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Register a new userQuantum Brownian motion in ratchet potentials: duality relation and its consequences
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Quantum Brownian motion in ratchet potentials is investigated by means of an approach based on a duality relation. This relation links the long-time dynamics in a tilted ratchet potential in the presence of dissipation with the one in a driven dissipative tight-binding model. The application to quantum ratchet yields a simple expression for the ratchet current in terms of the transition rates in the tight-binding system.
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We investigate the dynamics of quantum particles in a ratchet potential subject to an ac force field. We develop a perturbative approach for weak ratchet potentials and force fields. Within this approach, we obtain an analytic description of dc current rectification and current reversals. Transport characteristics for various limiting cases -- such as the classical limit, limit of high or low frequencies, and/or high temperatures -- are derived explicitly. To gain insight into the intricate dependence of the rectified current on the relevant parameters, we identify characteristic scales and obtain the response of the ratchet system in terms of scaling functions. We pay a special attention to inertial effects and show that they are often relevant, for example, at high temperatures. We find that the high temperature decay of the rectified current follows an algebraic law with a non-trivial exponent, $jpropto T^{-17/6}$.
We study the effects of an intermittent harmonic potential of strength $mu = mu_0 u$ -- that switches on and off stochastically at a constant rate $gamma$, on an overdamped Brownian particle with damping coefficient $ u$. This can be thought of as a realistic model for realisation of stochastic resetting. We show that this dynamics admits a stationary solution in all parameter regimes and compute the full time dependent variance for the position distribution and find the characteristic relaxation time. We find the exact non-equilibrium stationary state distributions in the limits -- (i) $gammallmu_0 $ which shows a non-trivial distribution, in addition as $mu_0toinfty$, we get back the result for resetting with refractory period; (ii) $gammaggmu_0$ where the particle relaxes to a Boltzmann distribution of an Ornstein-Uhlenbeck process with half the strength of the original potential and (iii) intermediate $gamma=2nmu_0$ for $n=1, 2$. The mean first passage time (MFPT) to find a target exhibits an optimisation with the switching rate, however unlike instantaneous resetting the MFPT does not diverge but reaches a stationary value at large rates. MFPT also shows similar behavior with respect to the potential strength. Our results can be verified in experiments on colloids using optical tweezers.
We study the stochastic motion of particles driven by long-range correlated fractional Gaussian noise in a superharmonic external potential of the form $U(x)propto x^{2n}$ ($ninmathbb{N}$). When the noise is considered to be external, the resulting overdamped motion is described by the non-Markovian Langevin equation for fractional Brownian motion. For this case we show the existence of long time, stationary probability density functions (PDFs) the shape of which strongly deviates from the naively expected Boltzmann PDF in the confining potential $U(x)$. We analyse in detail the temporal approach to stationarity as well as the shape of the non-Boltzmann stationary PDF. A typical characteristic is that subdiffusive, antipersistent (with negative autocorrelation) motion tends to effect an accumulation of probability close to the origin as compared to the corresponding Boltzmann distribution while the opposite trend occurs for superdiffusive (persistent) motion. For this latter case this leads to distinct bimodal shapes of the PDF. This property is compared to a similar phenomenon observed for Markovian L{e}vy flights in superharmonic potentials. We also demonstrate that the motion encoded in the fractional Langevin equation driven by fractional Gaussian noise always relaxes to the Boltzmann distribution, as in this case the fluctuation-dissipation theorem is fulfilled.
We consider an overdamped Brownian particle moving in a confining asymptotically logarithmic potential, which supports a normalized Boltzmann equilibrium density. We derive analytical expressions for the two-time correlation function and the fluctuations of the time-averaged position of the particle for large but finite times. We characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential, and support our predictions with extensive Langevin simulations. While the Boltzmann measure is used to obtain stationary correlation functions, we show how the non-normalizable infinite covariant density is related to the super-aging behavior.
The exact formulae for spectra of equilibrium diffusion in a fixed bistable piecewise linear potential and in a randomly flipping monostable potential are derived. Our results are valid for arbitrary intensity of driving white Gaussian noise and arbitrary parameters of potential profiles. We find: (i) an exponentially rapid narrowing of the spectrum with increasing height of the potential barrier, for fixed bistable potential; (ii) a nonlinear phenomenon, which manifests in the narrowing of the spectrum with increasing mean rate of flippings, and (iii) a nonmonotonic behaviour of the spectrum at zero frequency, as a function of the mean rate of switchings, for randomly switching potential. The last feature is a new characterization of resonant activation phenomenon.
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