one cell to another is obtained from the multidimensional rate theory of Langer. We find that enhanced dimensionality plays an important role. To quantify the effects of noise on the potential-imposed coherence on the trajectories, we calculate the Lyapunov exponent. At small friction values, the Lyapunov exponent mimics the friction dependence of the rate.

High friction limits of Euler–Navier–Stokes–Korteweg equations for multicomponent models

High friction limits of Euler–Navier–Stokes–Korteweg equations for multicomponent models

Asymptotic derivation of multicomponent compressible flows with heat conduction and mass diffusion

A Type-I model of a multicomponent system of fluids with non-constant temperature is derived as the high-friction limit of a Type-II model via a Chapman-Enskog expansion. The asymptotic model is shown to fit into the general theory of hyperbolic-parabolic systems, by exploiting the entropy structure inherited through the asymptotic procedure. Finally, by deriving the relative entropy identity for the Type-I model, two convergence results for smooth solutions are presented, from the system with mass-diffusion and heat conduction to the corresponding system without mass-diffusion but including heat conduction and to its hyperbolic counterpart.

Molecular Origin of Internal Friction in Intrinsically Disordered Proteins.

Protein folding and dynamics are controlled by an interplay of thermal and viscosity effects. The effect of viscous drag through the solvent molecules is described by the classic Kramers theory in the high friction limit, which considers the dampening of the reactant molecules in the solution and quantifies the dependence of the reaction rate on the frictional drag. In addition to the external energy dissipation originating from the surrounding solvent molecules, there is an additional mode of internal energy dissipative force operative within the polypeptide chain reflecting the internal resistance of the chain to its conformational alterations. This dry, solvent-independent intrinsic frictional drag is termed internal friction. In the case of natively folded proteins, the physical origin of internal friction is primarily attributed to the intrachain interactions or other nonnative interactions in their compact states. However, the molecular origin of internal friction in intrinsically disordered proteins (IDPs) remains elusive.In this Account, we address this fundamental issue: what are the principal drivers of viscosity-independent (dry) friction in highly solvated, expanded, conformationally flexible, rapidly fluctuating IDPs that do not possess persistent intrachain interactions? IDPs exhibit diffusive conformational dynamics that is predominantly dominated by the segmental motion of the backbone arising due to the dihedral rotations in the Ramachandran Φ-Ψ space. The physical origin of friction in a complex biopolymeric system such as IDPs can be described by classic polymer models, namely, Rouse/Zimm models with internal friction. These one-dimensional models do not invoke torsional fluctuation components. Kuhn's classic description includes the connection between internal friction and microscopic dihedral hopping. Based on our time-resolved fluorescence anisotropy results, we describe that the sequence-dependent, collective, short-range backbone dihedral rotations govern localized internal friction in an archetypal IDP, namely, α-synuclein. The highly sensitive, residue-specific fluorescence depolarization kinetics offers a potent methodology to characterize and quantify the directional decorrelation engendered due to the short-range dihedral relaxation of the polypeptide backbone in the dihedral space. We utilized this characteristic relaxation time scale as our dynamic readout to quantify the site-specific frictional component. Our linear viscosity-dependent model of torsional relaxation time scale furnished a finite nonzero time constant at the zero solvent viscosity representing the solvent-independent internal friction. These results unveil the effect of the degree of dihedral restraining parameter on the internal friction component by showing that a restrained proline residue imparts higher torsional stiffness in the chain segments and, therefore, exhibits higher internal friction. This Account sheds light on the molecular underpinning of the sequence-specific internal friction in IDPs and will be of interest to unmask the role of internal friction in a diverse range of biomolecular processes involving binding-induced folding, allosteric interaction, protein misfolding and aggregation, and biomolecular condensation via phase separation.

Probing the Internal Dynamics and Shape of Simple Peptides in Urea, Guanidinium Hydrochloride, and Proline Solutions with Time-Resolved Fluorescence Anisotropy and Atomistic Cosolvent Simulations.

Picosecond time-resolved fluorescence anisotropy was used to measure the effect of denaturants and osmolytes on the reorientation dynamics of the simplest dipeptide. The solvent denaturants guanidinium hydrochloride (gdm), urea, and the osmolyte proline were used at several concentrations. Analysis of the concentration dependence of denaturants at a fixed temperature showed faster and slower reorientation time in two different denaturants at a nearly identical solvent viscosity (η). The reorientation time τ significantly deviates from Kramers' theory (τ ∝ η1) in the high friction limit for guanidinium and urea with r ≈ 0.4 and r ≈ 0.6 at pH 7.2, respectively. In proline, τ is nearly proportional to η. Atomistic molecular dynamics simulations of the dipeptide in identical cosolvents showed excellent agreement with the measured rotational orientation time. The dipeptide dihedral (ϕ, ψ) isomerization times in water and 6 M urea are almost identical and significantly slower in guanidinium. If a faster and slower reorientation time can be associated with the compact and expanded shapes, the fractional viscosity dependence for guanidinium and urea may result from the fact that internal dynamics of peptides in these cosolvents involve higher and lower internal friction within the dynamic elements.

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.

Effect of Memory and Inertial Contribution on Transition-Time Distributions: Theory and Simulations.

Transition paths refer to the time taken by molecules to cross a barrier separating two molecular conformations. In this work, we study how memory, as well as inertial contribution in the dynamics along a reaction coordinate, can affect the distribution of the transition-path time. We use a simple model of dynamics governed by a generalized Langevin equation with a power-law memory along with the inertial term, which was neglected in previous studies, where memory effects were explored only in the overdamped limit. We derive an approximate expression for the transit-time distribution and discuss our results for the short- and long-time limits and also compare it with known results in the high friction (overdamped) limit as well as in the Markovian limit. We have developed a numerical algorithm to test our theoretical results against extensive numerical simulations.

The Reconstruction of Non-Minimal Derivative Coupling Inflationary Potentials

We derive the reconstruction formulae for the inflation model with the non-minimal derivative coupling term. If reconstructing the potential from the tensor-to-scalar ratio r, we could obtain the potential without using the high friction limit. As an example, we reconstruct the potential from the parameterization r=8α/(N+β)γ, which is a general form of the α-attractor. The reconstructed potential has the same asymptotic behavior as the T- and E-model if we choose γ=2 and α≪1. We also discuss the constraints from the reheating phase by assuming the parameter wre of state equation during reheating is a constant. The scale of big-bang nucleosynthesis could put an upper limit on ns if wre=2/3 and a low limit on ns if wre=1/6.

Development of Sliding Regimes in Faults and Slow Strain Waves

The paper investigates a model simulating crustal fault dynamics and strain wave generation in a fault block geological medium, the parameters determining sliding regimes in faults, and the physics of transitions between different deformation regimes. The model comprises the most important mechanisms responsible for the interaction of fault walls: friction, geometric irregularities (roughness and asperities on the fault surface), and external load, which govern sliding along the fault. The results of field and laboratory studies of deformation migration on the macro/mesoscale are consistent with the concept of localized deformation propagation in the form of solitary waves (kinks, solitons) and autowaves. The conditions are defined which make possible the transition from the model simulating solitary waves in a conservative medium with low “friction” (soliton-like behavior of the system) toward the model of solitary waves in an active medium with diffusion (autowave-like behavior of the system). Two possible deformation regimes of the fault block structure in the high-friction limit are considered. The fault wall displacement is stopped due to this friction, but the adjacent blocks move relative to each other in the core of the fault. It is shown that in the high-friction limit a perturbed sine-Gordon equation applied for fault dynamics modeling is reduced to a reaction-diffusion equation, whereas the system goes from the soliton regime to the autowave regime. In the case of high friction and a lack of energy supply to the fault from an external source, the transfer of localized deformation is changed by a diffusive dissipation of stress.

Non-Markovian barrier crossing with two-time-scale memory is dominated by the faster memory component.

We investigate non-Markovian barrier-crossing kinetics of a massive particle in one dimension in the presence of a memory function that is the sum of two exponentials with different memory times, [Formula: see text] and [Formula: see text] . Our Langevin simulations for the special case where both exponentials contribute equally to the total friction show that the barrier-crossing time becomes independent of the longer memory time if at least one of the two memory times is larger than the intrinsic diffusion time. When we associate memory effects with coupled degrees of freedom that are orthogonal to a one-dimensional reaction coordinate, this counterintuitive result shows that the faster orthogonal degrees of freedom dominate barrier-crossing kinetics in the non-Markovian limit and that the slower orthogonal degrees become negligible, quite contrary to the standard time-scale separation assumption and with important consequences for the proper setup of coarse-graining procedures in the non-Markovian case. By asymptotic matching and symmetry arguments, we construct a crossover formula for the barrier crossing time that is valid for general multi-exponential memory kernels. This formula can be used to estimate barrier-crossing times for general memory functions for high friction, i.e. in the overdamped regime, as well as for low friction, i.e. in the inertial regime. Typical examples where our results are important include protein folding in the high-friction limit and chemical reactions such as proton-transfer reactions in the low-friction limit.

High-friction limits of Euler flows for multicomponent systems

The high-friction limit in Euler–Korteweg equations for fluid mixtures is analyzed. The convergence of the solutions towards the zeroth-order limiting system and the first-order correction is shown, assuming suitable uniform bounds. Three results are proved: the first-order correction system is shown to be of Maxwell–Stefan type and its diffusive part is parabolic in the sense of Petrovskii. The high-friction limit towards the first-order Chapman–Enskog approximate system is proved in the weak-strong solution context for general Euler–Korteweg systems. Finally, the limit towards the zeroth-order system is shown for smooth solutions in the isentropic case and for weak-strong solutions in the Euler–Korteweg case. These results include the case of constant capillarities and multicomponent quantum hydrodynamic models.

Diffusion in a bistable system: The eigenvalue spectrum of the Fokker-Planck operator and Kramers' reaction rate theory.

The time-dependent solution of the Fokker-Planck equation with bistable potentials is considered in terms of the eigenfunctions and eigenvalues of the linear Fokker-Planck operator. The Fokker-Planck equation is the high friction limit of the corresponding Kramers' equation. Two different potentials are considered defined with a constant diffusion coefficient, ε, and position-dependent drift coefficients. The smallest nonzero eigenvalue of the Fokker-Planck operator, λ_{1}, provides the long-time rate coefficient for the transformation of the different species in the two stable states. A novel pseudospectral method with nonclassical polynomials is applied to this class of systems. The convergence of the eigenvalues and eigenfunctions of the Fokker-Planck operator versus the number of basis functions is studied and compared with previous results. The results are consistent with Kramers' theory, and a linear relationship between lnλ_{1} and 1/ε for sufficiently small ε values is verified. A comparison with analytic approximations to λ_{1} is provided.

From the Liouville to the Smoluchowski equation for a colloidal solute particle in a solvent

We show how the Smoluchowski dynamics of a colloidal Brownian particle suspended in a molecular solvent can be reached starting from the microscopic Liouvillian evolution of the full classical model in the high friction limit. The integration of the solvent degrees of freedom goes through a multiple time scale perturbation expansion which removes the secular divergences. A simple dynamical Monte Carlo scheme is then proposed to solve the resulting evolution equation for the colloid solute particle. In particular we study the approach to the equilibrium Boltzmann distribution at late times and its resilience behavior at shorter times as influenced by the steepness of the external potential and the friction coefficient around their respective minima. This is very important to understand the fate of the Brownian particle’s random walk and its evolution history.

Fast-forward Langevin dynamics with momentum flips.

Stochastic thermostats based on the Langevin equation, in which a system is coupled to an external heat bath, are popular methods for temperature control in molecular dynamics simulations due to their ergodicity and their ease of implementation. Traditionally, these thermostats suffer from sluggish behavior in the limit of high friction, unlike thermostats of the Nosé-Hoover family whose performance degrades more gently in the strong coupling regime. We propose a simple and easy-to-implement modification to the integration scheme of the Langevin algorithm that addresses the fundamental source of the overdamped behavior of high-friction Langevin dynamics: if the action of the thermostat causes the momentum of a particle to change direction, it is flipped back. This fast-forward Langevin equation preserves the momentum distribution and so guarantees the correct equilibrium sampling. It mimics the quadratic behavior of Nosé-Hoover thermostats and displays similarly good performance in the strong coupling limit. We test the efficiency of this scheme by applying it to a 1-dimensional harmonic oscillator, as well as to water and Lennard-Jones polymers. The sampling efficiency of the fast-forward Langevin equation thermostat, measured by the correlation time of relevant system variables, is at least as good as the traditional Langevin thermostat, and in the overdamped regime, the fast-forward thermostat performs much better, improving the efficiency by an order of magnitude at the highest frictions we considered.