Over-damped Limit Research Articles

Mass Changes the Diffusion Coefficient of Particles with Ligand-Receptor Contacts in the Overdamped Limit.

Inertia does not generally affect the long-time diffusion of passive overdamped particles in fluids. Yet a model starting from the Langevin equation predicts a surprising property of particles coated with ligands that bind reversibly to surface receptors: heavy particles diffuse more slowly than light ones of the same size. We show this by simulation and by deriving an analytic formula for the mass-dependent diffusion coefficient in the overdamped limit. We estimate the magnitude of this effect for a range of biophysical ligand-receptor systems, and find it is potentially observable for tailored micronscale DNA-coated colloids.

Active motion of passive asymmetric dumbbells in a non-equilibrium bath

Persistent motion of passive asymmetric bodies in non-equilibrium media has been experimentally observed in a variety of settings. However, fundamental constraints on the efficiency of such motion are not fully explored. Understanding such limits, and ways to circumvent them, is important for efficient utilization of energy stored in agitated surroundings for purposes of taxis and transport. Here, we examine such issues in the context of erratic movements of a passive asymmetric dumbbell driven by non-equilibrium noise. For uncorrelated (white) noise, we find a (non-Boltzmann) joint probability distribution for the velocity and orientation, which indicates that the dumbbell preferentially moves along its symmetry axis. The dumbbell thus behaves as an Ornstein-Uhlenbeck walker, a prototype of active matter. Exploring the efficiency of this active motion, we show that in the over-damped limit, the persistence length l of the dumbbell is bound from above by half its mean size, while the propulsion speed v∥ is proportional to its inverse size. The persistence length can be increased by exploiting inertial effects beyond the over-damped regime, but this improvement always comes at the price of smaller propulsion speeds. This limitation is explained by noting that the diffusivity of a dumbbell, related to the product v∥l, is always less than that of its components, thus severely constraining the usefulness of passive dumbbells as active particles.

Phase transition and emergence of active temperature in an active Brownian system in underdamped background

We explore the role of inertia in the properties of active Brownian particles (ABPs) immersed in an underdamped background in two dimensions using Langevin dynamics computer simulation. Similar to an equilibrium two-dimensional passive interacting particle system, the system of ABPs transits from a liquid phase to a solid phase with the change in the coupling parameter, which is the ratio of interaction potential energy and thermal energy of the background solvent. Important qualitative and quantitative differences are found in the liquid-solid phase transition with increasing strength of activity as compared to those found in the conventional overdamped background limit. In the underdamped background, inherent activity is found to lead to a temperature, called the active temperature and defined by average velocity fluctuations of the ABPs, that is different from the fixed background solvent temperature. A new scaling law for active temperature as a function of activity strength is found near the liquid-solid boundary. Active temperature, which behaves similar to the thermodynamic equilibrium temperature, is also found to depend upon the interaction strength between the active particles and the strength of the background dissipation. With an increase in background dissipation, the difference between active temperature and the background solvent temperature decreases and the difference is found to eventually vanish in the overdamped limit, demonstrating the correctness of the calculation.

Generalization of Stokes–Einstein relation to coordinate dependent damping and diffusivity: an apparent conflict

Brownian motion with coordinate dependent damping and diffusivity is ubiquitous. Understanding equilibrium of a Brownian particle with coordinate dependent diffusion and damping is a contentious area. In this paper, we present an alternative approach to this problem based on already established methods. We solve for the equilibrium distribution of the over-damped dynamics using Kramers–Moyal expansion. We compare this with the over-damped limit of the generalized/modified Maxwell–Boltzmann distribution. We investigate two distinct possibilities of the Stokes–Einstein relation not holding locally and holding locally everywhere. In the former case we get a local proportional relation between the coordinate dependent diffusivity and damping which is consistent with other requirements of equilibrium. The latter case requires restrictions on the upper limit of the local velocity of the Brownian particle to make the modified Maxwell–Boltzmann relation obtain the correct over-damped limit.

Dynamics of a Self-Propelled Particle in a Harmonic Trap.

The dynamics of an active walker in a harmonic potential is studied experimentally, numerically, and theoretically. At odds with usual models of self-propelled particles, we identify two dynamical states for which the particle condensates at a finite distance from the trap center. In the first state, also found in other systems, the particle points radially outward from the trap, while diffusing along the azimuthal direction. In the second state, the particle performs circular orbits around the center of the trap. We show that self-alignment, taking the form of a torque coupling the particle orientation and velocity, is responsible for the emergence of this second dynamical state. The transition between the two states is controlled by the persistence of the particle orientation. At low inertia, the transition is continuous. For large inertia, the transition is discontinuous and a coexistence regime with intermittent dynamics develops. The two states survive in the overdamped limit or when the particle is confined by a curved hard wall.

Erratum to: Effective temperature of active fluids and sheared soft glassy materials.

The authors were notified by their collaborators Golan Bel and Dan Wexler that the expression for the over-damped limit of the kinetic energy of the trapped particle, that they have used in the orignal paper, was in error. They provide the correct expressions in this erratum.

Point-source dispersion of quasi-neutrally-buoyant inertial particles.

We analyze the evolution of the distribution, both in the phase space and in the physical space, of inertial particles released by a spatially-localized (punctual) source and advected by an incompressible flow. The difference in mass density between fluid and particles is assumed to be small, and represents the basic parameter for a regular perturbative expansion. By means of analytical techniques such as Hermitianization, we derive a chain of equations of the advection-diffusion-reaction type, easily solvable at least numerically. Our procedure provides results also for finite particle inertia, away from the over-damped limit of quasi-tracer dynamics.

Heat Out of Equilibrium Driven by Potential Pulling Beyond the Overdamped Limit

We investigate heat accumulated for a long period time in nonequilibrium motion of a colloid under a potential with center pulled by a constant velocity. Heat and work have the same average value in steady state, but the distribution functions for thermal fluctuations are different. Difference comes from everlasting memory of initial high energy on total amount of accumulated heat. As a result, the positive tail of the heat distribution exhibits a distinct correction to the large deviation function, compared to a usual logarithmic correction. Mathematically, difference originates from the singularity in the generating function, the Fourier transform, of the distribution function and makes usual saddle point approximation not valid. We use the modified saddle point approximation technique to compute the heat distribution function. We find the heat distribution for the underdamped case and find the correction to large deviation function more predominant than for the overdamped case.

Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.

The long time behavior of Brownian motion in tilted periodic potentials

A variety of phenomena in physics and other fields can be modeled as Brownian motion in a heat bath under tilted periodic potentials. We are interested in the long time average velocity considered as a function of the external force, that is, the tilt of the potential. In many cases, the long time behavior–pinning and de-pinning phenomenon–has been observed. We use the method of stochastic differential equation to study the Langevin equation describing such diffusion. In the over-damped limit, we show the convergence of the long time average velocity to that of the Smoluchowski–Kramers approximation, and carry out asymptotic analysis based on Risken’s and Reimann et al.’s formula. In the under-damped limit, applying Freidlin et al.’s theory, we first show the existence of three pinning and de-pinning thresholds of the normalized tilt, corresponding to the bi-stability phenomenon; and second, as noise approaches zero, derive formulas of the mean transition times between the pinning and running states.

Brownian dynamics simulation of peptides with the University of Houston Brownian Dynamics (UHBD) program.

This chapter provides the background theory and a practical protocol for performing Brownian dynamics simulation of peptides. Brownian dynamics simulation represents a complementary approach to Monte Carlo and molecular dynamics methods. Unlike Monte Carlo methods, it could provide dynamical information in a timescale longer than the momentum relaxation time. On the other hand, it is faster than molecular dynamics by approximating the solvent by a continuum and by operating in the over-damped limit. This chapter introduces the use of the University of Houston Brownian Dynamics (UHBD) program [1, 2] to perform Brownian dynamics simulation on peptides.

On extremals of the entropy production by ‘Langevin–Kramers’ dynamics

We refer as ‘Langevin–Kramers’ dynamics to a class of stochastic differential systems exhibiting a degenerate ‘metriplectic’ structure. This means that the drift field can be decomposed into a symplectic and a gradient-like component with respect to a pseudo-metric tensor associated with random fluctuations affecting increments of only a sub-set of the degrees of freedom. Systems in this class are often encountered in applications as elementary models of Hamiltonian dynamics in a heat bath eventually relaxing to a Boltzmann steady state. Entropy production control in Langevin–Kramers models differs from the now well-understood case of Langevin–Smoluchowski dynamics for two reasons. First, the definition of entropy production stemming from fluctuation theorems specifies a cost functional which does not act coercively on all degrees of freedom of control protocols. Second, the presence of a symplectic structure imposes a non-local constraint on the class of admissible controls. Using Pontryagin control theory and restricting the attention to additive noise, we show that smooth protocols attaining extremal values of the entropy production appear generically in continuous parametric families as a consequence of a trade-off between smoothness of the admissible protocols and non-coercivity of the cost functional. Uniqueness is, however, always recovered in the over-damped limit as extremal equations reduce at leading order to the Monge–Ampère–Kantorovich optimal mass-transport equations.

Entropy production in quantum Brownian motion

We investigate how to coherently define entropy production for a process of transient relaxation in the quantum Brownian motion model for the harmonic potential. We compare a form, referred to as ‘poised’ (P), which after non-Markovian transients corresponds to a definition of heat as the change in the system Hamiltonian of mean force, with a recent proposal by Esposito et al (ELB) based on a definition of heat as the energy change in the bath. Both expressions yield a positive-definite entropy production and they coincide for vanishing coupling strength, but their difference is proved to be always positive (after non-Markovian transients disappear) and to grow as the coupling strength increases. In the classical over-damped limit the ‘poised’ entropy production converges to the entropy production used in stochastic thermodynamics. We also investigate the effects of the system size, and of the ensuing Poincaré recurrences, and how the classical limit is approached. We close by discussing the strong-coupling limit, in which the ideal canonical equilibrium of the bath is violated.

Characterization of Viscous and Excluded Volume effects on the Folding Kinetics of the Tetraloop-Receptor Motif: Single Molecule Tests of Kramer's Theory

It is widely known that there exists a strong relationship between the solvent, including co-solutes (e.g. cations, proteins, nucleic acids), and RNA that cooperatively defines the kinetics of tertiary structure formation. Viscosity, a solvent property that is coupled to dynamics, is critical in RNA folding due to the propensity for populating kinetically trapped species on a rough free energy landscape. Also of importance is the role of macromolecular co-solutes that limit the volume available for molecules to sample (excluded volume effects). The effects of these properties are highlighted by observed in vivo diffusion coefficients that are 4-200x smaller than in aqueous solvents. In this work, Kramers' rate theory is used to describe the viscosity dependence of tetraloop-receptor docking kinetics at the single molecule level. Both rate constants, kfold and kunfold, decrease with increasing viscosity (increasing glycerol %), trends that are predicted by Kramers' theory in the over-damped limit. However, the same measurements made in high molecular weight PEG solutions showed folding rate constants are accelerated by 1-2 orders of magnitude. Scaled particle theory, describing a hard spheres PEG-RNA interaction, quantitatively predicts the stabilizing effect of excluded volume. Temperature dependent measurements show that the thermodynamics of docking are not perturbed (ΔH°= −23(2) kcal/mol and ΔS°= −76(6) cal/mol∗K) even in up to 50% glycerol (∼6 cP) and that solvent activation accounts for 4-6 kcal/mol of the folding enthalpy. In PEG solutions thermodynamics reveal that the folding is stabilized by a reduction in the entropy of the unfolded RNA ensemble (ΔΔS°>0). Thus, these studies validate physical models describing characteristic kinetic and thermodynamic trends for intramolecular structure formation attributed to viscosity and excluded volume.

Nucleation of colloids and macromolecules in a finite volume

A recently formulated description of homogeneous nucleation for Brownian particles in the over-damped limit based on fluctuating hydrodynamics is used to determine the nucleation pathway, characterized as the most likely path (MLP), for the nucleation of a dense-concentration droplet of globular protein from a dilute solution in a small, finite container. The calculations are performed by directly discretizing the equations for the MLP and it is found that they confirm previous results obtained for infinite systems: the process of homogeneous nucleation begins with a long-wavelength, spatially-extended concentration fluctuation that condenses to form the pre-critical cluster. This is followed by a classical growth processes. The calculations show that the post-critical growth involves the formation of a depletion zone around the cluster whereas no such depletion is observed in the pre-critical cluster. The approach therefore captures dynamical effects not found in classical density functional theory studies while consistently describing the formation of the pre-critical cluster.

Extinction rates of established spatial populations

This paper deals with extinction of an isolated population caused by intrinsic noise. We model the population dynamics in a "refuge" as a Markov process which involves births and deaths on discrete lattice sites and random migrations between neighboring sites. In extinction scenario I, the zero population size is a repelling fixed point of the on-site deterministic dynamics. In extinction scenario II, the zero population size is an attracting fixed point, corresponding to what is known in ecology as the Allee effect. Assuming a large population size, we develop a WKB (Wentzel-Kramers-Brillouin) approximation to the master equation. The resulting Hamilton's equations encode the most probable path of the population toward extinction and the mean time to extinction. In the fast-migration limit these equations coincide, up to a canonical transformation, with those obtained, in a different way, by Elgart and Kamenev [Phys. Rev. E 70, 041106 (2004)]. We classify possible regimes of population extinction with and without an Allee effect and for different types of refuge, and solve several examples analytically and numerically. For a very strong Allee effect, the extinction problem can be mapped into the overdamped limit of the theory of homogeneous nucleation due to Langer [Ann. Phys. (NY) 54, 258 (1969)]. In this regime, and for very long systems, we predict an optimal refuge size that maximizes the mean time to extinction.

Exact time-average distribution for a stationary non-Markovian massive Brownian particle coupled to two heat baths

Using a time-averaging technique we obtain exactly the probability distribution for position and velocity of a Brownian particle under the influence of two heat baths at different temperatures. These baths are expressed by a white noise term, representing the fast dynamics, and a colored noise term, representing the slow dynamics. Our exact solution scheme accounts for inertial effects that are not present in approaches that assume the Brownian particle in the overdamped limit. We are also able to obtain the contributions associated with the fast noise that are usually neglected by other approaches.

Nonequilibrium fluctuation induced escape from a metastable state

Based on a simple microscopic model where the bath is in a non-equilibrium state we study the escape from a metastable state in the over-damped limit. Making use of Fokker-Planck-Smoluchowski description we derive the time dependent escape rate in the non-stationary regime in closed analytical form which brings on to fore a strong non-exponential kinetic of the system mode.

Continuum simulation of dislocation dynamics: predictions for internal friction response

The amplitude dependent mechanical loss due to bowing of an idealized Frank–Read source is studied using both simulation and analytical techniques. Dislocations are modeled within isotropic elasticity theory, and are assumed to be in the over-damped limit. The dynamics incorporated in the model are tested directly, by comparison with the exact solution to the linearized isotropic elasticity theory of the mechanical losses. The model is then applied to the study of the mechanical losses from two coupled dislocations. It is concluded that the implemented dislocation dynamics method represents an accurate solution to the initially stated problem, that the elastic string model often employed to model losses is inadequate, and that dislocation interactions may alter substantially the dislocation component of the spectrum observed during internal friction experiments.

Effect of Stationary Particles on the Phase Separation of Binary Fluids

ABSTRACTPhase separating binary fluids with the addition of immobile particles, which act as osmotic force centres, were simulated using a Lattice Boltzmann model in two dimensions. In the hydrodynamic over-damped limit, where the flow is entirely driven by capillary effects, the presence of particles that are preferentially wetted by one of the fluid components significantly affects the kinetics of the growth of the fluid domains. The late time dynamics is governed by the wetting interactions and the final size of the domains can be tailored by varying the strength of the particles-fluid interaction as well as the particles concentration. These features are predicted within a simple theoretical model and are amenable of experimental checks.