We present a numerical study of classical particles diffusing on a solid surface. The particles' motion is modeled by an underdamped Langevin equation with ordinary thermal noise. The particlesurface interaction is described by a periodic or a random two dimensional potential. The model leads to a rich variety of different transport regimes, some of which correspond to anomalous diffusion such as has recently been observed in experiments and Monte Carlo simulations. We show that this anomalous behavior is controlled by the friction coefficient, and stress that it emerges naturally in a system described by ordinary canonical Maxwell-Boltzmann statistics.

We have carried out a detailed study of the motion of particles driven by a constant external force over a landscape consisting of a periodic potential corrugated by a small amount of spatial disorder. We observe anomalous behavior in the form of subdiffusion and superdiffusion and even subtransport over very long time scales. Recent studies of transport over slightly random landscapes have focused only on parameters leading to normal behavior, and while enhanced diffusion has been identified when the external force approaches the critical value associated with the transition from locked to running solutions, the regime of anomalous behavior had not been recognized. We provide a qualitative explanation for the origin of these anomalies. Solid state surfaces frequently present periodic potentials marred by some disorder. Herein we show that an overdamped particle moving over such a potential in one dimension (1D) may exhibit anomalous behavior in the form of superdiffusion, subdiffusion, and even subtransport. Although we cannot prove that these are steady state regimes, our numerical simulation data show them to be present over time spans of several orders of magnitude.That diffusion of particles over both periodic and random surfaces lead to some forms of anomalous behavior is of course well known and continues to attract a great deal of attention both theoretically and experimentally [1][2][3][4][5][6][7][8][9][10]. In periodic potentials with low friction, extremely long (in time) dispersionless transport regimes can be observed when forces exceed a critical force [6]. Moreover, in these same systems, in both overdamped and underdamped regimes, the diffusion coefficient versus the applied force presents a pronounced peak around the critical force that allows the coexistence of locked and running states [1,[4][5][6][7]. The The enhancement is quantitatively larger than the free particle diffusion coefficient. This behavior has been observed experimentally when tracking the motion of colloidal spheres through a periodic potential created with optical vortex traps [9].The enhancement of the diffusion coefficient is even more pronounced when disorder is also present [9]. This phenomenon has been tested by numerical simulations on a surface in which a small amount of spatial disorder in the form of a random potential is added to the periodic potential [10]. Dramatic diffusive enhancement occurs even for very small amounts of disorder, e.g., when the amplitude of the random contribution of the potential is as small as ∼ 5% of that of the periodic contribution.Although dramatic, diffusive enhancement turns out to be only a limited aspect of the story because it is not the only manifestation of disorder. Here we present a range of additional anomalous transport and diffusion phenomena arising from weak disorder that have not been previously noted. Our model and the behaviors it exhibits are inspired by [9,10]. We consider the overdamped motion of identical noninteracting Brownian particles moving in a 1D potential la...

We study the effects of external noise in a one-dimensional model of front propagation. Noise is introduced through the fluctuations of a control parameter leading to a multiplicative stochastic partial differential equation. Analytical and numerical results for the front shape and velocity are presented. The linear-marginal-stability theory is found to increase its range of validity in the presence of external noise. As a consequence noise can stabilize fronts not allowed by the deterministic equation. [S0031-9007(96) PACS numbers: 03.40. Kf, 05.40.+j, 47.20.Ky, 47.54.+r The problem of front propagation has been receiving a great deal of attention in recent years due to its relevance to a large variety of systems in nonlinear physics, chemistry, and biology [1]. Here we will focus on the simplest case in which a globally stable state invades an unstable or metastable state. This problem has been extensively studied in the recent literature [2][3][4][5][6][7][8] particularly concerning the issue of velocity selection.On the other hand, in the last few years there has been a growing interest in the theoretical study of the role of fluctuations in front propagation [7,[9][10][11][12][13][14], and in particular there have been some experiments on the effects of stochastic turbulence in front propagation in the context of chemical fronts [15]. These studies have been basically concerned with the modification of the front velocity and the spreading of the front due to fluctuations.Internal [9][10][11][12] and external [13,14] fluctuations have been introduced in particular models using both Langevin [9,11,13,14] and master equation formalisms [10,12], but no systematic studies have been carried out concerning the modification of the well established selection criteria of the deterministic case. For internal fluctuations mostly numerical studies of different situations have obtained distinct effects on the front propagation. The case with the most direct comparison with the present work [9] found no change in the front velocity. On the other hand, previous analytical approaches for external fluctuations [13,14] have been based on small noise perturbative expansions which turn out to have a rather small range of validity for our purposes.Here we will introduce a new approach which relies on a physically intuitive picture of the problem but which is nonperturbative. As the accompanying numerical simulations will show, our theoretical approach gives an accurate quantitative description for a very broad range of noise intensities and allows for a general discussion of selection criteria in the presence of external fluctuations.We focus our study on the simplest prototypical equation for front propagation dynamics, and we introduce fluctuations via a Langevin equation. In our study, noise is assumed to be of external origin and is thus introduced as a stochastic spatiotemporal variation of a control parameter. For example, in an experimental situation such as a nematic liquid crystal in the presence of a magnetic field...

We study and characterize a new dynamical regime of underdamped particles in a tilted washboard potential. We find that for small friction in a finite range of forces the particles move essentially nondispersively, that is, coherently, over long intervals of time. The associated distribution of the particle positions moves at an essentially constant velocity and is far from Gaussian-like. This new regime is complementary to, and entirely different from, well-known nonlinear response and large dispersion regimes observed for other values of the external force. Particle transport and diffusion in periodic potentials at finite temperatures has been addressed in so many contexts and over so many decades that one might think this to be a fully solved problem [1]. However, as modern experimental and numerical methods continually evolve, ever broader parameter regimes and time regimes become accessible to inquiry, and behaviors continue to be revealed that have not previously been explored or even noted [2 -10]. Intermediate time regimes are especially challenging. On the one hand, numerical methods need to be efficient to reach beyond relatively short time behavior. On the other, analytic methods usually deal with asymptotia. Yet, experiments often involve intermediate time regimes. In this Letter we report an unexplored dynamical regime, namely, particle transport that is essentially nondispersive or coherent over long time intervals.Consider a particle moving in a periodic potential Vx of amplitude V 0 and period , with coefficient of friction at temperature T, and subject to a constant external force f. The variables x and t, respectively, denote the position of the particle and the time. The equation of motion reads r ÿV 0 r ÿ _ r F ;where r x=, V 0 =m 1=2 t=, the dot and prime denote derivatives with respect to and r respectively, V r Vx=V 0 , and the noise obeys the fluctuationdissipation relation h 0 i 2T ÿ 0 . Equation (1) models the translational Brownian motion of a particle in a tilted periodic potential, and also the rotational Brownian motion of a damped pendulum driven by a constant torque. The pendulum provides the mathematical background underlying a number of applications, including mobility in superionic conductors, dynamics of chargedensity waves, ring laser gyroscopes, and phase-locking phenomena in radio engineering. Perhaps most directly relevant for experimental testing, it also models a resistively and capacitively shunted single Josephson junction.This latter correspondence has been invoked in some of the most recent work on transport in tilted periodic potentials [7,8]. An excellent table indicating the precise translation between the parameters of a number of physical systems including the quintessential Josephson junction test bed and our model equation can be found in [7].There are three independent parameters in the model: the scaled force F f=V 0 , the scaled temperature T k B T=V 0 , and the scaled dissipation =mV 0 1=2 . In our subsequent numerical simulations we choose V r ÿ1=2 cos2r,...

PACS. 47.55.Mh -Flows through porous media. PACS. 68.35.Ct -Interface structure and roughness. PACS. 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion..Abstract. -We study the forced fluid invasion of an air-filled model porous medium at constant flow rate, in 1+1 dimensions, both experimentally and theoretically. We focus on the non-local character of the interface dynamics, due to liquid conservation, and its effect on the scaling properties of the interface upon roughening. Specifically, we study the limit of large flow rates and weak capillary forces. Our theory predicts a roughening behaviour characterized at short times by a growth exponent β1 = 5/6, a roughness exponent α1 = 5/2, and a dynamic exponent z1 = 3, and by β2 = 1/2, α2 = 1/2, and z2 = 1 at long times, before saturation. This theoretical prediction is in good agreement with the experiments at long times.The ensemble of experiments, theory, and simulations provides evidence for a new universality class of interface roughening in 1 + 1 dimensions.