Constant External Force Research Articles

Anomalous transport in disordered systems under the influence of external fields

We discuss two models for the description of anomalous diffusion, these being the continuous time random walk scheme, and fractional diffusion equations. We show their interrelations, and combine both approaches for the description of anomalous transport in constant external velocity and force fields. For an arbitrary external force F( x), we introduce a fractional Fokker–Planck equation, which generalises the two Einstein relations.

Self-locking of a modulated single overlayer in a nanotribology simulation

Molecular dynamics simulations of a single layer with +10% mismatch deposited on a (001) surface of an fcc substrate are reported. Lennard–Jones pair interactions are used. At equilibrium, the overlayer exhibits a triangular structure, one edge of which is parallel to the [ 1 ̄ 10] direction of the fcc substrate lattice. This direction displays a modulation of the interatomic distances. Constant external forces are then applied parallel to the modulation axis, which is the easy sliding direction; an amplitude of 0.2 σ/ ε per atom is enough to initiate sliding motion. A spectacular reorganization of the monolayer then occurs with a 90° rotation of the modulation direction and a self-locking of the motion. A microscopic mechanism for this gradual reorganization is suggested.

Resonant steps and spatiotemporal dynamics in the damped dc-driven Frenkel-Kontorova chain

Kink dynamics of the damped Frenkel-Kontorova (discrete sine-Gordon) chain driven by a constant external force are investigated. Resonant steplike transitions of the average velocity occur due to the competitions between the moving kinks and their radiated phasonlike modes. A mean-field consideration is introduced to give a precise prediction of the resonant steps. Slip-stick motion and spatiotemporal dynamics on those resonant steps are discussed. Our results can be applied to studies of the fluxon dynamics of 1D Josephson-junction arrays and ladders, dislocations, tribology and other fields.

Matricial formalism of transient dynamics: I. The presence of constant external forces

An alternative method in terms of a matricial formulation of quasideterministic (QD) approach and its connection with the nonlinear relaxation times (NLRT) has recently been proposed by (Jiménez-Aquino, 1997), to characterize the decay of unstable states of nonlinear systems, in the context of Gaussian white noise (GWN). The objective of this paper is to extend the method to the study of transient stochastic dynamics in the presence of a constant external force. In this matricial scheme, besides the influence of internal fluctuations and of the external force in the dynamical relaxation of the system, we also study the effect of fluctuations of initial conditions. The theory is explicitly applied to general systems of the two physical variables, and particularly to the study of a laser system in the presence of weak optical signals.

Diffusion and creep of a particle in a random potential

We investigate the diffusive motion of an overdamped classical particle in a 1D random potential using the mean first-passage time formalism and demonstrate the efficiency of this method in the investigation of the large-time dynamics of the particle. We determine the $log$-time diffusion ${<{< x^{2}(t)>}_{th}>}_{dis}=A\ln^{\beta} \left ({t}/{t_{r}})$ and relate the prefactor $A,$ the relaxation time $t_{r},$ and the exponent $\beta$ to the details of the (generally non-gaussian) long-range correlated potential. Calculating the moments ${<{< t^{n}>}_{th}>}_{dis}$ of the first-passage time distribution $P(t),$ we reconstruct the large time distribution function itself and draw attention to the phenomenon of intermittency. The results can be easily interpreted in terms of the decay of metastable trapped states. In addition, we present a simple derivation of the mean velocity of a particle moving in a random potential in the presence of a constant external force.

Elasticity in strongly interacting soft solids: A polyelectrolyte network

This paper discusses the elastic behavior of a very long cross-linked polyelectrolyte chain (Debye-H\"uckel chain), which is weakly charged. Therefore the response of the cross-linked chain (network) on an external constant force $f$ acting on the ends of the chain is considered. A self-consistent variational computation of an effective field theory is employed. It is shown that the modulus of the polyelectrolyte network has two parts: the first term represents the usual entropy elasticity of connected flexible chains and the second term takes into account the electrostatic interaction of the monomers. It is proportional to the squared cross-link density and the Debye-screening parameter.

Stochastic potential for a periodically forced nonlinear oscillator

We investigate stationary and nonstationary probability densities for a weakly forced nonlinear physical or chemical system that displays self-oscillations in the absence of forcing. The period and amplitude of forcing are taken as adjustable constraints. We consider a homogeneous reaction system described by a master equation. Our method of solution is based on the Wentzel–Kramers–Brillouin (WKB) expansion of the probability density with the system size as the expansion parameter. The first term in this expansion is the stochastic potential (eikonal). In the absence of forcing, the probability density is logarithmically flat on the limit cycle. With periodic forcing, the phenomenon of phase locking can occur whereby a stable cycle, which is close to the unforced cycle, adopts a constant relative phase to the forcing. A saddle cycle also exists and has a different constant relative phase. For such phase-locked solutions, the distribution over the relative phases is peaked on the stable cycle and exhibits a logarithmically flat region (a plateau) that originates on the saddle cycle. This plateau is due to a nonzero relative phase slippage: large fluctuations from the stable cycle over the saddle cycle are overwhelmingly more probable in a certain relative phase direction, which depends upon the location of the parameters within an entrainment region. This distribution of relative phases is logarithmically equivalent to that of a Brownian particle in a periodic potential with a constant external force in the strong damping and weak noise limits. For parameter values outside of an entrainment region (for which a quasiperiodic solution exists), the distribution in relative phase is logarithmically flat. For this regime, we investigate the evolution of an initially localized density and show that the width grows proportionally with the square root of time. The proportionality factor depends upon both the position (phase) on the cross section of the peak of the density and the distance in parameter space from the boundary of the entrainment region. For parameter values that approach the boundary of an entrainment region, this proportionality factor tends to infinity. We also determine an expression for the first order correction to the stochastic potential for both entrained and quasiperiodic solutions. A thermodynamic interpretation of these results is made possible by the equality of the stochastic potential with an excess work function.

Dissipative Model for a Gas System with Constant External Force

The thermodynamic characteristics of a system consisting in a mixture of two gases affected by a constant external force (gravity, for example) with dissipation are derived. Assuming a long narrow-square shape pipe container for the gas, the pressure as a function of the altitude is determined using the partition function of the classical mechanics. The limit case for α and f going to zero is studied, where α and f are the parameters which characterize the dissipation and the constant force.

Comment on stochastic resonance in washboard potentials

We discuss the possibility of detecting stochastic resonance in a periodic potential driven by both a white noise source and an external constant force (washboard potential). Evidence of a noise controlled resonance is found in the limit of low damping, where inertia replaces the periodic drive.

Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams

We study the dynamics of ultracold atoms in a periodic optical potential submitted to a constant external force. Bloch oscillations in the fundamental and first excited bands of the potential are observed. In addition to a solid-state analysis we give a quantum-optics interpretation of this effect in terms of photon exchanges between the atoms and the laser waves. The dynamics of Bloch oscillations are equivalent to a sequence of adiabatic rapid passages between momentum states and can be described using the dressed-atom approach. We demonstrate efficient and dissipation-free acceleration of atoms by coherent transfer of a large number of photon momenta (\ensuremath{\approx}100). This technique produces atomic beams with a very small longitudinal velocity spread that may find applications in atom optics and precision measurements.

A viscoelastic material model to represent smooth muscle shortening

The mechanical properties of a contracting smooth muscle can be changed by changing its length. A viscoelastic material model was developed to predict the length-dependent stiffness changes when a constrained muscle is allowed to shorten under a constant external force. Three-dimensional finite element simulations were carried out to estimate the stiffness changes and compared to available experimental data. A good agreement was found indicating that the viscoelastic material model developed gives a valid representation of the length dependent stiffness changes of a smooth muscle. Sensitivity analysis was carried out to determine the relative effects of material constants in the model on the length dependent stiffness.

Initial-boundary value problems of the Boltzmann equation at zero temperature

For a general class of collision operators, we show that the stationary boundary valve problem of the one dimensional linear Boltzmann equation on a finite intcrvall including an arbitrarily high constant external force can be solved analytically (apart form an inverse Laplace transform) at zero temperature of the host medium. As a byproduct we obtain existence, positivity and uniqueness of the solution. Our method also applies to coupled initial-boundary value problems, for which we demonstrate the decay to the solution of the stationary boundary value problem.

Dispersion mechanism of coagulated particles in liquid flow

This paper reviews our recent research results for the dispersion mechanisms of agglomerated colloidal particles in acceleration flow and shear flow, using latex particles of 1.1–5.2 μm in diameter. (1) Experimental results of the dispersion of coagulated particles by an accelerating suspension in a convergent nozzle are well correlated by a dispersibility DS, which is defined as the ratio of the separation force to the interaction force between particles (van der Waals force and electrostatic force). (2) The dispersion is enhanced by an increase of the surface potential and electrolyte concentration under a constant external force, which can be explained on the basis of the DLVO theory. (3) In a shear flow field, both dispersion and shear coagulation occur simultaneously. A definition of dispersibility is proposed as DM = M b / M t , where M b is the bending moment induced by the drag force on the doublet, and M t is the moment required to separate the two adhering spheres. For DM < 1, doublets in the shear flow of a concentric rotating cylinder become dispersed. For DM < 1, shear coagulation overcomes dispersion and new doublets are formed.

An energy conserving co-rotational procedure for non-linear dynamics with finite elements

A new procedure is proposed for implicit dynamic analysis using the finite element method. The main aim is to give stable solutions with large time-steps in the presence of significant rigid body motions, in particular rotations. In contrast to most conventional approaches, the time integration strategy is closely linked to the “element technology” with the latter involving a form of co-rotational procedure. For the undamped situation, the solution procedure leads to an algorithm that exactly conserves energy when constant external forces are applied (i.e. with gravity loading).

A Comparison of Shaped Piezoelectric Actuators for Divergence Control

The characteristics and effectiveness of shaped piezo-actuators for use in controlling the divergence of a simplified forward swept wing model are analytically investigated. The forward swept wing is modeled as a simplified cantilever beam with surface mounted piezo-actuators on the upper surface of the beam. A constant electric field is applied to the actuator inducing a constant external force on the wing, and the effects of actuator location, size, thickness, and shape are evaluated using the principle of virtual work by maximizing the modal amplitude of the first bending mode. For control purposes, strain at the wing root is used as an error signal to minimize the beam deflection. Results suggest that the optimum actuator size should span the entire wing. The required actuator thickness for divergence control decreases with increasing airspeed due to the effectiveness softening of the wing in the presence of air loads. A uniform rectangular actuator is more effective for controlling divergence; however, a linearly shaped actutor is more efficient from the standpoint of control effectiveness per actuator weight. Application is currently limited by the large thicknesses and applied voltages required by the actuators to achieve effective divergence control. However, this investigation suggests that, because of the softening of the wing near divergence, only a small control effort is required to deform the wing thus allowing the airstream to act as the power source for control actuation.

How shear flow of a semidilute suspension modifies its self-mobility.

We investigate a semidilute hard sphere colloidal suspension with no hydrodynamic interactions. The suspension undergoes a steady shear flow. In addition, a small constant external force induces a tagged particle flux. In order to determine the shear-dependent self-mobility matrix (relating the flux to the force) we construct the approximate solution to the two-particle Smoluchowski equation at contact, applying the induced source method, combined with a power expansion in the square root of the shear rate. We evaluate the self-mobility tensor numerically by means of Pad\'e approximants for small and moderate shear rates.

The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation

We consider the flow of a gas in a channel whose walls are kept at fixed (different) temperatures. There is a constant external force parallel to the boundaries which may themselves also be moving. The system is described by the stationary Boltzmann equation to which are added Maxwellian boundary conditions with unit accommodation coefficient. We prove that when the temperature gap, the relative velocity of the planes, and the force are all sufficiently small, there is a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip voundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.

Perturbation analysis of a stationary nonequilibrium flow generated by an external force

The stationary flow of a gas in a slab under the action of a constant external force parallel to the walls is analyzed in the context of the Bhatnagar-Gross-Krook model kinetic equation. The force produces spatial gradients along the coordinate normal to the walls. By performing a perturbation expansion in powers of the force, we obtain the hydrodynamic fields up to fifth order in the force. Then the velocity distribution function and all its moments are evaluated to third order. The expansion coefficients are polynomials in the space variable of a degree increasing linearly with the expansion order. Although the series expansion is only asymptotic, it shows how the state of the system is modified by a variation of the external force beyond the linear regime.

A co‐rotational element/time‐integration strategy for non‐linear dynamics

AbstractNew procedures are proposed for implicit dynamic analysis using the finite element method. The main aim is to give stable solutions with significant rigid‐body motions, in particular rotations. In contrast to most conventional approaches, the time‐integration strategy is closely linked to the ‘element technologies’ with the latter involving a form of co‐rotational procedure. For the undamped situation, one of the solution procedures leads to an algorithm that exactly conserves energy when constant external forces are applied (i.e. with gravity loading).

Hydrodynamic limit of the stationary Boltzmann equation in a slab

We study the stationary solution of the Boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature. We prove that when the force is sufficiently small there exists a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip boundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.