Brownian ratchets: How stronger thermal noise can reduce diffusion

Memory-induced current reversal of Brownian motors

The kinetics of biological motors such as kinesin or dynein is notably influenced by a viscoelastic intracellular environment. The characteristic relaxation time of the cytosol is not separable from the colloidal timescale and therefore their dynamics is inherently non-Markovian. In this paper, we consider a variant of a Brownian motor model, namely, a Brownian ratchet immersed in a correlated thermal bath, and we analyze how memory influences its dynamics. In particular, we demonstrate the memory-induced current reversal effect and explain this phenomenon by applying the effective mass approximation as well as uncovering the memory-induced dynamical localization of the motor trajectories in the phase space. Our results reveal new aspects of the role of memory in microscopic systems out of thermal equilibrium.

Temperature anomalies of oscillating diffusion in ac-driven periodic systems

We analyze the impact of temperature on the diffusion coefficient of an inertial Brownian particle moving in a symmetric periodic potential and driven by a symmetric time-periodic force. Recent studies have revealed the low-friction regime in which the diffusion coefficient shows giant damped quasiperiodic oscillations as a function of the amplitude of the time-periodic force [I. G. Marchenko et al., Chaos 32, 113106 (2022)1054-150010.1063/5.0117902]. We find out that when temperature grows the diffusion coefficient increases at its minima; however, it decreases at the maxima within a finite temperature window. This curious behavior is explained in terms of the deterministic dynamics perturbed by thermal fluctuations and mean residence time of the particle in the locked and running trajectories. We demonstrate that temperature dependence of the diffusion coefficient can be accurately reconstructed from the stationary probability to occupy the running trajectories.

Diffusion Coefficient of a Brownian Particle in Equilibrium and Nonequilibrium: Einstein Model and Beyond

The diffusion of small particles is omnipresent in many processes occurring in nature. As such, it is widely studied and exerted in almost all branches of sciences. It constitutes such a broad and often rather complex subject of exploration that we opt here to narrow our survey to the case of the diffusion coefficient for a Brownian particle that can be modeled in the framework of Langevin dynamics. Our main focus centers on the temperature dependence of the diffusion coefficient for several fundamental models of diverse physical systems. Starting out with diffusion in equilibrium for which the Einstein theory holds, we consider a number of physical situations outside of free Brownian motion and end by surveying nonequilibrium diffusion for a time-periodically driven Brownian particle dwelling randomly in a periodic potential. For this latter situation the diffusion coefficient exhibits an intriguingly non-monotonic dependence on temperature.

Giant oscillations of diffusion in ac-driven periodic systems

We revisit the problem of diffusion in a driven system consisting of an inertial Brownian particle moving in a symmetric periodic potential and subjected to a symmetric time-periodic force. We reveal parameter domains in which diffusion is normal in the long time limit and exhibits intriguing giant damped quasiperiodic oscillations as a function of the external driving amplitude. As the mechanism behind this effect, we identify the corresponding oscillations of difference in the number of locked and running trajectories that carry the leading contribution to the diffusion coefficient. Our findings can be verified experimentally in a multitude of physical systems, including colloidal particles, Josephson junction, or cold atoms dwelling in optical lattices, to name only a few.

Roughness in the periodic potential induces absolute negative mobility in a driven Brownian ratchet

Absolute negative mobility, where particles move opposite to the direction as governed by the external load, is an anomalous transport property of a Brownian ratchet and has technological implications in mass separation and bioanalytical applications. We numerically investigated here the effect of roughness in symmetric periodic potential on the negative mobility of a driven inertial Brownian ratchet in the presence of an external load. We show that the microscopic spatial heterogeneity of the potential can generate negative mobility which would not otherwise be possible under smooth potential in the concerned parameter space. We determined the optimal condition in terms of parameter space for such anomalous behavior. Our calculations indicate that the shift of balance towards the negative velocity phase in the temporal oscillations of velocity and weakly chaotic dynamics are responsible factors for roughness-induced negative mobility. These calculations highlight a constructive role of roughness in the anomalous transport properties of Brownian ratchet.

Roughness in the periodic potential enhances transport in a driven inertial ratchet

We study the effects of roughness in the asymmetric periodic potential on the transport and diffusion of an inertial Brownian particle driven by a time-periodic force in a Gaussian environment. We find that moderate roughness leads to the loss of transient anomalous diffusion, and it helps to establish normal diffusion in the weak noise limit. We uncover a contrasting effect of roughness on the transport of particles in the weak and moderate to large noise limit. In the weak noise limit, small amplitude roughness results in the increase of directed transport, whereas in the moderate to large noise limit, roughness hinders transport. The deterministic dynamics of the system reveals that the purely periodic system under smooth potential transits into a chaotic system due to the moderate roughness in the potential. Therefore our calculations demonstrate the constructive role of roughness in the transport of particles in the inertial regime.

Diffusion in a biased washboard potential revisited

The celebrated Sutherland-Einstein relation for systems at thermal equilibrium states that spread of trajectories of Brownian particles is an increasing function of temperature. Here, we scrutinize the diffusion of underdamped Brownian motion in a biased periodic potential and analyze regimes in which a diffusion coefficient decreases with increasing temperature within a finite temperature window. Comprehensive numerical simulations of the corresponding Langevin equation performed with unprecedented resolution allow us to construct a phase diagram for the occurrence of the nonmonotonic temperature dependence of the diffusion coefficient. We discuss the relation of the later effect with the phenomenon of giant diffusion.

Noise-induced rectification in out-of-equilibrium structures

We consider the motion of overdamped particles over random potentials subjected to a Gaussian white noise and a time-dependent periodic external forcing. The random potential is modeled as the potential resulting from the interaction of a point particle with a random polymer. The random polymer is made up, by means of some stochastic process, from a finite set of possible monomer types. The process is assumed to reach a nonequilibrium stationary state, which means that every realization of a random polymer can be considered as an out-of-equilibrium structure. We show that the net flux of particles over this random medium is nonvanishing when the potential profile on every monomer is symmetric. We prove that this ratchetlike phenomenon is a consequence of the irreversibility of the stochastic process generating the polymer. On the contrary, when the process generating the polymer is at equilibrium (thus fulfilling the detailed balance condition) the system is unable to rectify the motion. We calculate the net flux of the particles in the adiabatic limit for a simple model and we test our theoretical predictions by means of Langevin dynamics simulations. We also show that, out of the adiabatic limit, the system also exhibits current reversals as well as nonmonotonic dependence of the diffusion coefficient as a function of forcing amplitude.

SQUID ratchet: Statistics of transitions in dynamical localization

We study occupation of certain regions of phase space of an asymmetric superconducting quantum interference device (SQUID) driven by thermal noise, subjected to an external ac current and threaded by a constant magnetic flux. Thermally activated transitions between the states which reflect three deterministic attractors are analyzed in the regime of the noise induced dynamical localization of the Josephson phase velocity, i.e., there is a temperature interval in which the conditional probability of the voltage to remain in one of the states is very close to one. Implications of this phenomenon on the dc voltage drop across the SQUID are discussed. We detect the emergence of the power law tails in a residence time probability distribution of the Josephson phase velocity and discuss the role of symmetry breaking in dynamical localization induced by thermal noise. This phenomenon illustrates how deterministic-like behavior may be extracted from randomness by stochasticity itself. It reveals another face of noise.

Least-rattling feedback from strong time-scale separation

In most interacting many-body systems associated with some "emergent phenomena," we can identify subgroups of degrees of freedom that relax on dramatically different time scales. Time-scale separation of this kind is particularly helpful in nonequilibrium systems where only the fast variables are subjected to external driving; in such a case, it may be shown through elimination of fast variables that the slow coordinates effectively experience a thermal bath of spatially varying temperature. In this paper, we investigate how such a temperature landscape arises according to how the slow variables affect the character of the driven quasisteady state reached by the fast variables. Brownian motion in the presence of spatial temperature gradients is known to lead to the accumulation of probability density in low-temperature regions. Here, we focus on the implications of attraction to low effective temperature for the long-term evolution of slow variables. After quantitatively deriving the temperature landscape for a general class of overdamped systems using a path-integral technique, we then illustrate in a simple dynamical system how the attraction to low effective temperature has a fine-tuning effect on the slow variable, selecting configurations that bring about exceptionally low force fluctuation in the fast-variable steady state. We furthermore demonstrate that a particularly strong effect of this kind can take place when the slow variable is tuned to bring about orderly, integrable motion in the fast dynamics that avoids thermalizing energy absorbed from the drive. We thus point to a potentially general feedback mechanism in multi-time-scale active systems, that leads to the exploration of slow variable space, as if in search of fine tuning for a "least-rattling" response in the fast coordinates.

Subdiffusion via dynamical localization induced by thermal equilibrium fluctuations

We reveal the mechanism of subdiffusion which emerges in a straightforward, one dimensional classical nonequilibrium dynamics of a Brownian ratchet driven by both a time-periodic force and Gaussian white noise. In a tailored parameter set for which the deterministic counterpart is in a non-chaotic regime, subdiffusion is a long-living transient whose lifetime can be many, many orders of magnitude larger than characteristic time scales of the setup thus being amenable to experimental observations. As a reason for this subdiffusive behaviour in the coordinate space we identify thermal noise induced dynamical localization in the velocity (momentum) space. This novel idea is distinct from existing knowledge and has never been reported for any classical or quantum system. It suggests reconsideration of generally accepted opinion that subdiffusion is due to broad distributions or strong correlations which reflect disorder, trapping, viscoelasticity of the medium or geometrical constraints.

Transports in a rough ratchet induced by Lévy noises

We study the transport of a particle subjected to a Lévy noise in a rough ratchet potential which is constructed by superimposing a fast oscillating trigonometric function on a common ratchet background. Due to the superposition of roughness, the transport process exhibits significantly different properties under the excitation of Lévy noises compared to smooth cases. The influence of the roughness on the directional motion is explored by calculating the mean velocities with respect to the Lévy stable index α and the spatial asymmetry parameter q of the ratchet. Variations in the splitting probability have been analyzed to illustrate how roughness affects the transport. In addition, we have examined the influences of roughness on the mean first passage time to know when it accelerates or slows down the first passage process. We find that the roughness can lead to a fast reduction of the absolute value of the mean velocity for small α, however the influence is small for large α. We have illustrated that the ladder-like roughness on the potential wall increases the possibility for particles to cross the gentle side of the ratchet, which results in an increase of the splitting probability to right for the right-skewed ratchet potential. Although the roughness increases the corresponding probability, it does not accelerate the mean first passage process to the right adjacent well. Our results show that the influences of roughness on the mean first passage time are sensitive to the combination of q and α. Hence, the proper q and α can speed up the passage process, otherwise it will slow down it.