Dispersionless Transport in a Washboard Potential

Abstract

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,...

Enhanced diffusion with abnormal temperature dependence in underdamped space-periodic systems subject to time-periodic driving

“…Definitions (8,9) differ from such in [16,22,25,[30][31][32] by extra factors of 2 appearing in the definition of t ′ , and consequently in T ′ and γ ′ as well. Current definition makes interpreting the results easier (t ′ = 1 corresponds to one oscillation period at small γ ′ ).…”

“…Physically τ 2 is the time at which the particle distribution function in velocities assumes its stationary form. The distribution in space is still strongly non-equilibrium at τ 2 , it has exponential tail and sharp front [16] (in the direction of F ). It takes till τ 3 for the distribution function to assume approximately Gaussian shape in space (if averaged over the potential spatial period).…”

“…This leads to the D(ω) derivation based on approximations Eqs. (13,16) progressively less accurate at higher ω's, the power-law decay first slowing down, and then going into a completely different regime at ω approaching 2π.…”

“…Timedependence of the particle ensemble dispersion was studied in [16]. It was shown for the first time that in underdamped systems a special regime of dispersionless transport was realized, in which dispersion virtually does not change with time, on certain limited interval of time.…”

“…Further progress in studies of the diffusion under the action of constant force was made in [16][17][18]. Timedependence of the particle ensemble dispersion was studied in [16].…”